Subgroups — C₇×#60458 #13237674

Order 1792.

2824 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3488	C₂	1	2	yes	maximal, complemented
4	C₇×#3062	C₂	1	6	yes	maximal, complemented
5	C₇×#3061	C₂	1	3	yes	maximal, complemented
6	C₇×#3058	C₂	1	3	yes	maximal, complemented
7	C₇×#3055	C₂	1	6	yes	maximal, complemented
8	C₇×#3050	C₂	1	3	yes	maximal, complemented
9	C₂×C₇×#466	C₂	1	3	yes	maximal, complemented
10	C₇×#2913	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#2897	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
13	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
14	C₇×#576	—	2	8	no	—
15	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
16	C₂×C₇×#135	C₂²	1	3	yes	complemented
17	C₂²×C₇×D₈	C₂²	1	3	yes	complemented
18	C₂×C₇×#131	—	2	8	no	—
19	C₂×C₇×#131	C₂²	1	1	yes	characteristic
20	C₂×C₇×#131	C₂²	1	1	yes	characteristic
21	C₂×C₇×#131	C₂²	1	3	yes	complemented
22	C₂×C₇×#131	C₂²	1	3	yes	complemented
23	C₂²×C₇×M₁₆	C₂²	1	3	yes	complemented
24	C₂³×C₇×C₈	C₂²	1	3	yes	complemented
25	C₇×#550	—	2	2	no	—
26	C₇×#548	—	2	2	no	—
27	C₇×#546	—	2	6	no	—
28	C₇×D₄²	—	2	6	no	—
29	C₇×#537	C₂²	1	6	yes	complemented
30	C₇×#535	C₂²	1	6	yes	complemented
31	C₇×#525	C₂²	1	6	yes	complemented
32	C₂×C₇×#121	C₂²	1	6	yes	complemented
33	C₇×#520	C₂²	1	2	yes	—
34	C₇×#469	C₂²	1	12	yes	complemented
35	C₇×#468	C₂²	1	12	yes	complemented
36	C₇×#467	C₂²	1	1	yes	characteristic, complemented
37	C₇×#467	C₂²	1	3	yes	complemented
38	C₇×#466	C₂²	1	3	yes	complemented
39	C₇×#466	C₂²	1	6	yes	complemented
40	C₇×#466	C₂²	1	6	yes	complemented
41	C₇×#466	C₂²	1	6	yes	complemented
42	C₇×#466	C₂²	1	3	yes	complemented
43	C₇×#465	C₂²	1	6	yes	complemented
44	C₇×#465	C₂²	1	6	yes	complemented
45	C₇×#428	C₂²	1	6	yes	complemented
46	C₇×#427	C₂²	1	3	yes	complemented
47	C₂×C₇×#107	C₂²	1	3	yes	complemented
48	C₇×#418	C₂²	1	6	yes	complemented
49	C₇×#417	C₂²	1	6	yes	complemented
50	C₇×#416	C₂²	1	6	yes	complemented
51	C₂×C₇×#102	C₂²	1	6	yes	complemented
52	#60458	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₇×H(5,𝔽₂)	—	2	8	no	—
54	C₇×H(5,𝔽₂)	—	2	8	no	—
55	C₂×C₇×#41	—	2	4	no	—
56	C₂×C₇×#41	—	2	4	no	—
57	C₂×C₇×#41	—	2	2	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	D₄	1	1	yes	characteristic
64	C₂²×C₇×D₄	—	2	6	no	—
65	C₂²×C₇×D₄	—	2	6	no	—
66	C₂²×C₇×D₄	D₄	1	3	yes	—
67	C₂²×C₇×D₄	C₂³	1	6	yes	—
68	C₂³×C₄×C₇	C₂³	1	3	yes	—
69	C₇×AGL(1,ℤ/8)	—	2	24	no	—
