Subgroups — C₇×#60462 #13237678

Order 1792.

2666 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3489	C₂	1	2	yes	maximal, complemented
4	C₇×#3063	C₂	1	6	yes	maximal, complemented
5	C₇×#3060	C₂	1	3	yes	maximal, complemented
6	C₇×#3059	C₂	1	3	yes	maximal, complemented
7	C₇×#3056	C₂	1	6	yes	maximal, complemented
8	C₇×#3054	C₂	1	3	yes	maximal, complemented
9	C₂×C₇×#470	C₂	1	3	yes	maximal, complemented
10	C₇×#2912	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#2898	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
14	C₇×#578	—	2	8	no	—
15	C₂×C₇×#137	C₂²	1	6	yes	complemented
16	C₂×C₇×#135	C₂²	1	3	yes	complemented
17	C₂²×C₇×Q₁₆	C₂²	1	3	yes	complemented
18	C₇×#568	—	2	8	no	—
19	C₂×C₇×#131	C₂²	1	3	yes	—
20	C₂×C₇×#131	C₂²	1	3	yes	—
21	C₂×C₇×#131	C₂²	1	1	yes	characteristic
22	C₂×C₇×#131	C₂²	1	1	yes	characteristic
23	C₂²×C₇×M₁₆	C₂²	1	3	yes	—
24	C₂²×C₇×M₁₆	C₂²	1	3	yes	—
25	C₇×D₄×Q₈	—	2	2	no	—
26	C₇×#548	—	2	2	no	—
27	C₇×#547	—	2	6	no	—
28	C₇×#546	—	2	6	no	—
29	C₇×#537	C₂²	1	6	yes	—
30	C₇×#536	C₂²	1	6	yes	—
31	C₇×#525	C₂²	1	6	yes	—
32	C₂×C₇×C₂²⋊Q₈	C₂²	1	6	yes	—
33	C₇×#520	C₂²	1	2	yes	—
34	C₇×#470	C₂²	1	3	yes	—
35	C₇×#470	C₂²	1	6	yes	—
36	C₇×#470	C₂²	1	6	yes	complemented
37	C₇×#470	C₂²	1	6	yes	complemented
38	C₇×#470	C₂²	1	3	yes	complemented
39	C₇×#469	C₂²	1	1	yes	characteristic
40	C₇×#469	C₂²	1	3	yes	complemented
41	C₇×#468	C₂²	1	6	yes	—
42	C₇×#468	C₂²	1	6	yes	complemented
43	C₇×#467	C₂²	1	12	yes	complemented
44	C₇×#465	C₂²	1	12	yes	complemented
45	C₇×#428	C₂²	1	6	yes	complemented
46	C₇×#427	C₂²	1	3	yes	complemented
47	C₂×C₇×#106	C₂²	1	3	yes	complemented
48	C₇×#419	C₂²	1	6	yes	complemented
49	C₇×#417	C₂²	1	6	yes	complemented
50	C₇×#416	C₂²	1	6	yes	complemented
51	C₂×C₇×#103	C₂²	1	6	yes	complemented
52	#60462	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₇×2₋¹⁺⁴	—	2	8	no	—
54	C₇×2₋¹⁺⁴	—	2	8	no	—
55	C₂×C₇×#41	—	2	4	no	—
56	C₂×C₇×#41	—	2	4	no	—
57	C₂×C₇×#41	—	2	6	no	—
58	C₂×C₇×#41	D₄	1	3	yes	—
59	C₂×C₇×#41	D₄	1	1	yes	characteristic
60	C₂×C₇×#41	C₂³	1	6	yes	—
61	C₂×C₇×#41	C₂³	1	3	yes	—
62	C₂×C₇×#41	C₂³	1	1	yes	characteristic
63	C₂²×C₇×Q₈	—	2	2	no	—
64	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
65	C₂²×C₇×Q₈	C₂³	1	6	yes	—
66	C₂²×C₇×D₄	D₄	1	3	yes	—
67	C₂³×C₄×C₇	C₂³	1	3	yes	—
68	C₇×#137	—	2	24	no	—
69	C₇×#135	—	2	12	no	—
70	C₂×C₇×Q₁₆	—	2	12	no	—
71	C₂×C₇×Q₁₆	C₂³	1	6	yes	—
72	C₂×C₇×Q₁₆	C₂³	1	3	yes	—
73	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
