Subgroups — C₇×#60450 #13237666

Order 1792.

2722 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3449	C₂	1	2	yes	maximal, complemented
4	C₇×#3061	C₂	1	4	yes	maximal, complemented
5	C₇×#3060	C₂	1	4	yes	maximal, complemented
6	C₇×#3059	C₂	1	2	yes	maximal, complemented
7	C₇×#3058	C₂	1	2	yes	maximal, complemented
8	C₇×#3057	C₂	1	4	yes	maximal, complemented
9	C₇×#3054	C₂	1	4	yes	maximal, complemented
10	C₇×#3050	C₂	1	4	yes	maximal, complemented
11	C₂×C₇×#427	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#2896	C₂	1	2	yes	maximal, complemented
13	C₂²×C₇×#41	C₂²	1	2	yes	—
14	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
15	C₂×C₇×#135	—	2	8	no	—
16	C₂×C₇×#135	C₂²	1	2	yes	—
17	C₂×C₇×#135	C₂²	1	2	yes	—
18	C₂×C₇×#135	C₂²	1	2	yes	complemented
19	C₂×C₇×#135	C₂²	1	2	yes	complemented
20	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₂²×C₇×SD₁₆	C₂²	1	2	yes	complemented
22	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₇×#131	—	2	8	no	—
24	C₂×C₇×#131	C₂²	1	2	yes	—
25	C₂×C₇×#131	C₂²	1	2	yes	—
26	C₂×C₇×#131	C₂²	1	2	yes	complemented
27	C₂×C₇×#131	C₂²	1	2	yes	complemented
28	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
29	C₂²×C₇×M₁₆	C₂²	1	2	yes	complemented
30	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₂³×C₇×C₈	C₂²	1	2	yes	complemented
32	C₇×#537	C₂²	1	8	yes	complemented
33	C₇×#536	—	2	2	no	—
34	C₇×#536	C₂²	1	4	yes	—
35	C₇×#535	—	2	2	no	—
36	C₇×#535	C₂²	1	4	yes	—
37	C₇×#534	—	2	4	no	—
38	C₇×#532	—	2	4	no	—
39	C₇×#525	—	2	4	no	—
40	C₇×#525	C₂²	1	8	yes	complemented
41	C₂×C₇×#117	C₂²	1	2	yes	—
42	C₇×#470	C₂²	1	8	yes	complemented
43	C₇×#469	C₂²	1	8	yes	complemented
44	C₇×#468	C₂²	1	16	yes	complemented
45	C₇×#467	C₂²	1	8	yes	complemented
46	C₇×#466	C₂²	1	8	yes	complemented
47	C₇×#465	C₂²	1	16	yes	complemented
48	C₇×#427	C₂²	1	2	yes	complemented
49	C₇×#427	C₂²	1	2	yes	complemented
50	C₇×#427	C₂²	1	2	yes	complemented
51	C₇×#427	C₂²	1	2	yes	complemented
52	C₂×C₇×#107	C₂²	1	2	yes	complemented
53	C₂×C₇×#106	C₂²	1	2	yes	complemented
54	C₇×#419	C₂²	1	4	yes	complemented
55	C₇×#418	C₂²	1	4	yes	complemented
56	C₇×#417	C₂²	1	8	yes	complemented
57	C₇×#416	C₂²	1	8	yes	complemented
58	#60450	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×C₇×#41	—	2	4	no	—
60	C₂×C₇×#41	—	2	2	no	—
61	C₂×C₇×#41	—	2	2	no	—
62	C₂×C₇×#41	—	2	4	no	—
63	C₂×C₇×#41	—	2	8	no	—
64	C₂×C₇×#41	—	2	8	no	—
65	C₂×C₇×#41	D₄	1	2	yes	—
66	C₂×C₇×#41	D₄	1	2	yes	—
67	C₂×C₇×#41	C₂³	1	4	yes	—
68	C₂×C₇×#41	C₂³	1	4	yes	—
69	C₂×C₇×#41	C₂³	1	2	yes	—
70	C₂×C₇×#41	C₂³	1	2	yes	—
71	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
72	C₂²×C₇×Q₈	C₂³	1	2	yes	—
73	C₂²×C₇×D₄	—	2	2	no	—
74	C₂²×C₇×D₄	D₄	1	2	yes	—
75	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
76	C₂²×C₇×D₄	C₂³	1	2	yes	—
77	C₂³×C₄×C₇	—	2	2	no	—
78	C₂³×C₄×C₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