70	C₇×#135	—	2	12	no	—
71	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
73	C₂×C₇×D₈	—	2	12	no	—
74	C₂×C₇×D₈	C₂³	1	3	yes	—
75	C₂×C₇×D₈	C₂³	1	6	yes	—
76	C₇×#131	—	2	4	no	—
77	C₇×#131	—	2	12	no	—
78	C₇×#131	—	2	12	no	—
79	C₇×#131	D₄	1	8	yes	complemented
80	C₂×C₇×M₁₆	—	2	12	no	—
81	C₂×C₇×M₁₆	C₂³	1	6	yes	—
82	C₂×C₇×M₁₆	C₂³	1	3	yes	—
83	C₂×C₇×M₁₆	C₂³	1	3	yes	—
84	C₂²×C₇×C₈	—	2	12	no	—
85	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₂²×C₇×C₈	C₂³	1	3	yes	—
87	C₂²×C₇×C₈	C₂³	1	3	yes	—
88	C₂²×C₇×C₈	C₂³	1	3	yes	—
89	C₂²×C₇×C₈	C₂³	1	3	yes	—
90	C₂²×C₇×C₈	C₂³	1	3	yes	—
91	C₇×#127	—	2	6	no	—
92	C₇×#124	—	2	6	no	—
93	C₇×#123	—	2	6	no	—
94	C₇×C₂²⋊Q₈	—	2	6	no	—
95	C₇×C₂²⋊Q₈	C₂³	1	6	yes	—
96	C₇×C₂²⋊Q₈	C₂³	1	2	yes	—
97	C₇×#121	—	2	12	no	—
98	C₇×#121	—	2	6	no	—
99	C₇×#121	—	2	6	no	—
100	C₇×#121	—	2	6	no	—
101	C₇×#121	C₂³	1	6	yes	—
102	C₇×#121	C₂³	1	12	yes	—
103	C₇×#121	C₂³	1	6	yes	—
104	C₇×#120	—	2	6	no	—
105	C₇×#120	—	2	12	no	—
106	C₄×C₇×Q₈	—	2	2	no	—
107	C₄×C₇×D₄	—	2	6	no	—
108	C₄×C₇×D₄	—	2	6	no	—
109	C₄×C₇×D₄	—	2	6	no	—
110	C₇×#117	C₂³	1	6	yes	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
112	C₂×C₇×C₂²⋊C₄	—	2	6	no	—
113	C₇×#107	C₂³	1	3	yes	—
114	C₇×#107	C₂³	1	1	yes	characteristic
115	C₇×#107	C₂³	1	6	yes	—
116	C₇×#106	C₂³	1	6	yes	—
117	C₇×#103	C₂³	1	6	yes	—
118	C₇×#103	C₂³	1	2	yes	—
119	C₇×#102	C₂³	1	6	yes	—
120	C₇×#102	C₂³	1	12	yes	—
121	C₇×#102	C₂³	1	6	yes	—
122	C₂×#576	C₂×C₇	1	1	yes	characteristic, complemented
123	C₂²×#131	C₂×C₇	1	1	yes	characteristic, complemented
124	#3488	C₂×C₇	1	2	yes	complemented
125	#3062	C₂×C₇	1	6	yes	complemented
126	#3061	C₂×C₇	1	3	yes	complemented
127	#3058	C₂×C₇	1	3	yes	complemented
128	#3055	C₂×C₇	1	6	yes	complemented
129	#3050	C₂×C₇	1	3	yes	complemented
130	C₂×#466	C₂×C₇	1	3	yes	complemented
131	#2913	C₂×C₇	1	1	yes	characteristic, complemented
132	#2897	C₂×C₇	1	2	yes	complemented
133	C₂⁴×C₇	—	2	6	no	—
134	C₂⁴×C₇	C₂×D₄	1	3	yes	—
135	C₇×#41	—	2	4	no	—
136	C₇×#41	—	2	12	no	—
137	C₇×#41	—	2	6	no	—
138	C₇×#41	—	2	6	no	—
139	C₇×#41	—	2	12	no	—
140	C₇×#41	—	2	12	no	—
141	C₇×#41	—	4	8	no	—
142	C₇×#41	C₂×D₄	1	4	yes	—
143	C₇×#41	#41	1	4	yes	—
144	C₂×C₇×Q₈	—	2	4	no	—
145	C₂×C₇×Q₈	C₂×D₄	1	3	yes	—
146	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
147	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
148	C₂×C₇×D₄	—	2	12	no	—
149	C₂×C₇×D₄	—	2	6	no	—
150	C₂×C₇×D₄	—	2	6	no	—