74	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
75	C₇×#131	—	2	4	no	—
76	C₇×#131	—	2	12	no	—
77	C₇×#131	—	2	12	no	—
78	C₇×#131	D₄	1	8	yes	complemented
79	C₂×C₇×M₁₆	—	2	12	no	—
80	C₂×C₇×M₁₆	—	2	12	no	—
81	C₂×C₇×M₁₆	C₂³	1	3	yes	—
82	C₂×C₇×M₁₆	C₂³	1	3	yes	—
83	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
84	C₂×C₇×M₁₆	C₂³	1	3	yes	—
85	C₂×C₇×M₁₆	C₂³	1	3	yes	—
86	C₂×C₇×M₁₆	C₂³	1	3	yes	—
87	C₂²×C₇×C₈	C₂³	1	6	yes	—
88	C₂²×C₇×C₈	C₂³	1	3	yes	—
89	C₂²×C₇×C₈	C₂³	1	3	yes	—
90	C₇×C₄⋊Q₈	—	2	6	no	—
91	C₇×#124	—	2	6	no	—
92	C₇×#123	—	2	12	no	—
93	C₇×#123	—	2	6	no	—
94	C₇×C₂²⋊Q₈	—	2	6	no	—
95	C₇×C₂²⋊Q₈	—	2	6	no	—
96	C₇×C₂²⋊Q₈	—	2	6	no	—
97	C₇×C₂²⋊Q₈	C₂³	1	6	yes	—
98	C₇×C₂²⋊Q₈	C₂³	1	12	yes	—
99	C₇×C₂²⋊Q₈	C₂³	1	6	yes	—
100	C₇×#121	—	2	12	no	—
101	C₇×#121	—	2	6	no	—
102	C₇×#121	C₂³	1	2	yes	—
103	C₇×#121	C₂³	1	6	yes	—
104	C₇×#120	—	2	6	no	—
105	C₄×C₇×Q₈	—	2	2	no	—
106	C₄×C₇×D₄	—	2	6	no	—
107	C₄×C₇×D₄	—	2	6	no	—
108	C₄×C₇×D₄	—	2	6	no	—
109	C₇×#117	C₂³	1	6	yes	—
110	C₂×C₇×C₄⋊C₄	—	2	6	no	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
112	C₂×C₇×C₂²⋊C₄	—	2	6	no	—
113	C₇×#107	C₂³	1	6	yes	—
114	C₇×#106	C₂³	1	1	yes	characteristic
115	C₇×#106	C₂³	1	3	yes	—
116	C₇×#106	C₂³	1	6	yes	—
117	C₇×#103	C₂³	1	6	yes	—
118	C₇×#103	C₂³	1	12	yes	—
119	C₇×#103	C₂³	1	6	yes	—
120	C₇×#102	C₂³	1	6	yes	—
121	C₇×#102	C₂³	1	2	yes	—
122	C₂×#578	C₂×C₇	1	1	yes	characteristic, complemented
123	C₂×#568	C₂×C₇	1	1	yes	characteristic, complemented
124	#3489	C₂×C₇	1	2	yes	complemented
125	#3063	C₂×C₇	1	6	yes	complemented
126	#3060	C₂×C₇	1	3	yes	complemented
127	#3059	C₂×C₇	1	3	yes	complemented
128	#3056	C₂×C₇	1	6	yes	complemented
129	#3054	C₂×C₇	1	3	yes	complemented
130	C₂×#470	C₂×C₇	1	3	yes	complemented
131	#2912	C₂×C₇	1	1	yes	characteristic, complemented
132	#2898	C₂×C₇	1	2	yes	complemented
133	C₂⁴×C₇	C₂×D₄	1	3	yes	—
134	C₇×#41	—	2	4	no	—
135	C₇×#41	—	2	12	no	—
136	C₇×#41	—	2	6	no	—
137	C₇×#41	—	2	6	no	—
138	C₇×#41	—	2	12	no	—
139	C₇×#41	—	2	12	no	—
140	C₇×#41	—	4	24	no	—
141	C₇×#41	C₂×D₄	1	4	yes	—
142	C₇×#41	#41	1	4	yes	—
143	C₂×C₇×Q₈	—	2	4	no	—
144	C₂×C₇×Q₈	—	2	6	no	—
145	C₂×C₇×Q₈	—	2	12	no	—
146	C₂×C₇×Q₈	—	2	12	no	—
147	C₂×C₇×Q₈	—	4	8	no	—
148	C₂×C₇×Q₈	C₂×D₄	1	3	yes	—
149	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
150	C₂×C₇×Q₈	C₂⁴	1	6	yes	—
151	C₂×C₇×D₄	—	2	12	no	—
152	C₂×C₇×D₄	—	2	2	no	—
153	C₂×C₇×D₄	—	2	6	no	—
154	C₂×C₇×D₄	—	2	6	no	—
155	C₂×C₇×D₄	C₂×D₄	1	6	yes	—
156	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
