Subgroups — C₇×#60624 #13237840

Order 1792.

2038 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3500	C₂	1	1	yes	maximal, characteristic
3	C₇×#3142	C₂	1	3	yes	maximal, complemented
4	C₇×#3140	C₂	1	3	yes	maximal, complemented
5	C₇×#3137	C₂	1	6	yes	maximal, complemented
6	C₇×#3136	C₂	1	6	yes	maximal, complemented
7	C₇×#3131	C₂	1	3	yes	maximal, complemented
8	C₂×C₇×#488	C₂	1	3	yes	maximal, complemented
9	C₇×#2898	C₂	1	2	yes	maximal, complemented
10	C₇×#2897	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
13	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
14	C₂×C₇×#131	C₂²	1	2	yes	—
15	C₇×#557	—	2	2	no	—
16	C₇×#554	—	2	6	no	—
17	C₇×#548	—	2	2	no	—
18	C₇×#546	—	2	6	no	—
19	C₇×#544	C₂²	1	6	yes	complemented
20	C₇×#536	C₂²	1	3	yes	—
21	C₇×#535	C₂²	1	3	yes	—
22	C₇×#533	C₂²	1	3	yes	complemented
23	C₇×#532	C₂²	1	3	yes	—
24	C₂×C₇×#125	C₂²	1	3	yes	complemented
25	C₂×C₇×#124	C₂²	1	3	yes	—
26	C₇×#520	C₂²	1	2	yes	—
27	C₄×C₇×#41	C₂²	1	1	yes	characteristic
28	C₇×#491	C₂²	1	6	yes	complemented
29	C₇×#490	C₂²	1	6	yes	complemented
30	C₇×#489	C₂²	1	12	yes	complemented
31	C₇×#488	C₂²	1	3	yes	complemented
32	C₇×#488	C₂²	1	12	yes	complemented
33	C₇×#488	C₂²	1	6	yes	complemented
34	C₇×#488	C₂²	1	1	yes	characteristic, complemented
35	C₇×#488	C₂²	1	3	yes	complemented
36	C₇×#488	C₂²	1	3	yes	complemented
37	C₇×#487	C₂²	1	6	yes	complemented
38	C₇×#486	C₂²	1	6	yes	complemented
39	C₇×#419	C₂²	1	6	yes	complemented
40	C₇×#418	C₂²	1	6	yes	—
41	C₇×#417	C₂²	1	6	yes	—
42	C₇×#417	C₂²	1	6	yes	complemented
43	C₇×#416	C₂²	1	6	yes	—
44	C₇×#416	C₂²	1	6	yes	complemented
45	C₂×C₇×#103	C₂²	1	6	yes	complemented
46	C₂×C₇×#102	C₂²	1	6	yes	—
47	C₇×#405	C₂²	1	6	yes	complemented
48	C₄×C₇×M₁₆	C₂²	1	3	yes	complemented
49	C₂×C₄×C₇×C₈	C₂²	1	3	yes	complemented
50	#60624	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₇×2₋¹⁺⁴	—	2	8	no	—
52	C₇×H(5,𝔽₂)	—	2	8	no	—
53	C₂×C₇×#41	—	2	1	no	—
54	C₂×C₇×#41	—	2	3	no	—
55	C₂×C₇×#41	C₂³	1	3	yes	—
56	C₂×C₇×#41	C₂³	1	3	yes	—
57	C₂×C₇×#41	C₂³	1	1	yes	characteristic
58	C₂²×C₇×Q₈	—	2	1	no	—
59	C₂²×C₇×Q₈	C₂³	1	3	yes	—
60	C₂²×C₇×D₄	—	2	3	no	—
61	C₂²×C₇×D₄	C₂³	1	3	yes	—
62	C₇×#131	—	2	8	no	—
63	C₂×C₇×M₁₆	C₂³	1	6	yes	—
64	C₂²×C₇×C₈	C₂³	1	6	yes	—
65	C₇×C₄⋊Q₈	—	2	6	no	—
66	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
67	C₇×C₄⋊Q₈	C₂³	1	6	yes	—
68	C₇×#127	C₂³	1	3	yes	—
69	C₇×#125	—	2	6	no	—
70	C₇×#125	—	2	6	no	—
71	C₇×#125	C₂³	1	3	yes	—
72	C₇×#125	C₂³	1	1	yes	characteristic
73	C₇×#125	C₂³	1	6	yes	—
