Subgroups — C₇×#21286 #13198502

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#442	C₂	1	1	yes	maximal, characteristic
2	C₇×#2293	C₂	1	2	yes	maximal, complemented
3	C₇×#1972	C₂	1	1	yes	maximal, characteristic
4	C₇×#1970	C₂	1	1	yes	maximal, characteristic
5	C₇×#1944	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1941	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1936	C₂	1	2	yes	maximal, complemented
8	C₇×#1934	C₂	1	2	yes	maximal, complemented
9	C₇×#1776	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1762	C₂	1	2	yes	maximal, complemented
11	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
14	C₂×C₇×#117	—	2	2	no	—
15	C₇×#442	C₄	1	8	yes	complemented
16	C₂×C₇×#107	C₂²	1	1	yes	characteristic
17	C₂×C₇×#106	C₂²	1	1	yes	characteristic
18	C₂×C₇×#106	C₂²	1	1	yes	characteristic
19	C₂×C₇×#102	C₂²	1	2	yes	—
20	C₂×C₇×#102	C₂²	1	2	yes	—
21	C₂×C₇×#102	C₂²	1	2	yes	—
22	C₂×C₇×#97	C₂²	1	1	yes	characteristic
23	C₂×C₇×#97	C₂²	1	1	yes	characteristic
24	C₂×C₇×#97	C₂²	1	1	yes	characteristic
25	C₂×C₇×#97	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
28	C₇×#390	C₂²	1	2	yes	—
29	C₇×#390	C₂²	1	2	yes	—
30	C₇×#389	C₂²	1	2	yes	—
31	C₇×#389	C₂²	1	2	yes	—
32	C₇×#388	—	2	2	no	—
33	C₇×#388	—	2	2	no	—
34	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
35	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
36	C₄³×C₇	C₂²	1	1	yes	characteristic
37	C₇×#340	C₂²	1	2	yes	—
38	C₇×#340	C₂²	1	2	yes	—
39	C₇×#340	C₂²	1	2	yes	—
40	#21286	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×C₇×D₄	C₂³	1	2	yes	—
42	C₂³×C₄×C₇	—	2	2	no	—
43	C₂×C₇×D₈	—	2	2	no	—
44	C₂×C₇×D₈	—	2	2	no	—
45	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
46	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₇×D₄	—	2	4	no	—
51	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
52	C₇×#117	—	4	8	no	—
53	C₂×C₇×C₄⋊C₄	—	2	2	no	—
54	C₂×C₇×C₄⋊C₄	—	2	2	no	—
55	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
57	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
58	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
59	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
60	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
61	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
62	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
63	C₂×C₄²×C₇	—	2	2	no	—
64	C₂×C₄²×C₇	D₄	1	2	yes	—
65	C₂×C₄²×C₇	D₄	1	2	yes	—
66	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
69	C₇×#107	—	2	2	no	—
70	C₇×#106	—	2	2	no	—
71	C₇×#106	C₂×C₄	1	4	yes	—
72	C₇×#102	—	2	4	no	—
73	C₇×#102	—	2	4	no	—
74	C₇×#102	C₂×C₄	1	8	yes	—
75	C₇×#97	—	2	2	no	—
76	C₇×#97	—	2	2	no	—
77	C₇×#97	—	2	2	no	—
78	C₇×#97	C₂×C₄	1	4	yes	—
79	C₄×C₇×C₈	—	2	2	no	—
80	C₄×C₇×C₈	—	2	2	no	—
81	C₇×#95	—	2	2	no	—
82	C₇×#95	—	2	2	no	—
83	C₇×#95	—	2	2	no	—
84	C₂×#442	C₂×C₇	1	1	yes	characteristic
85	#2293	C₂×C₇	1	2	yes	complemented
86	#1972	C₂×C₇	1	1	yes	characteristic
87	#1970	C₂×C₇	1	1	yes	characteristic
88	#1944	C₂×C₇	1	1	yes	characteristic, complemented
89	#1941	C₂×C₇	1	1	yes	characteristic, complemented