80	C₇×#135	—	2	8	no	—
81	C₇×#135	—	2	8	no	—
82	C₇×#135	—	2	8	no	—
83	C₇×#135	—	2	8	no	—
84	C₂×C₇×Q₁₆	—	2	4	no	—
85	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
86	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
87	C₂×C₇×SD₁₆	—	2	8	no	—
88	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
89	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
90	C₂×C₇×D₈	—	2	4	no	—
91	C₂×C₇×D₈	C₂³	1	2	yes	—
92	C₂×C₇×D₈	C₂³	1	2	yes	—
93	C₇×#131	—	2	8	no	—
94	C₇×#131	—	2	8	no	—
95	C₇×#131	—	2	8	no	—
96	C₇×#131	—	2	8	no	—
97	C₂×C₇×M₁₆	—	2	4	no	—
98	C₂×C₇×M₁₆	—	2	8	no	—
99	C₂×C₇×M₁₆	C₂³	1	4	yes	—
100	C₂×C₇×M₁₆	C₂³	1	4	yes	—
101	C₂×C₇×M₁₆	C₂³	1	2	yes	—
102	C₂×C₇×M₁₆	C₂³	1	2	yes	—
103	C₂²×C₇×C₈	—	2	8	no	—
104	C₂²×C₇×C₈	D₄	1	8	yes	complemented
105	C₂²×C₇×C₈	C₂³	1	4	yes	—
106	C₂²×C₇×C₈	C₂³	1	2	yes	—
107	C₂²×C₇×C₈	C₂³	1	4	yes	—
108	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
109	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
110	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
111	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
112	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
113	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
114	C₇×C₄⋊Q₈	—	2	4	no	—
115	C₇×#127	—	2	4	no	—
116	C₇×#124	—	2	4	no	—
117	C₇×#124	—	2	4	no	—
118	C₇×#123	—	2	8	no	—
119	C₇×#123	—	2	8	no	—
120	C₇×C₂²⋊Q₈	—	2	8	no	—
121	C₇×C₂²⋊Q₈	C₂³	1	16	yes	—
122	C₇×#121	—	2	8	no	—
123	C₇×#121	—	2	16	no	—
124	C₇×#121	C₂³	1	16	yes	—
125	C₇×#120	—	2	8	no	—
126	C₇×#120	—	2	8	no	—
127	C₄×C₇×D₄	—	2	16	no	—
128	C₇×#117	—	2	2	no	—
129	C₇×#117	—	2	2	no	—
130	C₇×#117	C₂³	1	4	yes	—
131	C₇×#117	C₂³	1	4	yes	—
132	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
133	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
134	C₂×C₄²×C₇	—	2	2	no	—
135	C₇×#107	C₂³	1	4	yes	—
136	C₇×#107	C₂³	1	4	yes	—
137	C₇×#106	C₂³	1	4	yes	—
138	C₇×#106	C₂³	1	4	yes	—
139	C₇×#103	C₂³	1	16	yes	—
140	C₇×#102	C₂³	1	16	yes	—
141	C₂²×#135	C₂×C₇	1	1	yes	characteristic, complemented
142	C₂²×#131	C₂×C₇	1	1	yes	characteristic, complemented
143	#3449	C₂×C₇	1	2	yes	complemented
144	#3061	C₂×C₇	1	4	yes	complemented
145	#3060	C₂×C₇	1	4	yes	complemented
146	#3059	C₂×C₇	1	2	yes	complemented
147	#3058	C₂×C₇	1	2	yes	complemented
148	#3057	C₂×C₇	1	4	yes	complemented
149	#3054	C₂×C₇	1	4	yes	complemented
150	#3050	C₂×C₇	1	4	yes	complemented
151	C₂×#427	C₂×C₇	1	1	yes	characteristic, complemented
152	#2896	C₂×C₇	1	2	yes	complemented
153	C₂⁴×C₇	—	2	2	no	—
154	C₂⁴×C₇	C₂×D₄	1	2	yes	—
155	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
156	C₇×#41	—	2	8	no	—
157	C₇×#41	—	2	8	no	—
158	C₇×#41	—	2	4	no	—
159	C₇×#41	—	2	4	no	—
160	C₇×#41	—	2	4	no	—
161	C₇×#41	—	2	4	no	—
162	C₇×#41	—	2	8	no	—
163	C₇×#41	—	2	8	no	—
164	C₇×#41	—	2	8	no	—
165	C₇×#41	—	2	8	no	—
166	C₇×#41	—	4	8	no	—