74	C₇×#124	—	2	6	no	—
75	C₇×#124	C₂³	1	3	yes	—
76	C₇×#124	C₂³	1	1	yes	characteristic
77	C₇×#124	C₂³	1	6	yes	—
78	C₇×#123	—	2	6	no	—
79	C₇×C₂²⋊Q₈	—	2	6	no	—
80	C₇×C₂²⋊Q₈	—	2	6	no	—
81	C₇×C₂²⋊Q₈	—	2	12	no	—
82	C₇×#121	—	2	6	no	—
83	C₇×#121	—	2	6	no	—
84	C₇×#120	—	2	6	no	—
85	C₄×C₇×Q₈	—	2	2	no	—
86	C₄×C₇×Q₈	—	2	6	no	—
87	C₄×C₇×Q₈	D₄	1	2	yes	—
88	C₄×C₇×D₄	—	2	6	no	—
89	C₄×C₇×D₄	—	2	6	no	—
90	C₄×C₇×D₄	D₄	1	6	yes	—
91	C₇×#117	C₂³	1	6	yes	—
92	C₇×#117	C₂³	1	3	yes	—
93	C₂×C₇×C₄⋊C₄	—	2	6	no	—
94	C₂×C₇×C₄⋊C₄	C₂³	1	6	yes	—
95	C₂×C₇×C₂²⋊C₄	—	2	6	no	—
96	C₂×C₄²×C₇	C₂³	1	3	yes	—
97	C₇×#103	C₂³	1	2	yes	—
98	C₇×#103	C₂³	1	6	yes	—
99	C₇×#103	C₂³	1	6	yes	—
100	C₇×#103	C₂³	1	6	yes	—
101	C₇×#103	C₂³	1	12	yes	—
102	C₇×#102	C₂³	1	2	yes	—
103	C₇×#102	C₂³	1	6	yes	—
104	C₇×#102	C₂³	1	6	yes	—
105	C₇×#102	C₂³	1	6	yes	—
106	C₇×#102	C₂³	1	12	yes	—
107	C₇×#97	C₂³	1	6	yes	—
108	C₄×C₇×C₈	C₂³	1	6	yes	—
109	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
110	C₄×C₇×C₈	C₂³	1	3	yes	—
111	#3504	C₂×C₇	1	1	yes	characteristic, complemented
112	#3500	C₂×C₇	1	1	yes	characteristic
113	#3142	C₂×C₇	1	3	yes	complemented
114	#3140	C₂×C₇	1	3	yes	complemented
115	#3137	C₂×C₇	1	6	yes	complemented
116	#3136	C₂×C₇	1	6	yes	complemented
117	#3131	C₂×C₇	1	3	yes	complemented
118	C₂×#488	C₂×C₇	1	3	yes	complemented
119	#2898	C₂×C₇	1	2	yes	complemented
120	#2897	C₂×C₇	1	2	yes	complemented
121	C₄×#131	C₂×C₇	1	1	yes	characteristic, complemented
122	C₂⁴×C₇	—	2	3	no	—
123	C₇×#41	—	2	12	no	—
124	C₇×#41	—	2	12	no	—
125	C₇×#41	—	4	4	no	—
126	C₇×#41	—	4	12	no	—
127	C₇×#41	#41	1	8	yes	—
128	C₂×C₇×Q₈	—	2	3	no	—
129	C₂×C₇×Q₈	—	2	6	no	—
130	C₂×C₇×Q₈	—	2	12	no	—
131	C₂×C₇×Q₈	—	4	4	no	—
132	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₇×Q₈	C₂⁴	1	3	yes	—
134	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
135	C₂×C₇×D₄	—	2	3	no	—
136	C₂×C₇×D₄	—	2	3	no	—
137	C₂×C₇×D₄	—	2	3	no	—
138	C₂×C₇×D₄	—	2	6	no	—
139	C₂×C₇×D₄	—	2	12	no	—
140	C₂×C₇×D₄	—	4	12	no	—
141	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
142	C₂×C₇×D₄	C₂⁴	1	3	yes	—
143	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₇	—	2	3	no	—
145	C₂²×C₄×C₇	—	2	6	no	—
146	C₂²×C₄×C₇	—	2	3	no	—
147	C₂²×C₄×C₇	—	2	3	no	—
148	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
149	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
150	C₇×M₁₆	—	2	12	no	—
151	C₂×C₇×C₈	—	2	12	no	—
152	C₂×C₇×C₈	C₂⁴	1	2	yes	—
153	C₂×C₇×C₈	C₂⁴	1	6	yes	—
154	C₇×C₄⋊C₄	—	2	6	no	—
155	C₇×C₄⋊C₄	—	2	12	no	—