90	#1936	C₂×C₇	1	2	yes	complemented
91	#1934	C₂×C₇	1	2	yes	complemented
92	#1776	C₂×C₇	1	1	yes	characteristic, complemented
93	#1762	C₂×C₇	1	2	yes	complemented
94	C₄×#97	C₂×C₇	1	1	yes	characteristic, complemented
95	C₂⁴×C₇	—	2	2	no	—
96	C₂×C₇×D₄	—	2	2	no	—
97	C₂×C₇×D₄	—	2	2	no	—
98	C₂×C₇×D₄	—	2	2	no	—
99	C₂×C₇×D₄	—	2	2	no	—
100	C₂×C₇×D₄	—	2	2	no	—
101	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₇	—	2	2	no	—
103	C₂²×C₄×C₇	—	2	8	no	—
104	C₂²×C₄×C₇	—	2	2	no	—
105	C₂²×C₄×C₇	—	2	2	no	—
106	C₂²×C₄×C₇	—	4	4	no	—
107	C₂²×C₄×C₇	—	4	4	no	—
108	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
113	C₇×D₈	—	2	8	no	—
114	C₂×C₇×C₈	—	2	1	no	—
115	C₂×C₇×C₈	—	2	1	no	—
116	C₂×C₇×C₈	—	2	1	no	—
117	C₂×C₇×C₈	—	2	1	no	—
118	C₂×C₇×C₈	—	2	1	no	—
119	C₂×C₇×C₈	—	2	1	no	—
120	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
121	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
122	C₂×C₇×C₈	#41	1	2	yes	—
123	C₂×C₇×C₈	#41	1	2	yes	—
124	C₂×C₇×C₈	#41	1	2	yes	—
125	C₂×C₇×C₈	#41	1	2	yes	—
126	C₇×C₄⋊C₄	—	2	2	no	—
127	C₇×C₄⋊C₄	—	2	2	no	—
128	C₇×C₄⋊C₄	—	2	2	no	—
129	C₇×C₄⋊C₄	—	2	2	no	—
130	C₇×C₄⋊C₄	—	2	2	no	—
131	C₇×C₄⋊C₄	—	4	4	no	—
132	C₇×C₄⋊C₄	—	4	4	no	—
133	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
134	C₇×C₂²⋊C₄	—	2	8	no	—
135	C₇×C₂²⋊C₄	—	4	4	no	—
136	C₇×C₂²⋊C₄	—	4	4	no	—
137	C₇×C₂²⋊C₄	—	4	4	no	—
138	C₇×C₂²⋊C₄	—	4	4	no	—
139	C₄²×C₇	—	2	4	no	—
140	C₄²×C₇	—	2	1	no	—
141	C₄²×C₇	—	2	1	no	—
142	C₄²×C₇	—	2	1	no	—
143	C₄²×C₇	—	2	1	no	—
144	C₄²×C₇	—	4	2	no	—
145	C₄²×C₇	—	4	4	no	—
146	C₄²×C₇	C₂²×C₄	1	2	yes	—
147	C₄²×C₇	#41	1	2	yes	—
148	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
149	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
150	C₂×#117	—	2	2	no	—
151	#442	C₄×C₇	1	8	yes	complemented
152	C₂×#107	C₂²×C₇	1	1	yes	characteristic
153	C₂×#106	C₂²×C₇	1	1	yes	characteristic
154	C₂×#106	C₂²×C₇	1	1	yes	characteristic
155	C₂×#102	C₂²×C₇	1	2	yes	—
156	C₂×#102	C₂²×C₇	1	2	yes	—
157	C₂×#102	C₂²×C₇	1	2	yes	—
158	C₂×#97	C₂²×C₇	1	1	yes	characteristic
159	C₂×#97	C₂²×C₇	1	1	yes	characteristic
160	C₂×#97	C₂²×C₇	1	1	yes	characteristic
161	C₂×#97	C₂²×C₇	1	1	yes	characteristic
162	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
164	#390	C₂²×C₇	1	2	yes	—
165	#390	C₂²×C₇	1	2	yes	—
166	#389	C₂²×C₇	1	2	yes	—
167	#389	C₂²×C₇	1	2	yes	—
168	#388	—	2	2	no	—
169	#388	—	2	2	no	—
170	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
171	C₄×C₂²⋊C₄	—	2	2	no	—
172	C₄³	C₂²×C₇	1	1	yes	characteristic
173	#340	C₂²×C₇	1	2	yes	—
174	#340	C₂²×C₇	1	2	yes	—
175	#340	C₂²×C₇	1	2	yes	—
176	C₂³×C₇	—	2	4	no	—
177	C₂³×C₇	—	4	2	no	—
178	C₂³×C₇	—	4	2	no	—
179	C₂³×C₇	—	4	2	no	—
180	C₂³×C₇	—	4	2	no	—
181	C₂³×C₇	—	4	2	no	—
182	C₂³×C₇	—	4	2	no	—
183	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
184	C₇×D₄	—	2	4	no	—
185	C₇×D₄	—	2	4	no	—
186	C₇×D₄	—	4	2	no	—
187	C₇×D₄	—	4	2	no	—
188	C₂×C₄×C₇	—	2	2	no	—
189	C₂×C₄×C₇	—	2	2	no	—
190	C₂×C₄×C₇	—	2	2	no	—
191	C₂×C₄×C₇	—	2	2	no	—
192	C₂×C₄×C₇	—	2	4	no	—
193	C₂×C₄×C₇	—	2	1	no	—
194	C₂×C₄×C₇	—	2	1	no	—
195	C₂×C₄×C₇	—	2	2	no	—