151	C₂×C₇×D₄	—	2	6	no	—
152	C₂×C₇×D₄	—	2	12	no	—
153	C₂×C₇×D₄	—	2	12	no	—
154	C₂×C₇×D₄	—	2	6	no	—
155	C₂×C₇×D₄	—	2	12	no	—
156	C₂×C₇×D₄	—	4	24	no	—
157	C₂×C₇×D₄	—	8	12	no	—
158	C₂×C₇×D₄	C₂×D₄	1	6	yes	—
159	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
160	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
161	C₂×C₇×D₄	C₂⁴	1	6	yes	—
162	C₂²×C₄×C₇	—	2	6	no	—
163	C₂²×C₄×C₇	—	2	6	no	—
164	C₂²×C₄×C₇	—	2	6	no	—
165	C₂²×C₄×C₇	—	2	6	no	—
166	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
167	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
168	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
169	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
171	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
172	C₇×Q₁₆	—	2	2	no	—
173	C₇×SD₁₆	—	2	12	no	—
174	C₇×D₈	—	2	6	no	—
175	C₇×D₈	—	2	12	no	—
176	C₇×M₁₆	—	2	6	no	—
177	C₇×M₁₆	—	2	12	no	—
178	C₇×M₁₆	C₂×D₄	1	12	yes	complemented
179	C₂×C₇×C₈	—	2	6	no	—
180	C₂×C₇×C₈	—	2	2	no	—
181	C₂×C₇×C₈	—	2	6	no	—
182	C₂×C₇×C₈	—	2	6	no	—
183	C₂×C₇×C₈	—	2	6	no	—
184	C₂×C₇×C₈	C₂×D₄	1	12	yes	complemented
185	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
186	C₂×C₇×C₈	C₂⁴	1	3	yes	—
187	C₂×C₇×C₈	C₂⁴	1	3	yes	—
188	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
189	C₇×C₄⋊C₄	—	2	6	no	—
190	C₇×C₄⋊C₄	—	2	2	no	—
191	C₇×C₄⋊C₄	—	2	6	no	—
192	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
193	C₇×C₄⋊C₄	C₂⁴	1	6	yes	—
194	C₇×C₂²⋊C₄	—	2	6	no	—
195	C₇×C₂²⋊C₄	—	2	2	no	—
196	C₇×C₂²⋊C₄	—	2	6	no	—
197	C₇×C₂²⋊C₄	—	2	6	no	—
198	C₇×C₂²⋊C₄	—	2	12	no	—
199	C₇×C₂²⋊C₄	—	2	6	no	—
200	C₄²×C₇	—	2	6	no	—
201	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
202	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
203	#576	—	2	8	no	—
204	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	6	yes	complemented
205	C₂×#135	C₂²×C₇	1	3	yes	complemented
206	C₂²×D₈	C₂²×C₇	1	3	yes	complemented
207	C₂×#131	—	2	8	no	—
208	C₂×#131	C₂²×C₇	1	1	yes	characteristic
209	C₂×#131	C₂²×C₇	1	1	yes	characteristic
210	C₂×#131	C₂²×C₇	1	3	yes	complemented
211	C₂×#131	C₂²×C₇	1	3	yes	complemented
212	C₂²×M₁₆	C₂²×C₇	1	3	yes	complemented
213	C₂³×C₈	C₂²×C₇	1	3	yes	complemented
214	#550	—	2	2	no	—
215	#548	—	2	2	no	—
216	#546	—	2	6	no	—
217	D₄²	—	2	6	no	—
218	#537	C₂²×C₇	1	6	yes	complemented
219	#535	C₂²×C₇	1	6	yes	complemented
220	#525	C₂²×C₇	1	6	yes	complemented
221	C₂×#121	C₂²×C₇	1	6	yes	complemented