156	C₇×C₄⋊C₄	—	2	6	no	—
157	C₇×C₄⋊C₄	—	2	6	no	—
158	C₇×C₄⋊C₄	—	2	2	no	—
159	C₇×C₄⋊C₄	—	2	6	no	—
160	C₇×C₄⋊C₄	C₂×D₄	1	6	yes	—
161	C₇×C₄⋊C₄	C₂⁴	1	6	yes	—
162	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
163	C₇×C₂²⋊C₄	—	2	6	no	—
164	C₇×C₂²⋊C₄	—	2	6	no	—
165	C₇×C₂²⋊C₄	—	2	12	no	—
166	C₇×C₂²⋊C₄	—	2	2	no	—
167	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
168	C₄²×C₇	—	2	6	no	—
169	C₄²×C₇	C₂×D₄	1	6	yes	—
170	C₄²×C₇	C₂⁴	1	1	yes	characteristic
171	C₄²×C₇	C₂⁴	1	3	yes	—
172	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
173	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic
174	C₂×#131	C₂²×C₇	1	2	yes	—
175	#557	—	2	2	no	—
176	#554	—	2	6	no	—
177	#548	—	2	2	no	—
178	#546	—	2	6	no	—
179	#544	C₂²×C₇	1	6	yes	complemented
180	#536	C₂²×C₇	1	3	yes	—
181	#535	C₂²×C₇	1	3	yes	—
182	#533	C₂²×C₇	1	3	yes	complemented
183	#532	C₂²×C₇	1	3	yes	—
184	C₂×#125	C₂²×C₇	1	3	yes	complemented
185	C₂×#124	C₂²×C₇	1	3	yes	—
186	#520	C₂²×C₇	1	2	yes	—
187	C₄×#41	C₂²×C₇	1	1	yes	characteristic
188	#491	C₂²×C₇	1	6	yes	complemented
189	#490	C₂²×C₇	1	6	yes	complemented
190	#489	C₂²×C₇	1	12	yes	complemented
191	#488	C₂²×C₇	1	3	yes	complemented
192	#488	C₂²×C₇	1	12	yes	complemented
193	#488	C₂²×C₇	1	6	yes	complemented
194	#488	C₂²×C₇	1	1	yes	characteristic, complemented
195	#488	C₂²×C₇	1	3	yes	complemented
196	#488	C₂²×C₇	1	3	yes	complemented
197	#487	C₂²×C₇	1	6	yes	complemented
198	#486	C₂²×C₇	1	6	yes	complemented
199	#419	C₂²×C₇	1	6	yes	complemented
200	#418	C₂²×C₇	1	6	yes	—
201	#417	C₂²×C₇	1	6	yes	—
202	#417	C₂²×C₇	1	6	yes	complemented
203	#416	C₂²×C₇	1	6	yes	—
204	#416	C₂²×C₇	1	6	yes	complemented
205	C₂×#103	C₂²×C₇	1	6	yes	complemented
206	C₂×#102	C₂²×C₇	1	6	yes	—
207	#405	C₂²×C₇	1	6	yes	complemented
208	C₄×M₁₆	C₂²×C₇	1	3	yes	complemented
209	C₂×C₄×C₈	C₂²×C₇	1	3	yes	complemented
210	C₂³×C₇	—	2	3	no	—
211	C₂³×C₇	—	2	1	no	—
212	C₂³×C₇	—	4	6	no	—
213	C₂³×C₇	—	8	6	no	—
214	C₂³×C₇	C₂²×D₄	1	3	yes	—
215	C₇×Q₈	—	2	6	no	—
216	C₇×Q₈	—	2	2	no	—
217	C₇×Q₈	—	4	6	no	—
218	C₇×Q₈	—	8	6	no	—
219	C₇×Q₈	#124	1	4	yes	—
220	C₇×D₄	—	2	6	no	—
221	C₇×D₄	—	2	2	no	—
222	C₇×D₄	—	4	6	no	—
223	C₇×D₄	—	4	6	no	—
224	C₇×D₄	—	4	6	no	—
225	C₇×D₄	—	8	6	no	—
226	C₇×D₄	#124	1	12	yes	—
227	C₂×C₄×C₇	—	2	2	no	—
228	C₂×C₄×C₇	—	2	6	no	—
229	C₂×C₄×C₇	—	2	3	no	—
230	C₂×C₄×C₇	—	2	1	no	—
231	C₂×C₄×C₇	—	4	6	no	—
232	C₂×C₄×C₇	—	4	6	no	—
233	C₂×C₄×C₇	—	4	6	no	—
234	C₂×C₄×C₇	—	4	6	no	—
235	C₂×C₄×C₇	—	8	6	no	—
236	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