196	C₂×C₄×C₇	—	2	2	no	—
197	C₂×C₄×C₇	—	4	16	no	—
198	C₂×C₄×C₇	—	4	2	no	—
199	C₂×C₄×C₇	—	4	2	no	—
200	C₂×C₄×C₇	—	4	2	no	—
201	C₂×C₄×C₇	—	4	2	no	—
202	C₂×C₄×C₇	—	4	2	no	—
203	C₂×C₄×C₇	—	4	2	no	—
204	C₂×C₄×C₇	—	4	2	no	—
205	C₂×C₄×C₇	—	4	2	no	—
206	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
207	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
208	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
214	C₇×C₈	—	2	2	no	—
215	C₇×C₈	—	2	2	no	—
216	C₇×C₈	—	2	2	no	—
217	C₇×C₈	#117	1	4	yes	complemented
218	C₂²×D₄	C₂³×C₇	1	2	yes	—
219	C₂³×C₄	—	2	2	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
223	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
227	C₄×D₄	—	2	4	no	—
228	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
229	#117	—	4	8	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
233	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
234	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₂²⋊C₄	—	2	2	no	—
239	C₂×C₂²⋊C₄	—	2	2	no	—
240	C₂×C₄²	—	2	2	no	—
241	C₂×C₄²	C₇×D₄	1	2	yes	—
242	C₂×C₄²	C₇×D₄	1	2	yes	—
243	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
244	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
245	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
246	#107	—	2	2	no	—
247	#106	—	2	2	no	—
248	#106	C₂×C₄×C₇	1	4	yes	—
249	#102	—	2	4	no	—
250	#102	—	2	4	no	—
251	#102	C₂×C₄×C₇	1	8	yes	—
252	#97	—	2	2	no	—
253	#97	—	2	2	no	—
254	#97	—	2	2	no	—
255	#97	C₂×C₄×C₇	1	4	yes	—
256	C₄×C₈	—	2	2	no	—
257	C₄×C₈	—	2	2	no	—
258	#95	—	2	2	no	—
259	#95	—	2	2	no	—
260	#95	—	2	2	no	—
261	C₂²×C₇	—	4	4	no	—
262	C₂²×C₇	—	4	4	no	—
263	C₂²×C₇	—	4	4	no	—
264	C₂²×C₇	—	8	2	no	—
265	C₂²×C₇	—	8	2	no	—
266	C₂²×C₇	—	8	2	no	—
267	C₂²×C₇	—	8	2	no	—
268	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
269	C₂²×C₇	#532	1	1	yes	characteristic
270	C₂²×C₇	#532	1	1	yes	characteristic
271	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
272	C₂²×C₇	#575	1	2	yes	—
273	C₄×C₇	—	2	4	no	—
274	C₄×C₇	—	2	4	no	—
275	C₄×C₇	—	4	1	no	—
276	C₄×C₇	—	4	1	no	—
277	C₄×C₇	—	4	4	no	—
278	C₄×C₇	—	8	2	no	—
279	C₄×C₇	—	8	2	no	—
280	C₄×C₇	C₂×#117	1	1	yes	characteristic
281	C₄×C₇	C₄×#41	1	1	yes	characteristic
282	C₄×C₇	#529	1	1	yes	characteristic
283	C₄×C₇	#529	1	1	yes	characteristic
284	C₂⁴	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	8	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	4	4	no	—
296	C₂²×C₄	—	4	4	no	—
297	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
300	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
301	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
302	D₈	—	2	8	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
310	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
311	C₂×C₈	C₇×#41	1	2	yes	—
312	C₂×C₈	C₇×#41	1	2	yes	—
313	C₂×C₈	C₇×#41	1	2	yes	—
314	C₂×C₈	C₇×#41	1	2	yes	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	4	4	no	—
321	C₄⋊C₄	—	4	4	no	—
322	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
323	C₂²⋊C₄	—	2	8	no	—
324	C₂²⋊C₄	—	4	4	no	—
325	C₂²⋊C₄	—	4	4	no	—
326	C₂²⋊C₄	—	4	4	no	—