157	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
158	C₂×C₇×D₄	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₇	—	2	6	no	—
160	C₂²×C₄×C₇	—	2	6	no	—
161	C₂²×C₄×C₇	—	2	6	no	—
162	C₂²×C₄×C₇	—	2	6	no	—
163	C₂²×C₄×C₇	—	2	6	no	—
164	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
165	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
166	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
167	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
168	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
170	C₇×Q₁₆	—	2	6	no	—
171	C₇×Q₁₆	—	2	12	no	—
172	C₇×SD₁₆	—	2	12	no	—
173	C₇×D₈	—	2	2	no	—
174	C₇×M₁₆	—	2	6	no	—
175	C₇×M₁₆	—	2	6	no	—
176	C₇×M₁₆	—	2	2	no	—
177	C₇×M₁₆	—	2	6	no	—
178	C₇×M₁₆	—	2	6	no	—
179	C₇×M₁₆	C₂×D₄	1	12	yes	—
180	C₂×C₇×C₈	—	2	6	no	—
181	C₂×C₇×C₈	—	2	12	no	—
182	C₂×C₇×C₈	C₂×D₄	1	12	yes	—
183	C₂×C₇×C₈	C₂⁴	1	3	yes	—
184	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
185	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
186	C₂×C₇×C₈	C₂⁴	1	3	yes	—
187	C₇×C₄⋊C₄	—	2	6	no	—
188	C₇×C₄⋊C₄	—	2	12	no	—
189	C₇×C₄⋊C₄	—	2	6	no	—
190	C₇×C₄⋊C₄	—	2	6	no	—
191	C₇×C₄⋊C₄	C₂⁴	1	6	yes	—
192	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
193	C₇×C₂²⋊C₄	—	2	6	no	—
194	C₇×C₂²⋊C₄	—	2	12	no	—
195	C₇×C₂²⋊C₄	—	2	6	no	—
196	C₇×C₂²⋊C₄	—	2	6	no	—
197	C₇×C₂²⋊C₄	—	2	2	no	—
198	C₇×C₂²⋊C₄	—	2	6	no	—
199	C₄²×C₇	—	2	6	no	—
200	C₂×2₋¹⁺⁴	C₂²×C₇	1	2	yes	—
201	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
202	#578	—	2	8	no	—
203	C₂×#137	C₂²×C₇	1	6	yes	complemented
204	C₂×#135	C₂²×C₇	1	3	yes	complemented
205	C₂²×Q₁₆	C₂²×C₇	1	3	yes	complemented
206	#568	—	2	8	no	—
207	C₂×#131	C₂²×C₇	1	1	yes	characteristic
208	C₂×#131	C₂²×C₇	1	3	yes	—
209	C₂×#131	C₂²×C₇	1	3	yes	—
210	C₂×#131	C₂²×C₇	1	1	yes	characteristic
211	C₂²×M₁₆	C₂²×C₇	1	3	yes	—
212	C₂²×M₁₆	C₂²×C₇	1	3	yes	—
213	D₄×Q₈	—	2	2	no	—
214	#548	—	2	2	no	—
215	#547	—	2	6	no	—
216	#546	—	2	6	no	—
217	#537	C₂²×C₇	1	6	yes	—
218	#536	C₂²×C₇	1	6	yes	—
219	#525	C₂²×C₇	1	6	yes	—
220	C₂×C₂²⋊Q₈	C₂²×C₇	1	6	yes	—
221	#520	C₂²×C₇	1	2	yes	—
222	#470	C₂²×C₇	1	3	yes	—
223	#470	C₂²×C₇	1	6	yes	—
224	#470	C₂²×C₇	1	6	yes	complemented
225	#470	C₂²×C₇	1	6	yes	complemented
226	#470	C₂²×C₇	1	3	yes	complemented
227	#469	C₂²×C₇	1	1	yes	characteristic
228	#469	C₂²×C₇	1	3	yes	complemented
229	#468	C₂²×C₇	1	6	yes	—
230	#468	C₂²×C₇	1	6	yes	complemented
231	#467	C₂²×C₇	1	12	yes	complemented
232	#465	C₂²×C₇	1	12	yes	complemented
233	#428	C₂²×C₇	1	6	yes	complemented