238	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
239	C₂×C₄×C₇	C₂×#41	1	12	yes	—
240	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
241	C₇×C₈	—	2	2	no	—
242	C₇×C₈	—	2	6	no	—
243	2₋¹⁺⁴	—	2	8	no	—
244	H(5,𝔽₂)	—	2	8	no	—
245	C₂×#41	—	2	1	no	—
246	C₂×#41	—	2	3	no	—
247	C₂×#41	C₂³×C₇	1	3	yes	—
248	C₂×#41	C₂³×C₇	1	3	yes	—
249	C₂×#41	C₂³×C₇	1	1	yes	characteristic
250	C₂²×Q₈	—	2	1	no	—
251	C₂²×Q₈	C₂³×C₇	1	3	yes	—
252	C₂²×D₄	—	2	3	no	—
253	C₂²×D₄	C₂³×C₇	1	3	yes	—
254	#131	—	2	8	no	—
255	C₂×M₁₆	C₂³×C₇	1	6	yes	—
256	C₂²×C₈	C₂³×C₇	1	6	yes	—
257	C₄⋊Q₈	—	2	6	no	—
258	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
259	C₄⋊Q₈	C₂³×C₇	1	6	yes	—
260	#127	C₂³×C₇	1	3	yes	—
261	#125	—	2	6	no	—
262	#125	—	2	6	no	—
263	#125	C₂³×C₇	1	3	yes	—
264	#125	C₂³×C₇	1	6	yes	—
265	#125	C₂³×C₇	1	1	yes	characteristic
266	#124	—	2	6	no	—
267	#124	C₂³×C₇	1	3	yes	—
268	#124	C₂³×C₇	1	6	yes	—
269	#124	C₂³×C₇	1	1	yes	characteristic
270	#123	—	2	6	no	—
271	C₂²⋊Q₈	—	2	6	no	—
272	C₂²⋊Q₈	—	2	6	no	—
273	C₂²⋊Q₈	—	2	12	no	—
274	#121	—	2	6	no	—
275	#121	—	2	6	no	—
276	#120	—	2	6	no	—
277	C₄×Q₈	—	2	2	no	—
278	C₄×Q₈	—	2	6	no	—
279	C₄×Q₈	C₇×D₄	1	2	yes	—
280	C₄×D₄	—	2	6	no	—
281	C₄×D₄	—	2	6	no	—
282	C₄×D₄	C₇×D₄	1	6	yes	—
283	#117	C₂³×C₇	1	6	yes	—
284	#117	C₂³×C₇	1	3	yes	—
285	C₂×C₄⋊C₄	—	2	6	no	—
286	C₂×C₄⋊C₄	C₂³×C₇	1	6	yes	—
287	C₂×C₂²⋊C₄	—	2	6	no	—
288	C₂×C₄²	C₂³×C₇	1	3	yes	—
289	#103	C₂³×C₇	1	6	yes	—
290	#103	C₂³×C₇	1	12	yes	—
291	#103	C₂³×C₇	1	2	yes	—
292	#103	C₂³×C₇	1	6	yes	—
293	#103	C₂³×C₇	1	6	yes	—
294	#102	C₂³×C₇	1	6	yes	—
295	#102	C₂³×C₇	1	12	yes	—
296	#102	C₂³×C₇	1	6	yes	—
297	#102	C₂³×C₇	1	2	yes	—
298	#102	C₂³×C₇	1	6	yes	—
299	#97	C₂³×C₇	1	6	yes	—
300	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
301	C₄×C₈	C₂³×C₇	1	3	yes	—
302	C₄×C₈	C₂³×C₇	1	6	yes	—
303	C₂²×C₇	—	2	6	no	—
304	C₂²×C₇	—	4	2	no	—
305	C₂²×C₇	—	4	6	no	—
306	C₂²×C₇	—	4	1	no	—
307	C₂²×C₇	—	4	3	no	—
308	C₂²×C₇	—	16	6	no	—
309	C₂²×C₇	C₂×#124	1	6	yes	—
310	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
311	C₄×C₇	—	2	2	no	—
312	C₄×C₇	—	2	6	no	—
313	C₄×C₇	—	4	6	no	—
314	C₄×C₇	—	4	2	no	—
315	C₄×C₇	—	4	1	no	—
316	C₄×C₇	—	4	3	no	—
317	C₄×C₇	C₂×#124	1	6	yes	—
318	C₄×C₇	C₂²×#41	1	2	yes	—
319	C₂⁴	—	2	3	no	—
320	#41	—	2	12	no	—
321	#41	—	2	12	no	—
322	#41	—	4	4	no	—
323	#41	—	4	12	no	—
324	#41	C₇×#41	1	8	yes	—
325	C₂×Q₈	—	2	3	no	—
326	C₂×Q₈	—	2	6	no	—
327	C₂×Q₈	—	2	12	no	—
328	C₂×Q₈	—	4	4	no	—
329	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