222	#520	C₂²×C₇	1	2	yes	—
223	#469	C₂²×C₇	1	12	yes	complemented
224	#468	C₂²×C₇	1	12	yes	complemented
225	#467	C₂²×C₇	1	1	yes	characteristic, complemented
226	#467	C₂²×C₇	1	3	yes	complemented
227	#466	C₂²×C₇	1	3	yes	complemented
228	#466	C₂²×C₇	1	6	yes	complemented
229	#466	C₂²×C₇	1	6	yes	complemented
230	#466	C₂²×C₇	1	6	yes	complemented
231	#466	C₂²×C₇	1	3	yes	complemented
232	#465	C₂²×C₇	1	6	yes	complemented
233	#465	C₂²×C₇	1	6	yes	complemented
234	#428	C₂²×C₇	1	6	yes	complemented
235	#427	C₂²×C₇	1	3	yes	complemented
236	C₂×#107	C₂²×C₇	1	3	yes	complemented
237	#418	C₂²×C₇	1	6	yes	complemented
238	#417	C₂²×C₇	1	6	yes	complemented
239	#416	C₂²×C₇	1	6	yes	complemented
240	C₂×#102	C₂²×C₇	1	6	yes	complemented
241	C₂³×C₇	—	2	6	no	—
242	C₂³×C₇	—	2	6	no	—
243	C₂³×C₇	—	2	6	no	—
244	C₂³×C₇	—	4	3	no	—
245	C₂³×C₇	—	4	12	no	—
246	C₂³×C₇	—	8	6	no	—
247	C₂³×C₇	—	8	6	no	—
248	C₂³×C₇	C₂²×D₄	1	3	yes	—
249	C₂³×C₇	C₂²×D₄	1	3	yes	—
250	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
251	C₇×Q₈	—	2	6	no	—
252	C₇×Q₈	—	2	2	no	—
253	C₇×Q₈	—	2	2	no	—
254	C₇×Q₈	#121	1	4	yes	—
255	C₇×D₄	—	2	12	no	—
256	C₇×D₄	—	2	6	no	—
257	C₇×D₄	—	2	6	no	—
258	C₇×D₄	—	2	6	no	—
259	C₇×D₄	—	4	12	no	—
260	C₇×D₄	—	4	12	no	—
261	C₇×D₄	—	4	12	no	—
262	C₇×D₄	—	8	6	no	—
263	C₇×D₄	—	8	12	no	—
264	C₇×D₄	#121	1	12	yes	—
265	C₂×C₄×C₇	—	2	6	no	—
266	C₂×C₄×C₇	—	2	6	no	—
267	C₂×C₄×C₇	—	2	6	no	—
268	C₂×C₄×C₇	—	2	1	no	—
269	C₂×C₄×C₇	—	2	3	no	—
270	C₂×C₄×C₇	—	2	1	no	—
271	C₂×C₄×C₇	—	2	3	no	—
272	C₂×C₄×C₇	—	2	2	no	—
273	C₂×C₄×C₇	—	2	6	no	—
274	C₂×C₄×C₇	—	2	2	no	—
275	C₂×C₄×C₇	—	4	12	no	—
276	C₂×C₄×C₇	—	8	6	no	—
277	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
279	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
280	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
281	C₂×C₄×C₇	C₂×#41	1	6	yes	—
282	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
283	C₇×C₈	—	2	1	no	—
284	C₇×C₈	—	2	3	no	—
285	C₇×C₈	C₂²×D₄	1	2	yes	—
286	C₇×C₈	C₂²×D₄	1	6	yes	—
287	H(5,𝔽₂)	—	2	8	no	—
288	H(5,𝔽₂)	—	2	8	no	—
289	C₂×#41	—	2	4	no	—
290	C₂×#41	—	2	4	no	—
291	C₂×#41	—	2	2	no	—
292	C₂×#41	C₇×D₄	1	3	yes	—
293	C₂×#41	C₇×D₄	1	1	yes	characteristic
294	C₂×#41	C₂³×C₇	1	6	yes	—
295	C₂×#41	C₂³×C₇	1	3	yes	—
296	C₂×#41	C₂³×C₇	1	1	yes	characteristic
297	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
298	C₂²×D₄	—	2	6	no	—
299	C₂²×D₄	—	2	6	no	—