234	#427	C₂²×C₇	1	3	yes	complemented
235	C₂×#106	C₂²×C₇	1	3	yes	complemented
236	#419	C₂²×C₇	1	6	yes	complemented
237	#417	C₂²×C₇	1	6	yes	complemented
238	#416	C₂²×C₇	1	6	yes	complemented
239	C₂×#103	C₂²×C₇	1	6	yes	complemented
240	C₂³×C₇	—	2	6	no	—
241	C₂³×C₇	—	2	2	no	—
242	C₂³×C₇	—	4	6	no	—
243	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
244	C₂³×C₇	C₂²×D₄	1	3	yes	—
245	C₂³×C₇	C₂²×D₄	1	3	yes	—
246	C₇×Q₈	—	2	6	no	—
247	C₇×Q₈	—	2	6	no	—
248	C₇×Q₈	—	2	6	no	—
249	C₇×Q₈	—	4	12	no	—
250	C₇×Q₈	#121	1	4	yes	—
251	C₇×D₄	—	2	12	no	—
252	C₇×D₄	—	2	6	no	—
253	C₇×D₄	—	2	2	no	—
254	C₇×D₄	—	2	2	no	—
255	C₇×D₄	—	4	4	no	—
256	C₇×D₄	—	4	12	no	—
257	C₇×D₄	—	8	6	no	—
258	C₇×D₄	#121	1	12	yes	—
259	C₂×C₄×C₇	—	2	6	no	—
260	C₂×C₄×C₇	—	2	6	no	—
261	C₂×C₄×C₇	—	2	6	no	—
262	C₂×C₄×C₇	—	2	3	no	—
263	C₂×C₄×C₇	—	2	1	no	—
264	C₂×C₄×C₇	—	2	1	no	—
265	C₂×C₄×C₇	—	2	3	no	—
266	C₂×C₄×C₇	—	2	2	no	—
267	C₂×C₄×C₇	—	2	6	no	—
268	C₂×C₄×C₇	—	2	6	no	—
269	C₂×C₄×C₇	—	4	12	no	—
270	C₂×C₄×C₇	—	4	12	no	—
271	C₂×C₄×C₇	—	8	6	no	—
272	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
274	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
275	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
276	C₂×C₄×C₇	C₂×#41	1	6	yes	—
277	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
278	C₇×C₈	—	2	1	no	—
279	C₇×C₈	—	2	3	no	—
280	C₇×C₈	C₂²×D₄	1	6	yes	—
281	C₇×C₈	C₂²×D₄	1	2	yes	—
282	2₋¹⁺⁴	—	2	8	no	—
283	2₋¹⁺⁴	—	2	8	no	—
284	C₂×#41	—	2	4	no	—
285	C₂×#41	—	2	4	no	—
286	C₂×#41	—	2	6	no	—
287	C₂×#41	C₇×D₄	1	3	yes	—
288	C₂×#41	C₇×D₄	1	1	yes	characteristic
289	C₂×#41	C₂³×C₇	1	6	yes	—
290	C₂×#41	C₂³×C₇	1	3	yes	—
291	C₂×#41	C₂³×C₇	1	1	yes	characteristic
292	C₂²×Q₈	—	2	2	no	—
293	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
294	C₂²×Q₈	C₂³×C₇	1	6	yes	—
295	C₂²×D₄	C₇×D₄	1	3	yes	—
296	C₂³×C₄	C₂³×C₇	1	3	yes	—
297	#137	—	2	24	no	—
298	#135	—	2	12	no	—
299	C₂×Q₁₆	—	2	12	no	—
300	C₂×Q₁₆	C₂³×C₇	1	3	yes	—
301	C₂×Q₁₆	C₂³×C₇	1	6	yes	—
302	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
303	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
304	#131	—	2	4	no	—
305	#131	—	2	12	no	—
306	#131	—	2	12	no	—
307	#131	C₇×D₄	1	8	yes	complemented
308	C₂×M₁₆	—	2	12	no	—
309	C₂×M₁₆	—	2	12	no	—
310	C₂×M₁₆	C₂³×C₇	1	3	yes	—
311	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
312	C₂×M₁₆	C₂³×C₇	1	3	yes	—
313	C₂×M₁₆	C₂³×C₇	1	3	yes	—
314	C₂×M₁₆	C₂³×C₇	1	3	yes	—
315	C₂×M₁₆	C₂³×C₇	1	3	yes	—
316	C₂²×C₈	C₂³×C₇	1	3	yes	—