167	C₇×#41	—	4	8	no	—
168	C₂×C₇×Q₈	—	2	4	no	—
169	C₂×C₇×Q₈	—	2	2	no	—
170	C₂×C₇×Q₈	—	2	4	no	—
171	C₂×C₇×Q₈	—	2	4	no	—
172	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
173	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
174	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
175	C₂×C₇×D₄	—	2	8	no	—
176	C₂×C₇×D₄	—	2	4	no	—
177	C₂×C₇×D₄	—	2	4	no	—
178	C₂×C₇×D₄	—	2	16	no	—
179	C₂×C₇×D₄	—	2	4	no	—
180	C₂×C₇×D₄	—	2	4	no	—
181	C₂×C₇×D₄	—	2	2	no	—
182	C₂×C₇×D₄	—	2	4	no	—
183	C₂×C₇×D₄	—	4	8	no	—
184	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
185	C₂×C₇×D₄	C₂×D₄	1	4	yes	—
186	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
187	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
188	C₂×C₇×D₄	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₇	—	2	4	no	—
190	C₂²×C₄×C₇	—	2	4	no	—
191	C₂²×C₄×C₇	—	2	2	no	—
192	C₂²×C₄×C₇	—	2	4	no	—
193	C₂²×C₄×C₇	—	2	4	no	—
194	C₂²×C₄×C₇	—	2	4	no	—
195	C₂²×C₄×C₇	—	4	8	no	—
196	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
198	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
202	C₂²×C₄×C₇	#41	1	4	yes	—
203	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
204	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
206	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
207	C₇×Q₁₆	—	2	4	no	—
208	C₇×Q₁₆	—	2	4	no	—
209	C₇×SD₁₆	—	2	8	no	—
210	C₇×SD₁₆	—	2	8	no	—
211	C₇×D₈	—	2	4	no	—
212	C₇×D₈	—	2	4	no	—
213	C₇×M₁₆	—	2	4	no	—
214	C₇×M₁₆	—	2	4	no	—
215	C₇×M₁₆	—	2	8	no	—
216	C₇×M₁₆	—	2	8	no	—
217	C₂×C₇×C₈	—	2	4	no	—
218	C₂×C₇×C₈	—	2	8	no	—
219	C₂×C₇×C₈	—	2	8	no	—
220	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
221	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
222	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
223	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
224	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
225	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
226	C₂×C₇×C₈	C₂⁴	1	4	yes	—
227	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
228	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
229	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
230	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
231	C₇×C₄⋊C₄	—	2	4	no	—
232	C₇×C₄⋊C₄	—	2	16	no	—
233	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
234	C₇×C₂²⋊C₄	—	2	4	no	—
235	C₇×C₂²⋊C₄	—	2	4	no	—
236	C₇×C₂²⋊C₄	—	2	16	no	—
237	C₇×C₂²⋊C₄	—	2	16	no	—
238	C₄²×C₇	—	2	4	no	—
239	C₄²×C₇	—	2	4	no	—
240	C₂²×#41	C₂²×C₇	1	2	yes	—
241	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
242	C₂×#135	—	2	8	no	—
243	C₂×#135	C₂²×C₇	1	2	yes	—
244	C₂×#135	C₂²×C₇	1	2	yes	—
245	C₂×#135	C₂²×C₇	1	2	yes	complemented
246	C₂×#135	C₂²×C₇	1	2	yes	complemented
247	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
248	C₂²×SD₁₆	C₂²×C₇	1	2	yes	complemented