300	C₂²×D₄	C₇×D₄	1	3	yes	—
301	C₂²×D₄	C₂³×C₇	1	6	yes	—
302	C₂³×C₄	C₂³×C₇	1	3	yes	—
303	AGL(1,ℤ/8)	—	2	24	no	—
304	#135	—	2	12	no	—
305	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
306	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
307	C₂×D₈	—	2	12	no	—
308	C₂×D₈	C₂³×C₇	1	3	yes	—
309	C₂×D₈	C₂³×C₇	1	6	yes	—
310	#131	—	2	4	no	—
311	#131	—	2	12	no	—
312	#131	—	2	12	no	—
313	#131	C₇×D₄	1	8	yes	complemented
314	C₂×M₁₆	—	2	12	no	—
315	C₂×M₁₆	C₂³×C₇	1	3	yes	—
316	C₂×M₁₆	C₂³×C₇	1	6	yes	—
317	C₂×M₁₆	C₂³×C₇	1	3	yes	—
318	C₂²×C₈	—	2	12	no	—
319	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₇	1	3	yes	—
321	C₂²×C₈	C₂³×C₇	1	3	yes	—
322	C₂²×C₈	C₂³×C₇	1	3	yes	—
323	C₂²×C₈	C₂³×C₇	1	3	yes	—
324	C₂²×C₈	C₂³×C₇	1	3	yes	—
325	#127	—	2	6	no	—
326	#124	—	2	6	no	—
327	#123	—	2	6	no	—
328	C₂²⋊Q₈	—	2	6	no	—
329	C₂²⋊Q₈	C₂³×C₇	1	2	yes	—
330	C₂²⋊Q₈	C₂³×C₇	1	6	yes	—
331	#121	—	2	12	no	—
332	#121	—	2	6	no	—
333	#121	—	2	6	no	—
334	#121	—	2	6	no	—
335	#121	C₂³×C₇	1	6	yes	—
336	#121	C₂³×C₇	1	12	yes	—
337	#121	C₂³×C₇	1	6	yes	—
338	#120	—	2	6	no	—
339	#120	—	2	12	no	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×D₄	—	2	6	no	—
342	C₄×D₄	—	2	6	no	—
343	C₄×D₄	—	2	6	no	—
344	#117	C₂³×C₇	1	6	yes	—
345	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
346	C₂×C₂²⋊C₄	—	2	6	no	—
347	#107	C₂³×C₇	1	3	yes	—
348	#107	C₂³×C₇	1	6	yes	—
349	#107	C₂³×C₇	1	1	yes	characteristic
350	#106	C₂³×C₇	1	6	yes	—
351	#103	C₂³×C₇	1	2	yes	—
352	#103	C₂³×C₇	1	6	yes	—
353	#102	C₂³×C₇	1	6	yes	—
354	#102	C₂³×C₇	1	12	yes	—
355	#102	C₂³×C₇	1	6	yes	—
356	C₂²×C₇	—	2	1	no	—
357	C₂²×C₇	—	2	3	no	—
358	C₂²×C₇	—	2	1	no	—
359	C₂²×C₇	—	2	3	no	—
360	C₂²×C₇	—	2	3	no	—
361	C₂²×C₇	—	2	3	no	—
362	C₂²×C₇	—	2	3	no	—
363	C₂²×C₇	—	4	12	no	—
364	C₂²×C₇	—	4	6	no	—
365	C₂²×C₇	—	4	6	no	—
366	C₂²×C₇	—	4	6	no	—
367	C₂²×C₇	—	16	12	no	—
368	C₂²×C₇	C₂×#121	1	6	yes	—
369	C₂²×C₇	#576	1	2	yes	complemented
370	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
371	C₄×C₇	—	2	1	no	—
372	C₄×C₇	—	2	3	no	—
373	C₄×C₇	—	4	2	no	—
374	C₄×C₇	—	4	2	no	—
375	C₄×C₇	—	4	6	no	—
376	C₄×C₇	C₂×#121	1	6	yes	—
377	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
378	C₄×C₇	C₂²×#41	1	1	yes	characteristic
379	C₂⁴	—	2	6	no	—
380	C₂⁴	C₂×C₇×D₄	1	3	yes	—
381	#41	—	2	4	no	—
382	#41	—	2	12	no	—
383	#41	—	2	6	no	—
384	#41	—	2	6	no	—
385	#41	—	2	12	no	—
386	#41	—	2	12	no	—
387	#41	—	4	8	no	—
388	#41	C₂×C₇×D₄	1	4	yes	—
