Subgroups — C₇×#2417 #11741726

Order 896.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#398	C₂	1	2	yes	maximal, complemented
6	C₇×#396	C₂	1	2	yes	maximal, complemented
7	C₇×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#386	C₂	1	2	yes	maximal
10	C₇×#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#380	C₂	1	2	yes	maximal, complemented
12	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₇	C₂²	1	2	yes	—
16	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
17	C₇×C₂²⋊Q₈	—	2	4	no	—
18	C₇×C₂²⋊Q₈	—	2	4	no	—
19	C₄×C₇×D₄	—	2	4	no	—
20	C₂×C₇×C₄⋊C₄	—	2	4	no	—
21	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
27	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
35	C₇×#95	C₂²	1	2	yes	—
36	C₇×#95	C₂²	1	2	yes	—
37	C₇×#95	C₂²	1	2	yes	—
38	C₇×#95	C₂²	1	2	yes	—
39	#2417	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂⁴×C₇	C₂³	1	1	yes	characteristic
41	C₂⁴×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₇×Q₈	—	2	2	no	—
43	C₂×C₇×Q₈	—	2	2	no	—
44	C₂×C₇×Q₈	—	2	2	no	—
45	C₂×C₇×D₄	—	2	2	no	—
46	C₂×C₇×D₄	—	2	2	no	—
47	C₂×C₇×D₄	Q₈	1	4	yes	complemented
48	C₂²×C₄×C₇	—	2	2	no	—
49	C₂²×C₄×C₇	—	2	2	no	—
50	C₂²×C₄×C₇	—	2	4	no	—
51	C₂²×C₄×C₇	—	2	4	no	—
52	C₂²×C₄×C₇	—	2	2	no	—
53	C₂²×C₄×C₇	—	2	2	no	—
54	C₂²×C₄×C₇	—	2	2	no	—
55	C₂²×C₄×C₇	—	4	2	no	—
56	C₂²×C₄×C₇	D₄	1	4	yes	—
57	C₂²×C₄×C₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂³	1	2	yes	—
63	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
66	C₇×C₄⋊C₄	—	2	4	no	—
67	C₇×C₄⋊C₄	—	2	2	no	—
68	C₇×C₄⋊C₄	—	2	2	no	—
69	C₇×C₄⋊C₄	—	2	2	no	—
70	C₇×C₄⋊C₄	—	2	4	no	—
71	C₇×C₄⋊C₄	—	2	4	no	—
72	C₇×C₄⋊C₄	—	4	2	no	—
73	C₇×C₄⋊C₄	D₄	1	4	yes	complemented
74	C₇×C₂²⋊C₄	—	2	2	no	—
75	C₇×C₂²⋊C₄	—	2	4	no	—
76	C₇×C₂²⋊C₄	—	2	2	no	—
77	C₇×C₂²⋊C₄	—	2	4	no	—
78	C₇×C₂²⋊C₄	—	4	2	no	—
79	C₄²×C₇	—	2	2	no	—
80	C₂×C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
81	C₂×C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
82	C₂×C₄×D₄	C₂×C₇	1	1	yes	characteristic
83	C₂²×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
84	#398	C₂×C₇	1	2	yes	complemented
85	#396	C₂×C₇	1	2	yes	complemented
86	#391	C₂×C₇	1	1	yes	characteristic, complemented
87	#389	C₂×C₇	1	1	yes	characteristic, complemented
88	#386	C₂×C₇	1	2	yes	—
89	#380	C₂×C₇	1	1	yes	characteristic, complemented
90	#380	C₂×C₇	1	2	yes	complemented
91	C₂³×C₇	—	2	1	no	—
92	C₂³×C₇	—	2	1	no	—
93	C₂³×C₇	—	2	1	no	—
94	C₂³×C₇	—	2	1	no	—
95	C₂³×C₇	—	2	1	no	—
96	C₂³×C₇	—	2	1	no	—
97	C₂³×C₇	—	2	1	no	—
98	C₂³×C₇	—	2	1	no	—
99	C₂³×C₇	—	2	1	no	—
100	C₂³×C₇	—	2	1	no	—
101	C₂³×C₇	C₂×D₄	1	2	yes	—
102	C₂³×C₇	C₂×Q₈	1	2	yes	—
103	C₂³×C₇	C₂×Q₈	1	2	yes	—
104	C₂³×C₇	#41	1	2	yes	—
105	C₂³×C₇	C₂⁴	1	1	yes	characteristic
106	C₇×Q₈	—	4	4	no	—
107	C₇×D₄	—	2	8	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	1	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	2	8	no	—
115	C₂×C₄×C₇	—	2	1	no	—
116	C₂×C₄×C₇	—	2	2	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	2	no	—
120	C₂×C₄×C₇	—	2	1	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	2	1	no	—
128	C₂×C₄×C₇	—	2	1	no	—
129	C₂×C₄×C₇	—	2	2	no	—
130	C₂×C₄×C₇	—	4	8	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
133	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
134	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
135	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
136	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
137	C₂×C₄×C₇	#41	1	2	yes	—
138	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
139	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
140	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
141	C₂³×C₄	C₂²×C₇	1	2	yes	—
142	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
143	C₂²⋊Q₈	—	2	4	no	—
144	C₂²⋊Q₈	—	2	4	no	—
145	C₄×D₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
153	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
160	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
161	#95	C₂²×C₇	1	2	yes	—
162	#95	C₂²×C₇	1	2	yes	—
163	#95	C₂²×C₇	1	2	yes	—
164	#95	C₂²×C₇	1	2	yes	—
165	C₂²×C₇	—	2	2	no	—
166	C₂²×C₇	—	2	2	no	—
167	C₂²×C₇	—	2	2	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	2	no	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	—	4	1	no	—