327	C₂²⋊C₄	—	4	4	no	—
328	C₄²	—	2	4	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	2	1	no	—
331	C₄²	—	2	1	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	4	2	no	—
334	C₄²	—	4	4	no	—
335	C₄²	C₂²×C₄×C₇	1	2	yes	—
336	C₄²	C₇×#41	1	2	yes	—
337	C₂×C₇	—	8	4	no	—
338	C₂×C₇	#2293	1	1	yes	characteristic
339	C₂×C₇	C₄×AGL(1,ℤ/8)	1	2	yes	—
340	C₂×C₇	#3180	1	2	yes	—
341	C₂×C₇	#3183	1	1	yes	characteristic
342	C₂×C₇	#3184	1	1	yes	characteristic
343	C₂³	—	2	4	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	2	no	—
350	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
351	D₄	—	2	4	no	—
352	D₄	—	2	4	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	4	16	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
374	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
375	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
376	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
377	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
378	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
379	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
380	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	2	2	no	—
383	C₈	—	2	2	no	—
384	C₈	C₇×#117	1	4	yes	complemented
385	C₇	#21286	1	1	yes	characteristic, direct product factor, Sylow
386	C₂²	—	4	4	no	—
387	C₂²	—	4	4	no	—
388	C₂²	—	4	4	no	—
389	C₂²	—	8	2	no	—
390	C₂²	—	8	2	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	2	no	—
393	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
394	C₂²	C₇×#532	1	1	yes	characteristic
395	C₂²	C₇×#532	1	1	yes	characteristic
396	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
397	C₂²	C₇×#575	1	2	yes	—
398	C₄	—	2	4	no	—
399	C₄	—	2	4	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	—	4	4	no	—
403	C₄	—	8	2	no	—
404	C₄	—	8	2	no	—
405	C₄	C₂×C₇×#117	1	1	yes	characteristic
406	C₄	C₄×C₇×#41	1	1	yes	characteristic
407	C₄	C₇×#529	1	1	yes	characteristic
408	C₄	C₇×#529	1	1	yes	characteristic
409	C₂	—	8	4	no	—
410	C₂	C₇×#2293	1	1	yes	characteristic
411	C₂	C₄×C₇×AGL(1,ℤ/8)	1	2	yes	—
412	C₂	C₇×#3180	1	2	yes	—
413	C₂	C₇×#3183	1	1	yes	characteristic
414	C₂	C₇×#3184	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2293	C₂	semidirect
5. C₇×#1944	C₂	semidirect
6. C₇×#1941	C₂	semidirect
7. C₇×#1936	C₂	semidirect
8. C₇×#1934	C₂	semidirect
9. C₇×#1776	C₂	semidirect
10. C₇×#1762	C₂	semidirect
11. C₄×C₇×#97	C₂	semidirect
15. C₇×#442	C₄	semidirect
40. #21286	C₇	direct (N × H)
85. #2293	C₂×C₇	semidirect
88. #1944	C₂×C₇	semidirect
89. #1941	C₂×C₇	semidirect
90. #1936	C₂×C₇	semidirect
91. #1934	C₂×C₇	semidirect
92. #1776	C₂×C₇	semidirect
93. #1762	C₂×C₇	semidirect
94. C₄×#97	C₂×C₇	semidirect
151. #442	C₄×C₇	semidirect
217. C₇×C₈	#117	semidirect
384. C₈	C₇×#117	semidirect
385. C₇	#21286	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×#442	2
C₇×#2293	2
C₇×#1972	2
C₇×#1970	2
C₇×#1944	2
C₇×#1941	2
C₇×#1936	2
C₇×#1934	2
C₇×#1776	2
C₇×#1762	2
C₄×C₇×#97	2
#21286	7

Subgroups — C7×#21286 #13198502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21286 #13198502