330	C₂×Q₈	C₂⁴×C₇	1	3	yes	—
331	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
332	C₂×D₄	—	2	3	no	—
333	C₂×D₄	—	2	3	no	—
334	C₂×D₄	—	2	3	no	—
335	C₂×D₄	—	2	6	no	—
336	C₂×D₄	—	2	12	no	—
337	C₂×D₄	—	4	12	no	—
338	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
339	C₂×D₄	C₂⁴×C₇	1	3	yes	—
340	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
341	C₂²×C₄	—	2	3	no	—
342	C₂²×C₄	—	2	6	no	—
343	C₂²×C₄	—	2	3	no	—
344	C₂²×C₄	—	2	3	no	—
345	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
346	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
347	M₁₆	—	2	12	no	—
348	C₂×C₈	—	2	12	no	—
349	C₂×C₈	C₂⁴×C₇	1	2	yes	—
350	C₂×C₈	C₂⁴×C₇	1	6	yes	—
351	C₄⋊C₄	—	2	6	no	—
352	C₄⋊C₄	—	2	12	no	—
353	C₄⋊C₄	—	2	6	no	—
354	C₄⋊C₄	—	2	6	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	6	no	—
357	C₄⋊C₄	C₂×C₇×D₄	1	6	yes	—
358	C₄⋊C₄	C₂⁴×C₇	1	6	yes	—
359	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
360	C₂²⋊C₄	—	2	6	no	—
361	C₂²⋊C₄	—	2	6	no	—
362	C₂²⋊C₄	—	2	12	no	—
363	C₂²⋊C₄	—	2	2	no	—
364	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
365	C₄²	—	2	6	no	—
366	C₄²	C₂×C₇×D₄	1	6	yes	—
367	C₄²	C₂⁴×C₇	1	1	yes	characteristic
368	C₄²	C₂⁴×C₇	1	3	yes	—
369	C₂×C₇	—	2	6	no	—
370	C₂×C₇	—	8	1	no	—
371	C₂×C₇	—	8	3	no	—
372	C₂×C₇	C₂²×#124	1	1	yes	characteristic
373	C₂×C₇	#3584	1	2	yes	—
374	C₂³	—	2	3	no	—
375	C₂³	—	2	1	no	—
376	C₂³	—	4	6	no	—
377	C₂³	—	8	6	no	—
378	C₂³	C₂²×C₇×D₄	1	3	yes	—
379	Q₈	—	2	6	no	—
380	Q₈	—	2	2	no	—
381	Q₈	—	4	6	no	—
382	Q₈	—	8	6	no	—
383	Q₈	C₇×#124	1	4	yes	—
384	D₄	—	2	6	no	—
385	D₄	—	2	2	no	—
386	D₄	—	4	6	no	—
387	D₄	—	4	6	no	—
388	D₄	—	4	6	no	—
389	D₄	—	8	6	no	—
390	D₄	C₇×#124	1	12	yes	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	6	no	—
393	C₂×C₄	—	2	3	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	4	6	no	—
396	C₂×C₄	—	4	6	no	—
397	C₂×C₄	—	4	6	no	—
398	C₂×C₄	—	4	6	no	—
399	C₂×C₄	—	8	6	no	—
400	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
401	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
402	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
403	C₂×C₄	C₂×C₇×#41	1	12	yes	—
404	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
405	C₈	—	2	2	no	—
406	C₈	—	2	6	no	—
407	C₇	#60624	1	1	yes	characteristic, direct product factor, Sylow
408	C₂²	—	2	6	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	6	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	3	no	—
413	C₂²	—	16	6	no	—
414	C₂²	C₂×C₇×#124	1	6	yes	—
415	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	6	no	—