389	#41	C₇×#41	1	4	yes	—
390	C₂×Q₈	—	2	4	no	—
391	C₂×Q₈	C₂×C₇×D₄	1	3	yes	—
392	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
393	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
394	C₂×D₄	—	2	12	no	—
395	C₂×D₄	—	2	6	no	—
396	C₂×D₄	—	2	6	no	—
397	C₂×D₄	—	2	6	no	—
398	C₂×D₄	—	2	12	no	—
399	C₂×D₄	—	2	12	no	—
400	C₂×D₄	—	2	6	no	—
401	C₂×D₄	—	2	12	no	—
402	C₂×D₄	—	4	24	no	—
403	C₂×D₄	—	8	12	no	—
404	C₂×D₄	C₂×C₇×D₄	1	6	yes	—
405	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
406	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
407	C₂×D₄	C₂⁴×C₇	1	6	yes	—
408	C₂²×C₄	—	2	6	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	—	2	6	no	—
411	C₂²×C₄	—	2	6	no	—
412	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
413	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
414	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
415	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
417	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
418	Q₁₆	—	2	2	no	—
419	SD₁₆	—	2	12	no	—
420	D₈	—	2	6	no	—
421	D₈	—	2	12	no	—
422	M₁₆	—	2	6	no	—
423	M₁₆	—	2	12	no	—
424	M₁₆	C₂×C₇×D₄	1	12	yes	complemented
425	C₂×C₈	—	2	6	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	6	no	—
428	C₂×C₈	—	2	6	no	—
429	C₂×C₈	—	2	6	no	—
430	C₂×C₈	C₂×C₇×D₄	1	12	yes	complemented
431	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
432	C₂×C₈	C₂⁴×C₇	1	3	yes	—
433	C₂×C₈	C₂⁴×C₇	1	3	yes	—
434	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
435	C₄⋊C₄	—	2	6	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	6	no	—
438	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
439	C₄⋊C₄	C₂⁴×C₇	1	6	yes	—
440	C₂²⋊C₄	—	2	6	no	—
441	C₂²⋊C₄	—	2	2	no	—
442	C₂²⋊C₄	—	2	6	no	—
443	C₂²⋊C₄	—	2	6	no	—
444	C₂²⋊C₄	—	2	12	no	—
445	C₂²⋊C₄	—	2	6	no	—
446	C₄²	—	2	6	no	—
447	C₂×C₇	—	2	2	no	—
448	C₂×C₇	—	2	6	no	—
449	C₂×C₇	—	4	3	no	—
450	C₂×C₇	—	8	6	no	—
451	C₂×C₇	C₂²×#121	1	1	yes	characteristic
452	C₂×C₇	C₂×#576	1	1	yes	characteristic
453	C₂×C₇	#3584	1	1	yes	characteristic
454	C₂³	—	2	6	no	—
455	C₂³	—	2	6	no	—
456	C₂³	—	2	6	no	—
457	C₂³	—	4	3	no	—
458	C₂³	—	4	12	no	—
459	C₂³	—	8	6	no	—
460	C₂³	—	8	6	no	—
461	C₂³	C₂²×C₇×D₄	1	3	yes	—
462	C₂³	C₂²×C₇×D₄	1	3	yes	—
463	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
464	Q₈	—	2	6	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	2	2	no	—
467	Q₈	C₇×#121	1	4	yes	—
468	D₄	—	2	12	no	—