249	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic, complemented
250	C₂×#131	—	2	8	no	—
251	C₂×#131	C₂²×C₇	1	2	yes	—
252	C₂×#131	C₂²×C₇	1	2	yes	—
253	C₂×#131	C₂²×C₇	1	2	yes	complemented
254	C₂×#131	C₂²×C₇	1	2	yes	complemented
255	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
256	C₂²×M₁₆	C₂²×C₇	1	2	yes	complemented
257	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
258	C₂³×C₈	C₂²×C₇	1	2	yes	complemented
259	#537	C₂²×C₇	1	8	yes	complemented
260	#536	—	2	2	no	—
261	#536	C₂²×C₇	1	4	yes	—
262	#535	—	2	2	no	—
263	#535	C₂²×C₇	1	4	yes	—
264	#534	—	2	4	no	—
265	#532	—	2	4	no	—
266	#525	—	2	4	no	—
267	#525	C₂²×C₇	1	8	yes	complemented
268	C₂×#117	C₂²×C₇	1	2	yes	—
269	#470	C₂²×C₇	1	8	yes	complemented
270	#469	C₂²×C₇	1	8	yes	complemented
271	#468	C₂²×C₇	1	16	yes	complemented
272	#467	C₂²×C₇	1	8	yes	complemented
273	#466	C₂²×C₇	1	8	yes	complemented
274	#465	C₂²×C₇	1	16	yes	complemented
275	#427	C₂²×C₇	1	2	yes	complemented
276	#427	C₂²×C₇	1	2	yes	complemented
277	#427	C₂²×C₇	1	2	yes	complemented
278	#427	C₂²×C₇	1	2	yes	complemented
279	C₂×#107	C₂²×C₇	1	2	yes	complemented
280	C₂×#106	C₂²×C₇	1	2	yes	complemented
281	#419	C₂²×C₇	1	4	yes	complemented
282	#418	C₂²×C₇	1	4	yes	complemented
283	#417	C₂²×C₇	1	8	yes	complemented
284	#416	C₂²×C₇	1	8	yes	complemented
285	C₂³×C₇	—	2	4	no	—
286	C₂³×C₇	—	2	4	no	—
287	C₂³×C₇	—	4	2	no	—
288	C₂³×C₇	—	4	4	no	—
289	C₂³×C₇	—	4	4	no	—
290	C₂³×C₇	—	8	4	no	—
291	C₂³×C₇	#121	1	4	yes	—
292	C₂³×C₇	C₂²×D₄	1	4	yes	—
293	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
294	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
295	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
296	C₇×Q₈	—	2	4	no	—
297	C₇×Q₈	—	2	4	no	—
298	C₇×Q₈	—	2	4	no	—
299	C₇×Q₈	—	2	4	no	—
300	C₇×Q₈	—	4	4	no	—
301	C₇×D₄	—	2	8	no	—
302	C₇×D₄	—	2	8	no	—
303	C₇×D₄	—	2	4	no	—
304	C₇×D₄	—	2	4	no	—
305	C₇×D₄	—	2	4	no	—
306	C₇×D₄	—	2	4	no	—
307	C₇×D₄	—	4	8	no	—
308	C₇×D₄	—	4	8	no	—
309	C₇×D₄	—	4	4	no	—
310	C₇×D₄	—	8	16	no	—
311	C₂×C₄×C₇	—	2	4	no	—
312	C₂×C₄×C₇	—	2	8	no	—
313	C₂×C₄×C₇	—	2	4	no	—
314	C₂×C₄×C₇	—	2	4	no	—
315	C₂×C₄×C₇	—	2	8	no	—
316	C₂×C₄×C₇	—	2	4	no	—
317	C₂×C₄×C₇	—	4	4	no	—
318	C₂×C₄×C₇	—	4	8	no	—
319	C₂×C₄×C₇	—	4	8	no	—
320	C₂×C₄×C₇	—	8	4	no	—
321	C₂×C₄×C₇	#121	1	4	yes	—
322	C₂×C₄×C₇	#121	1	4	yes	—
323	C₂×C₄×C₇	#121	1	4	yes	—
324	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
325	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
326	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
327	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
328	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
329	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
330	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
331	C₂×C₄×C₇	C₂×#41	1	2	yes	—
332	C₂×C₄×C₇	C₂×#41	1	2	yes	—
333	C₂×C₄×C₇	C₂×#41	1	2	yes	—