317	C₂²×C₈	C₂³×C₇	1	6	yes	—
318	C₂²×C₈	C₂³×C₇	1	3	yes	—
319	C₄⋊Q₈	—	2	6	no	—
320	#124	—	2	6	no	—
321	#123	—	2	12	no	—
322	#123	—	2	6	no	—
323	C₂²⋊Q₈	—	2	6	no	—
324	C₂²⋊Q₈	—	2	6	no	—
325	C₂²⋊Q₈	—	2	6	no	—
326	C₂²⋊Q₈	C₂³×C₇	1	6	yes	—
327	C₂²⋊Q₈	C₂³×C₇	1	12	yes	—
328	C₂²⋊Q₈	C₂³×C₇	1	6	yes	—
329	#121	—	2	12	no	—
330	#121	—	2	6	no	—
331	#121	C₂³×C₇	1	2	yes	—
332	#121	C₂³×C₇	1	6	yes	—
333	#120	—	2	6	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×D₄	—	2	6	no	—
336	C₄×D₄	—	2	6	no	—
337	C₄×D₄	—	2	6	no	—
338	#117	C₂³×C₇	1	6	yes	—
339	C₂×C₄⋊C₄	—	2	6	no	—
340	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
341	C₂×C₂²⋊C₄	—	2	6	no	—
342	#107	C₂³×C₇	1	6	yes	—
343	#106	C₂³×C₇	1	3	yes	—
344	#106	C₂³×C₇	1	6	yes	—
345	#106	C₂³×C₇	1	1	yes	characteristic
346	#103	C₂³×C₇	1	6	yes	—
347	#103	C₂³×C₇	1	12	yes	—
348	#103	C₂³×C₇	1	6	yes	—
349	#102	C₂³×C₇	1	2	yes	—
350	#102	C₂³×C₇	1	6	yes	—
351	C₂²×C₇	—	2	6	no	—
352	C₂²×C₇	—	2	2	no	—
353	C₂²×C₇	—	2	6	no	—
354	C₂²×C₇	—	2	1	no	—
355	C₂²×C₇	—	2	3	no	—
356	C₂²×C₇	—	4	2	no	—
357	C₂²×C₇	—	4	2	no	—
358	C₂²×C₇	—	4	2	no	—
359	C₂²×C₇	—	8	6	no	—
360	C₂²×C₇	C₂×#121	1	6	yes	—
361	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
362	C₄×C₇	—	2	1	no	—
363	C₄×C₇	—	2	3	no	—
364	C₄×C₇	—	4	6	no	—
365	C₄×C₇	—	4	2	no	—
366	C₄×C₇	—	4	6	no	—
367	C₄×C₇	C₂×#121	1	6	yes	—
368	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
369	C₄×C₇	C₂²×#41	1	1	yes	characteristic
370	C₂⁴	C₂×C₇×D₄	1	3	yes	—
371	#41	—	2	4	no	—
372	#41	—	2	12	no	—
373	#41	—	2	6	no	—
374	#41	—	2	6	no	—
375	#41	—	2	12	no	—
376	#41	—	2	12	no	—
377	#41	—	4	24	no	—
378	#41	C₂×C₇×D₄	1	4	yes	—
379	#41	C₇×#41	1	4	yes	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	2	6	no	—
382	C₂×Q₈	—	2	12	no	—
383	C₂×Q₈	—	2	12	no	—
384	C₂×Q₈	—	4	8	no	—
385	C₂×Q₈	C₂×C₇×D₄	1	3	yes	—
386	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
387	C₂×Q₈	C₂⁴×C₇	1	6	yes	—
388	C₂×D₄	—	2	12	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	6	no	—
391	C₂×D₄	—	2	6	no	—
392	C₂×D₄	C₂×C₇×D₄	1	6	yes	—
393	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
394	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
395	C₂×D₄	C₂⁴×C₇	1	2	yes	—
396	C₂²×C₄	—	2	6	no	—
397	C₂²×C₄	—	2	6	no	—
398	C₂²×C₄	—	2	6	no	—
399	C₂²×C₄	—	2	6	no	—
400	C₂²×C₄	—	2	6	no	—
401	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
402	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
403	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