469	D₄	—	2	6	no	—
470	D₄	—	2	6	no	—
471	D₄	—	2	6	no	—
472	D₄	—	4	12	no	—
473	D₄	—	4	12	no	—
474	D₄	—	4	12	no	—
475	D₄	—	8	6	no	—
476	D₄	—	8	12	no	—
477	D₄	C₇×#121	1	12	yes	—
478	C₂×C₄	—	2	6	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	3	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	3	no	—
483	C₂×C₄	—	2	6	no	—
484	C₂×C₄	—	2	6	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	6	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	4	12	no	—
489	C₂×C₄	—	8	6	no	—
490	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
491	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
492	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
493	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
494	C₂×C₄	C₂×C₇×#41	1	6	yes	—
495	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
496	C₈	—	2	1	no	—
497	C₈	—	2	3	no	—
498	C₈	C₂²×C₇×D₄	1	2	yes	—
499	C₈	C₂²×C₇×D₄	1	6	yes	—
500	C₇	#60458	1	1	yes	characteristic, direct product factor, Sylow
501	C₂²	—	2	3	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	3	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	2	3	no	—
506	C₂²	—	2	3	no	—
507	C₂²	—	2	3	no	—
508	C₂²	—	4	12	no	—
509	C₂²	—	4	6	no	—
510	C₂²	—	4	6	no	—
511	C₂²	—	4	6	no	—
512	C₂²	—	16	12	no	—
513	C₂²	C₂×C₇×#121	1	6	yes	—
514	C₂²	C₇×#576	1	2	yes	complemented
515	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
516	C₄	—	2	1	no	—
517	C₄	—	2	3	no	—
518	C₄	—	4	6	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	C₂×C₇×#121	1	6	yes	—
522	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
523	C₄	C₂²×C₇×#41	1	1	yes	characteristic
524	C₂	—	2	2	no	—
525	C₂	—	2	6	no	—
526	C₂	—	4	3	no	—
527	C₂	—	8	6	no	—
528	C₂	C₂²×C₇×#121	1	1	yes	characteristic
529	C₂	C₂×C₇×#576	1	1	yes	characteristic
530	C₂	C₇×#3584	1	1	yes	characteristic

	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×#576	C₂	semidirect
2. C₂²×C₇×#131	C₂	semidirect
3. C₇×#3488	C₂	semidirect
4. C₇×#3062	C₂	semidirect
5. C₇×#3061	C₂	semidirect
6. C₇×#3058	C₂	semidirect
7. C₇×#3055	C₂	semidirect
8. C₇×#3050	C₂	semidirect
9. C₂×C₇×#466	C₂	semidirect
10. C₇×#2913	C₂	semidirect
11. C₇×#2897	C₂	semidirect
15. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂×C₇×#135	C₂²	semidirect
17. C₂²×C₇×D₈	C₂²	semidirect
21. C₂×C₇×#131	C₂²	semidirect
22. C₂×C₇×#131	C₂²	semidirect
23. C₂²×C₇×M₁₆	C₂²	semidirect
24. C₂³×C₇×C₈	C₂²	semidirect