334	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
335	C₇×C₈	—	2	4	no	—
336	C₇×C₈	C₂²×D₄	1	2	yes	—
337	C₇×C₈	C₂²×D₄	1	2	yes	—
338	C₇×C₈	C₂²×D₄	1	2	yes	—
339	C₇×C₈	C₂²×D₄	1	2	yes	—
340	C₂×#41	—	2	4	no	—
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	—	2	4	no	—
344	C₂×#41	—	2	8	no	—
345	C₂×#41	—	2	8	no	—
346	C₂×#41	C₇×D₄	1	2	yes	—
347	C₂×#41	C₇×D₄	1	2	yes	—
348	C₂×#41	C₂³×C₇	1	4	yes	—
349	C₂×#41	C₂³×C₇	1	4	yes	—
350	C₂×#41	C₂³×C₇	1	2	yes	—
351	C₂×#41	C₂³×C₇	1	2	yes	—
352	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
353	C₂²×Q₈	C₂³×C₇	1	2	yes	—
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	C₇×D₄	1	2	yes	—
356	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
357	C₂²×D₄	C₂³×C₇	1	2	yes	—
358	C₂³×C₄	—	2	2	no	—
359	C₂³×C₄	C₂³×C₇	1	2	yes	—
360	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
361	#135	—	2	8	no	—
362	#135	—	2	8	no	—
363	#135	—	2	8	no	—
364	#135	—	2	8	no	—
365	C₂×Q₁₆	—	2	4	no	—
366	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
367	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
368	C₂×SD₁₆	—	2	8	no	—
369	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
370	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
371	C₂×D₈	—	2	4	no	—
372	C₂×D₈	C₂³×C₇	1	2	yes	—
373	C₂×D₈	C₂³×C₇	1	2	yes	—
374	#131	—	2	8	no	—
375	#131	—	2	8	no	—
376	#131	—	2	8	no	—
377	#131	—	2	8	no	—
378	C₂×M₁₆	—	2	4	no	—
379	C₂×M₁₆	—	2	8	no	—
380	C₂×M₁₆	C₂³×C₇	1	2	yes	—
381	C₂×M₁₆	C₂³×C₇	1	2	yes	—
382	C₂×M₁₆	C₂³×C₇	1	4	yes	—
383	C₂×M₁₆	C₂³×C₇	1	4	yes	—
384	C₂²×C₈	—	2	8	no	—
385	C₂²×C₈	C₇×D₄	1	8	yes	complemented
386	C₂²×C₈	C₂³×C₇	1	2	yes	—
387	C₂²×C₈	C₂³×C₇	1	4	yes	—
388	C₂²×C₈	C₂³×C₇	1	4	yes	—
389	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
390	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
391	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
392	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
393	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
394	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
395	C₄⋊Q₈	—	2	4	no	—
396	#127	—	2	4	no	—
397	#124	—	2	4	no	—
398	#124	—	2	4	no	—
399	#123	—	2	8	no	—
400	#123	—	2	8	no	—
401	C₂²⋊Q₈	—	2	8	no	—
402	C₂²⋊Q₈	C₂³×C₇	1	16	yes	—
403	#121	—	2	8	no	—
404	#121	—	2	16	no	—
405	#121	C₂³×C₇	1	16	yes	—
406	#120	—	2	8	no	—
407	#120	—	2	8	no	—
408	C₄×D₄	—	2	16	no	—
409	#117	—	2	2	no	—
410	#117	—	2	2	no	—
411	#117	C₂³×C₇	1	4	yes	—
412	#117	C₂³×C₇	1	4	yes	—
413	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
414	C₂×C₂²⋊C₄	—	2	2	no	—
415	C₂×C₄²	—	2	2	no	—
416	#107	C₂³×C₇	1	4	yes	—
417	#107	C₂³×C₇	1	4	yes	—
418	#106	C₂³×C₇	1	4	yes	—
419	#106	C₂³×C₇	1	4	yes	—
420	#103	C₂³×C₇	1	16	yes	—
421	#102	C₂³×C₇	1	16	yes	—
422	C₂²×C₇	—	2	2	no	—
423	C₂²×C₇	—	2	2	no	—
424	C₂²×C₇	—	2	2	no	—
425	C₂²×C₇	—	2	8	no	—
426	C₂²×C₇	—	2	4	no	—
427	C₂²×C₇	—	4	4	no	—
428	C₂²×C₇	—	4	8	no	—