175	C₂²×C₇	—	4	1	no	—
176	C₂²×C₇	—	4	1	no	—
177	C₂²×C₇	—	4	1	no	—
178	C₂²×C₇	C₂²⋊Q₈	1	4	yes	complemented
179	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
180	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
181	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
182	C₂²×C₇	C₂×#41	1	1	yes	characteristic
183	C₂²×C₇	C₂×#41	1	1	yes	characteristic
184	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
186	C₄×C₇	—	2	2	no	—
187	C₄×C₇	—	2	2	no	—
188	C₄×C₇	—	2	4	no	—
189	C₄×C₇	—	4	1	no	—
190	C₄×C₇	—	4	2	no	—
191	C₄×C₇	—	4	1	no	—
192	C₄×C₇	—	4	2	no	—
193	C₄×C₇	—	4	2	no	—
194	C₄×C₇	#121	1	4	yes	—
195	C₂⁴	C₂³×C₇	1	1	yes	characteristic
196	C₂⁴	C₂³×C₇	1	1	yes	characteristic
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	C₇×Q₈	1	4	yes	complemented
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₇×D₄	1	4	yes	—
212	C₂²×C₄	C₂³×C₇	1	2	yes	—
213	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
214	C₂²×C₄	C₂³×C₇	1	2	yes	—
215	C₂²×C₄	C₂³×C₇	1	2	yes	—
216	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
217	C₂²×C₄	C₂³×C₇	1	2	yes	—
218	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
219	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
220	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	C₇×D₄	1	4	yes	complemented
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₄²	—	2	2	no	—
235	C₂×C₇	—	2	4	no	—
236	C₂×C₇	—	4	2	no	—
237	C₂×C₇	C₂×#121	1	1	yes	characteristic
238	C₂×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
239	C₂×C₇	#537	1	1	yes	characteristic
240	C₂×C₇	#546	1	1	yes	characteristic
241	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
242	C₂×C₇	#552	1	1	yes	characteristic
243	C₂×C₇	#554	1	1	yes	characteristic
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	C₂×C₇×D₄	1	2	yes	—
255	C₂³	C₂×C₇×Q₈	1	2	yes	—
256	C₂³	C₂×C₇×Q₈	1	2	yes	—
257	C₂³	C₇×#41	1	2	yes	—
258	C₂³	C₂⁴×C₇	1	1	yes	characteristic
259	Q₈	—	4	4	no	—
260	D₄	—	2	8	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
286	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
287	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
288	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
289	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
290	C₂×C₄	C₇×#41	1	2	yes	—
291	C₇	#2417	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	C₇×C₂²⋊Q₈	1	4	yes	complemented
306	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
307	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
308	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
309	C₂²	C₂×C₇×#41	1	1	yes	characteristic
310	C₂²	C₂×C₇×#41	1	1	yes	characteristic
311	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
312	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	4	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	C₇×#121	1	4	yes	—
322	C₂	—	2	4	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂×C₇×#121	1	1	yes	characteristic
325	C₂	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
326	C₂	C₇×#537	1	1	yes	characteristic
327	C₂	C₇×#546	1	1	yes	characteristic
328	C₂	C₇×D₄×Q₈	1	1	yes	characteristic
329	C₂	C₇×#552	1	1	yes	characteristic
330	C₂	C₇×#554	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×C₂²⋊Q₈	C₂	semidirect
2. C₂×C₇×C₂²⋊Q₈	C₂	semidirect
4. C₂²×C₇×C₄⋊C₄	C₂	semidirect
5. C₇×#398	C₂	semidirect
6. C₇×#396	C₂	semidirect
7. C₇×#391	C₂	semidirect
8. C₇×#389	C₂	semidirect
10. C₇×#380	C₂	semidirect
11. C₇×#380	C₂	semidirect
39. #2417	C₇	direct (N × H)
47. C₂×C₇×D₄	Q₈	semidirect
73. C₇×C₄⋊C₄	D₄	semidirect
80. C₂×C₂²⋊Q₈	C₂×C₇	semidirect
81. C₂×C₂²⋊Q₈	C₂×C₇	semidirect
83. C₂²×C₄⋊C₄	C₂×C₇	semidirect
84. #398	C₂×C₇	semidirect
85. #396	C₂×C₇	semidirect
86. #391	C₂×C₇	semidirect
87. #389	C₂×C₇	semidirect
89. #380	C₂×C₇	semidirect
90. #380	C₂×C₇	semidirect
178. C₂²×C₇	C₂²⋊Q₈	semidirect
202. C₂×D₄	C₇×Q₈	semidirect
228. C₄⋊C₄	C₇×D₄	semidirect
291. C₇	#2417	direct (N × H)
305. C₂²	C₇×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₂²⋊Q₈	2
C₂×C₄×C₇×D₄	2
C₂²×C₇×C₄⋊C₄	2
C₇×#398	2
C₇×#396	2
C₇×#391	2
C₇×#389	2
C₇×#386	2
C₇×#380	2
#2417	7

Subgroups — C7×#2417 #11741726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2417 #11741726