29. C₇×#537	C₂²	semidirect
30. C₇×#535	C₂²	semidirect
31. C₇×#525	C₂²	semidirect
32. C₂×C₇×#121	C₂²	semidirect
34. C₇×#469	C₂²	semidirect
35. C₇×#468	C₂²	semidirect
36. C₇×#467	C₂²	semidirect
37. C₇×#467	C₂²	semidirect
38. C₇×#466	C₂²	semidirect
39. C₇×#466	C₂²	semidirect
40. C₇×#466	C₂²	semidirect
41. C₇×#466	C₂²	semidirect
42. C₇×#466	C₂²	semidirect
43. C₇×#465	C₂²	semidirect
44. C₇×#465	C₂²	semidirect
45. C₇×#428	C₂²	semidirect
46. C₇×#427	C₂²	semidirect
47. C₂×C₇×#107	C₂²	semidirect
48. C₇×#418	C₂²	semidirect
49. C₇×#417	C₂²	semidirect
50. C₇×#416	C₂²	semidirect
51. C₂×C₇×#102	C₂²	semidirect
52. #60458	C₇	direct (N × H)
79. C₇×#131	D₄	semidirect
122. C₂×#576	C₂×C₇	semidirect
123. C₂²×#131	C₂×C₇	semidirect
124. #3488	C₂×C₇	semidirect
125. #3062	C₂×C₇	semidirect
126. #3061	C₂×C₇	semidirect
127. #3058	C₂×C₇	semidirect
128. #3055	C₂×C₇	semidirect
129. #3050	C₂×C₇	semidirect
130. C₂×#466	C₂×C₇	semidirect
131. #2913	C₂×C₇	semidirect
132. #2897	C₂×C₇	semidirect
178. C₇×M₁₆	C₂×D₄	semidirect
184. C₂×C₇×C₈	C₂×D₄	semidirect
204. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
205. C₂×#135	C₂²×C₇	semidirect
206. C₂²×D₈	C₂²×C₇	semidirect
210. C₂×#131	C₂²×C₇	semidirect
211. C₂×#131	C₂²×C₇	semidirect
212. C₂²×M₁₆	C₂²×C₇	semidirect
213. C₂³×C₈	C₂²×C₇	semidirect
218. #537	C₂²×C₇	semidirect
219. #535	C₂²×C₇	semidirect
220. #525	C₂²×C₇	semidirect
221. C₂×#121	C₂²×C₇	semidirect
223. #469	C₂²×C₇	semidirect
224. #468	C₂²×C₇	semidirect
225. #467	C₂²×C₇	semidirect
226. #467	C₂²×C₇	semidirect
227. #466	C₂²×C₇	semidirect
228. #466	C₂²×C₇	semidirect
229. #466	C₂²×C₇	semidirect
230. #466	C₂²×C₇	semidirect
231. #466	C₂²×C₇	semidirect
232. #465	C₂²×C₇	semidirect
233. #465	C₂²×C₇	semidirect
234. #428	C₂²×C₇	semidirect
235. #427	C₂²×C₇	semidirect
236. C₂×#107	C₂²×C₇	semidirect
237. #418	C₂²×C₇	semidirect
238. #417	C₂²×C₇	semidirect
239. #416	C₂²×C₇	semidirect
240. C₂×#102	C₂²×C₇	semidirect
313. #131	C₇×D₄	semidirect
369. C₂²×C₇	#576	semidirect
424. M₁₆	C₂×C₇×D₄	semidirect
430. C₂×C₈	C₂×C₇×D₄	semidirect
500. C₇	#60458	direct (N × H)
514. C₂²	C₇×#576	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#576	2
C₂²×C₇×#131	2
C₇×#3488	2
C₇×#3062	2
C₇×#3061	2
C₇×#3058	2
C₇×#3055	2
C₇×#3050	2
C₂×C₇×#466	2
C₇×#2913	2
C₇×#2897	2
#60458	7

Subgroups — C7×#60458 #13237674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60458 #13237674