404	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
405	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
407	Q₁₆	—	2	6	no	—
408	Q₁₆	—	2	12	no	—
409	SD₁₆	—	2	12	no	—
410	D₈	—	2	2	no	—
411	M₁₆	—	2	6	no	—
412	M₁₆	—	2	6	no	—
413	M₁₆	—	2	2	no	—
414	M₁₆	—	2	6	no	—
415	M₁₆	—	2	6	no	—
416	M₁₆	C₂×C₇×D₄	1	12	yes	—
417	C₂×C₈	—	2	6	no	—
418	C₂×C₈	—	2	12	no	—
419	C₂×C₈	C₂×C₇×D₄	1	12	yes	—
420	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
421	C₂×C₈	C₂⁴×C₇	1	3	yes	—
422	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
423	C₂×C₈	C₂⁴×C₇	1	3	yes	—
424	C₄⋊C₄	—	2	6	no	—
425	C₄⋊C₄	—	2	12	no	—
426	C₄⋊C₄	—	2	6	no	—
427	C₄⋊C₄	—	2	6	no	—
428	C₄⋊C₄	C₂⁴×C₇	1	6	yes	—
429	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
430	C₂²⋊C₄	—	2	6	no	—
431	C₂²⋊C₄	—	2	12	no	—
432	C₂²⋊C₄	—	2	6	no	—
433	C₂²⋊C₄	—	2	6	no	—
434	C₂²⋊C₄	—	2	2	no	—
435	C₂²⋊C₄	—	2	6	no	—
436	C₄²	—	2	6	no	—
437	C₂×C₇	—	2	6	no	—
438	C₂×C₇	—	4	3	no	—
439	C₂×C₇	—	4	1	no	—
440	C₂×C₇	—	8	2	no	—
441	C₂×C₇	C₂²×#121	1	1	yes	characteristic
442	C₂×C₇	#3586	1	2	yes	—
443	C₂³	—	2	6	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	4	6	no	—
446	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
447	C₂³	C₂²×C₇×D₄	1	3	yes	—
448	C₂³	C₂²×C₇×D₄	1	3	yes	—
449	Q₈	—	2	6	no	—
450	Q₈	—	2	6	no	—
451	Q₈	—	2	6	no	—
452	Q₈	—	4	12	no	—
453	Q₈	C₇×#121	1	4	yes	—
454	D₄	—	2	12	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	2	6	no	—
458	D₄	—	4	4	no	—
459	D₄	—	4	12	no	—
460	D₄	—	8	6	no	—
461	D₄	C₇×#121	1	12	yes	—
462	C₂×C₄	—	2	6	no	—
463	C₂×C₄	—	2	3	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	3	no	—
467	C₂×C₄	—	2	6	no	—
468	C₂×C₄	—	2	6	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	6	no	—
471	C₂×C₄	—	2	6	no	—
472	C₂×C₄	—	4	12	no	—
473	C₂×C₄	—	4	12	no	—
474	C₂×C₄	—	8	6	no	—
475	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
476	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
477	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
478	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
479	C₂×C₄	C₂×C₇×#41	1	6	yes	—
480	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
481	C₈	—	2	1	no	—
482	C₈	—	2	3	no	—
483	C₈	C₂²×C₇×D₄	1	6	yes	—
484	C₈	C₂²×C₇×D₄	1	2	yes	—
485	C₇	#60462	1	1	yes	characteristic, direct product factor, Sylow
486	C₂²	—	2	6	no	—
487	C₂²	—	2	2	no	—
488	C₂²	—	2	6	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	3	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	8	6	no	—
495	C₂²	C₂×C₇×#121	1	6	yes	—