429	C₂²×C₇	—	4	8	no	—
430	C₂²×C₇	—	4	4	no	—
431	C₂²×C₇	—	8	8	no	—
432	C₂²×C₇	C₂×#121	1	2	yes	—
433	C₂²×C₇	C₂×#121	1	2	yes	—
434	C₂²×C₇	C₂×#121	1	2	yes	—
435	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
436	C₄×C₇	—	2	4	no	—
437	C₄×C₇	—	4	8	no	—
438	C₄×C₇	—	4	4	no	—
439	C₄×C₇	C₂×#121	1	2	yes	—
440	C₄×C₇	C₂×#121	1	2	yes	—
441	C₄×C₇	C₂×#121	1	2	yes	—
442	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
443	C₄×C₇	C₂²×#41	1	1	yes	characteristic
444	C₂⁴	—	2	2	no	—
445	C₂⁴	C₂×C₇×D₄	1	2	yes	—
446	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
447	#41	—	2	8	no	—
448	#41	—	2	8	no	—
449	#41	—	2	4	no	—
450	#41	—	2	4	no	—
451	#41	—	2	4	no	—
452	#41	—	2	4	no	—
453	#41	—	2	8	no	—
454	#41	—	2	8	no	—
455	#41	—	2	8	no	—
456	#41	—	2	8	no	—
457	#41	—	4	8	no	—
458	#41	—	4	8	no	—
459	C₂×Q₈	—	2	4	no	—
460	C₂×Q₈	—	2	2	no	—
461	C₂×Q₈	—	2	4	no	—
462	C₂×Q₈	—	2	4	no	—
463	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
464	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
465	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
466	C₂×D₄	—	2	8	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	16	no	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	4	8	no	—
475	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
476	C₂×D₄	C₂×C₇×D₄	1	4	yes	—
477	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
478	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
479	C₂×D₄	C₂⁴×C₇	1	4	yes	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	4	no	—
486	C₂²×C₄	—	4	8	no	—
487	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
488	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
489	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
490	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
491	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
492	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
493	C₂²×C₄	C₇×#41	1	4	yes	—
494	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
495	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
498	Q₁₆	—	2	4	no	—
499	Q₁₆	—	2	4	no	—
500	SD₁₆	—	2	8	no	—
501	SD₁₆	—	2	8	no	—
502	D₈	—	2	4	no	—
503	D₈	—	2	4	no	—
504	M₁₆	—	2	4	no	—
505	M₁₆	—	2	4	no	—
506	M₁₆	—	2	8	no	—
507	M₁₆	—	2	8	no	—
508	C₂×C₈	—	2	4	no	—
509	C₂×C₈	—	2	8	no	—
510	C₂×C₈	—	2	8	no	—
511	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
512	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
513	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
514	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
515	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
516	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
517	C₂×C₈	C₂⁴×C₇	1	4	yes	—
518	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
519	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
520	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