418	C₄	—	4	6	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	1	no	—
421	C₄	—	4	3	no	—
422	C₄	C₂×C₇×#124	1	6	yes	—
423	C₄	C₂²×C₇×#41	1	2	yes	—
424	C₂	—	2	6	no	—
425	C₂	—	8	3	no	—
426	C₂	—	8	1	no	—
427	C₂	C₂²×C₇×#124	1	1	yes	characteristic
428	C₂	C₇×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3500	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3504	C₂	semidirect
3. C₇×#3142	C₂	semidirect
4. C₇×#3140	C₂	semidirect
5. C₇×#3137	C₂	semidirect
6. C₇×#3136	C₂	semidirect
7. C₇×#3131	C₂	semidirect
8. C₂×C₇×#488	C₂	semidirect
9. C₇×#2898	C₂	semidirect
10. C₇×#2897	C₂	semidirect
11. C₄×C₇×#131	C₂	semidirect
19. C₇×#544	C₂²	semidirect
22. C₇×#533	C₂²	semidirect
24. C₂×C₇×#125	C₂²	semidirect
28. C₇×#491	C₂²	semidirect
29. C₇×#490	C₂²	semidirect
30. C₇×#489	C₂²	semidirect
31. C₇×#488	C₂²	semidirect
32. C₇×#488	C₂²	semidirect
33. C₇×#488	C₂²	semidirect
34. C₇×#488	C₂²	semidirect
35. C₇×#488	C₂²	semidirect
36. C₇×#488	C₂²	semidirect
37. C₇×#487	C₂²	semidirect
38. C₇×#486	C₂²	semidirect
39. C₇×#419	C₂²	semidirect
42. C₇×#417	C₂²	semidirect
44. C₇×#416	C₂²	semidirect
45. C₂×C₇×#103	C₂²	semidirect
47. C₇×#405	C₂²	semidirect
48. C₄×C₇×M₁₆	C₂²	semidirect
49. C₂×C₄×C₇×C₈	C₂²	semidirect
50. #60624	C₇	direct (N × H)
111. #3504	C₂×C₇	semidirect
113. #3142	C₂×C₇	semidirect
114. #3140	C₂×C₇	semidirect
115. #3137	C₂×C₇	semidirect
116. #3136	C₂×C₇	semidirect
117. #3131	C₂×C₇	semidirect
118. C₂×#488	C₂×C₇	semidirect
119. #2898	C₂×C₇	semidirect
120. #2897	C₂×C₇	semidirect
121. C₄×#131	C₂×C₇	semidirect
179. #544	C₂²×C₇	semidirect
182. #533	C₂²×C₇	semidirect
184. C₂×#125	C₂²×C₇	semidirect
188. #491	C₂²×C₇	semidirect
189. #490	C₂²×C₇	semidirect
190. #489	C₂²×C₇	semidirect
191. #488	C₂²×C₇	semidirect
192. #488	C₂²×C₇	semidirect
193. #488	C₂²×C₇	semidirect
194. #488	C₂²×C₇	semidirect
195. #488	C₂²×C₇	semidirect
196. #488	C₂²×C₇	semidirect
197. #487	C₂²×C₇	semidirect
198. #486	C₂²×C₇	semidirect
199. #419	C₂²×C₇	semidirect
202. #417	C₂²×C₇	semidirect
204. #416	C₂²×C₇	semidirect
205. C₂×#103	C₂²×C₇	semidirect
207. #405	C₂²×C₇	semidirect
208. C₄×M₁₆	C₂²×C₇	semidirect
209. C₂×C₄×C₈	C₂²×C₇	semidirect
407. C₇	#60624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3504	2
C₇×#3500	2
C₇×#3142	2
C₇×#3140	2
C₇×#3137	2
C₇×#3136	2
C₇×#3131	2
C₂×C₇×#488	2
C₇×#2898	2
C₇×#2897	2
C₄×C₇×#131	2
#60624	7

Subgroups — C7×#60624 #13237840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60624 #13237840