496	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
497	C₄	—	2	1	no	—
498	C₄	—	2	3	no	—
499	C₄	—	4	6	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	6	no	—
502	C₄	C₂×C₇×#121	1	6	yes	—
503	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
504	C₄	C₂²×C₇×#41	1	1	yes	characteristic
505	C₂	—	2	6	no	—
506	C₂	—	4	1	no	—
507	C₂	—	4	3	no	—
508	C₂	—	8	2	no	—
509	C₂	C₂²×C₇×#121	1	1	yes	characteristic
510	C₂	C₇×#3586	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#578	C₂	semidirect
2. C₂×C₇×#568	C₂	semidirect
3. C₇×#3489	C₂	semidirect
4. C₇×#3063	C₂	semidirect
5. C₇×#3060	C₂	semidirect
6. C₇×#3059	C₂	semidirect
7. C₇×#3056	C₂	semidirect
8. C₇×#3054	C₂	semidirect
9. C₂×C₇×#470	C₂	semidirect
10. C₇×#2912	C₂	semidirect
11. C₇×#2898	C₂	semidirect
15. C₂×C₇×#137	C₂²	semidirect
16. C₂×C₇×#135	C₂²	semidirect
17. C₂²×C₇×Q₁₆	C₂²	semidirect
36. C₇×#470	C₂²	semidirect
37. C₇×#470	C₂²	semidirect
38. C₇×#470	C₂²	semidirect
40. C₇×#469	C₂²	semidirect
42. C₇×#468	C₂²	semidirect
43. C₇×#467	C₂²	semidirect
44. C₇×#465	C₂²	semidirect
45. C₇×#428	C₂²	semidirect
46. C₇×#427	C₂²	semidirect
47. C₂×C₇×#106	C₂²	semidirect
48. C₇×#419	C₂²	semidirect
49. C₇×#417	C₂²	semidirect
50. C₇×#416	C₂²	semidirect
51. C₂×C₇×#103	C₂²	semidirect
52. #60462	C₇	direct (N × H)
78. C₇×#131	D₄	semidirect
122. C₂×#578	C₂×C₇	semidirect
123. C₂×#568	C₂×C₇	semidirect
124. #3489	C₂×C₇	semidirect
125. #3063	C₂×C₇	semidirect
126. #3060	C₂×C₇	semidirect
127. #3059	C₂×C₇	semidirect
128. #3056	C₂×C₇	semidirect
129. #3054	C₂×C₇	semidirect
130. C₂×#470	C₂×C₇	semidirect
131. #2912	C₂×C₇	semidirect
132. #2898	C₂×C₇	semidirect
203. C₂×#137	C₂²×C₇	semidirect
204. C₂×#135	C₂²×C₇	semidirect
205. C₂²×Q₁₆	C₂²×C₇	semidirect
224. #470	C₂²×C₇	semidirect
225. #470	C₂²×C₇	semidirect
226. #470	C₂²×C₇	semidirect
228. #469	C₂²×C₇	semidirect
230. #468	C₂²×C₇	semidirect
231. #467	C₂²×C₇	semidirect
232. #465	C₂²×C₇	semidirect
233. #428	C₂²×C₇	semidirect
234. #427	C₂²×C₇	semidirect
235. C₂×#106	C₂²×C₇	semidirect
236. #419	C₂²×C₇	semidirect
237. #417	C₂²×C₇	semidirect
238. #416	C₂²×C₇	semidirect
239. C₂×#103	C₂²×C₇	semidirect
307. #131	C₇×D₄	semidirect
485. C₇	#60462	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#578	2
C₂×C₇×#568	2
C₇×#3489	2
C₇×#3063	2
C₇×#3060	2
C₇×#3059	2
C₇×#3056	2
C₇×#3054	2
C₂×C₇×#470	2
C₇×#2912	2
C₇×#2898	2
#60462	7

Subgroups — C7×#60462 #13237678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60462 #13237678