521	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	16	no	—
524	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₂²⋊C₄	—	2	4	no	—
527	C₂²⋊C₄	—	2	16	no	—
528	C₂²⋊C₄	—	2	16	no	—
529	C₄²	—	2	4	no	—
530	C₄²	—	2	4	no	—
531	C₂×C₇	—	2	2	no	—
532	C₂×C₇	—	2	2	no	—
533	C₂×C₇	—	2	2	no	—
534	C₂×C₇	—	4	4	no	—
535	C₂×C₇	—	8	4	no	—
536	C₂×C₇	C₂²×#121	1	1	yes	characteristic
537	C₂×C₇	#3584	1	2	yes	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	4	2	no	—
541	C₂³	—	4	4	no	—
542	C₂³	—	4	4	no	—
543	C₂³	—	8	4	no	—
544	C₂³	C₇×#121	1	4	yes	—
545	C₂³	C₂²×C₇×D₄	1	4	yes	—
546	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
547	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
548	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
549	Q₈	—	2	4	no	—
550	Q₈	—	2	4	no	—
551	Q₈	—	2	4	no	—
552	Q₈	—	2	4	no	—
553	Q₈	—	4	4	no	—
554	D₄	—	2	8	no	—
555	D₄	—	2	8	no	—
556	D₄	—	2	4	no	—
557	D₄	—	2	4	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	4	no	—
560	D₄	—	4	8	no	—
561	D₄	—	4	8	no	—
562	D₄	—	4	4	no	—
563	D₄	—	8	16	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	8	no	—
568	C₂×C₄	—	2	8	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	8	no	—
572	C₂×C₄	—	4	8	no	—
573	C₂×C₄	—	8	4	no	—
574	C₂×C₄	C₇×#121	1	4	yes	—
575	C₂×C₄	C₇×#121	1	4	yes	—
576	C₂×C₄	C₇×#121	1	4	yes	—
577	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
578	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
579	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
580	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
581	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
582	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
583	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
584	C₂×C₄	C₂×C₇×#41	1	2	yes	—
585	C₂×C₄	C₂×C₇×#41	1	2	yes	—
586	C₂×C₄	C₂×C₇×#41	1	2	yes	—
587	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
588	C₈	—	2	4	no	—
589	C₈	C₂²×C₇×D₄	1	2	yes	—
590	C₈	C₂²×C₇×D₄	1	2	yes	—
591	C₈	C₂²×C₇×D₄	1	2	yes	—
592	C₈	C₂²×C₇×D₄	1	2	yes	—
593	C₇	#60450	1	1	yes	characteristic, direct product factor, Sylow
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	8	no	—
598	C₂²	—	2	4	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	4	8	no	—
601	C₂²	—	4	8	no	—
602	C₂²	—	4	4	no	—
603	C₂²	—	8	8	no	—
604	C₂²	C₂×C₇×#121	1	2	yes	—
605	C₂²	C₂×C₇×#121	1	2	yes	—
606	C₂²	C₂×C₇×#121	1	2	yes	—
607	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
608	C₄	—	2	4	no	—
609	C₄	—	4	8	no	—
610	C₄	—	4	4	no	—
611	C₄	C₂×C₇×#121	1	2	yes	—
612	C₄	C₂×C₇×#121	1	2	yes	—
613	C₄	C₂×C₇×#121	1	2	yes	—
614	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
615	C₄	C₂²×C₇×#41	1	1	yes	characteristic
616	C₂	—	2	2	no	—
617	C₂	—	2	2	no	—
618	C₂	—	2	2	no	—
619	C₂	—	4	4	no	—
620	C₂	—	8	4	no	—
621	C₂	C₂²×C₇×#121	1	1	yes	characteristic
622	C₂	C₇×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle	C₂²×C₇	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₇×#41 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₇×#135	C₂	semidirect
2. C₂²×C₇×#131	C₂	semidirect
3. C₇×#3449	C₂	semidirect
4. C₇×#3061	C₂	semidirect
5. C₇×#3060	C₂	semidirect
6. C₇×#3059	C₂	semidirect
7. C₇×#3058	C₂	semidirect
8. C₇×#3057	C₂	semidirect
9. C₇×#3054	C₂	semidirect
10. C₇×#3050	C₂	semidirect
11. C₂×C₇×#427	C₂	semidirect
12. C₇×#2896	C₂	semidirect
18. C₂×C₇×#135	C₂²	semidirect
19. C₂×C₇×#135	C₂²	semidirect
20. C₂²×C₇×Q₁₆	C₂²	semidirect
21. C₂²×C₇×SD₁₆	C₂²	semidirect
22. C₂²×C₇×D₈	C₂²	semidirect
26. C₂×C₇×#131	C₂²	semidirect
27. C₂×C₇×#131	C₂²	semidirect
29. C₂²×C₇×M₁₆	C₂²	semidirect
31. C₂³×C₇×C₈	C₂²	semidirect
32. C₇×#537	C₂²	semidirect
40. C₇×#525	C₂²	semidirect
42. C₇×#470	C₂²	semidirect
43. C₇×#469	C₂²	semidirect
44. C₇×#468	C₂²	semidirect
45. C₇×#467	C₂²	semidirect
46. C₇×#466	C₂²	semidirect
47. C₇×#465	C₂²	semidirect
48. C₇×#427	C₂²	semidirect
49. C₇×#427	C₂²	semidirect
50. C₇×#427	C₂²	semidirect
51. C₇×#427	C₂²	semidirect
52. C₂×C₇×#107	C₂²	semidirect
53. C₂×C₇×#106	C₂²	semidirect
54. C₇×#419	C₂²	semidirect
55. C₇×#418	C₂²	semidirect
56. C₇×#417	C₂²	semidirect
57. C₇×#416	C₂²	semidirect
58. #60450	C₇	direct (N × H)
104. C₂²×C₇×C₈	D₄	semidirect
141. C₂²×#135	C₂×C₇	semidirect
142. C₂²×#131	C₂×C₇	semidirect
143. #3449	C₂×C₇	semidirect
144. #3061	C₂×C₇	semidirect
145. #3060	C₂×C₇	semidirect
146. #3059	C₂×C₇	semidirect
147. #3058	C₂×C₇	semidirect
148. #3057	C₂×C₇	semidirect
149. #3054	C₂×C₇	semidirect
150. #3050	C₂×C₇	semidirect
151. C₂×#427	C₂×C₇	semidirect
152. #2896	C₂×C₇	semidirect
245. C₂×#135	C₂²×C₇	semidirect
246. C₂×#135	C₂²×C₇	semidirect
247. C₂²×Q₁₆	C₂²×C₇	semidirect
248. C₂²×SD₁₆	C₂²×C₇	semidirect
249. C₂²×D₈	C₂²×C₇	semidirect
253. C₂×#131	C₂²×C₇	semidirect
254. C₂×#131	C₂²×C₇	semidirect
256. C₂²×M₁₆	C₂²×C₇	semidirect
258. C₂³×C₈	C₂²×C₇	semidirect
259. #537	C₂²×C₇	semidirect
267. #525	C₂²×C₇	semidirect
269. #470	C₂²×C₇	semidirect
270. #469	C₂²×C₇	semidirect
271. #468	C₂²×C₇	semidirect
272. #467	C₂²×C₇	semidirect
273. #466	C₂²×C₇	semidirect
274. #465	C₂²×C₇	semidirect
275. #427	C₂²×C₇	semidirect
276. #427	C₂²×C₇	semidirect
277. #427	C₂²×C₇	semidirect
278. #427	C₂²×C₇	semidirect
279. C₂×#107	C₂²×C₇	semidirect
280. C₂×#106	C₂²×C₇	semidirect
281. #419	C₂²×C₇	semidirect
282. #418	C₂²×C₇	semidirect
283. #417	C₂²×C₇	semidirect
284. #416	C₂²×C₇	semidirect
385. C₂²×C₈	C₇×D₄	semidirect
593. C₇	#60450	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×#135	2
C₂²×C₇×#131	2
C₇×#3449	2
C₇×#3061	2
C₇×#3060	2
C₇×#3059	2
C₇×#3058	2
C₇×#3057	2
C₇×#3054	2
C₇×#3050	2
C₂×C₇×#427	2
C₇×#2896	2
#60450	7

Subgroups — C7×#60450 #13237666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60450 #13237666