Subgroups — #13337003

Order 1792.

774 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
2	#11740800	C₂	1	2	yes	maximal, complemented
3	C₂×#89467	C₂	1	1	yes	maximal, characteristic
4	#11728713	C₂	1	4	yes	maximal, complemented
5	#11728693	C₂	1	4	yes	maximal, complemented
6	#11728256	C₂	1	2	yes	maximal, complemented
7	#11727483	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#533	C₂²	1	1	yes	characteristic
9	C₂²×D₅₆	—	2	1	no	—
10	C₂²×D₅₆	—	2	1	no	—
11	C₂×#6437	C₂²	1	2	yes	—
12	C₇×#446	—	2	1	no	—
13	C₇×#446	C₂²	1	2	yes	complemented
14	C₇×C₈×Q₈	—	2	1	no	—
15	C₇×C₈×Q₈	C₂²	1	2	yes	complemented
16	C₇×#424	—	2	1	no	—
17	C₇×#423	C₂²	1	1	yes	characteristic
18	C₇×#405	—	2	1	no	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	#89860	—	2	2	no	—
21	#89857	—	2	2	no	—
22	#89818	—	2	4	no	—
23	#89799	C₂²	1	8	yes	complemented
24	#89697	C₂²	1	2	yes	—
25	#89467	—	2	1	no	—
26	#89467	—	2	1	no	—
27	#89467	C₂²	1	2	yes	—
28	#89467	C₂²	1	2	yes	—
29	#89280	C₂²	1	8	yes	complemented
30	#89253	C₂²	1	2	yes	complemented
31	#89252	C₂²	1	2	yes	complemented
32	#19203	—	7	1	no	maximal, Sylow
33	C₂²×D₂₈	—	2	2	no	—
34	C₂²×D₂₈	—	2	2	no	—
35	C₂²×D₂₈	—	2	2	no	—
36	C₂×C₇×M₁₆	—	2	1	no	—
37	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
39	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₇×#125	—	2	1	no	—
42	C₇×C₂²⋊Q₈	—	2	1	no	—
43	C₇×C₂²⋊Q₈	D₄	1	2	yes	—
44	C₄×C₇×Q₈	—	2	1	no	—
45	C₄×C₇×Q₈	C₂³	1	2	yes	—
46	C₇×#117	—	2	1	no	—
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₂×#1143	—	2	8	no	—
49	C₂×D₅₆	—	2	2	no	—
50	C₂×D₅₆	—	2	2	no	—
51	C₂×D₅₆	—	2	2	no	—
52	C₂×D₅₆	—	2	2	no	—
53	C₂×D₅₆	—	4	4	no	—
54	C₂×D₅₆	—	4	4	no	—
55	#6437	—	2	2	no	—
56	#6437	—	2	2	no	—
57	#6437	—	2	2	no	—
58	#6437	C₂³	1	4	yes	—
59	C₇×C₄⋊C₈	—	2	2	no	—
60	C₇×C₄⋊C₈	—	2	2	no	—
61	C₇×C₄⋊C₈	—	2	1	no	—
62	C₇×C₄⋊C₈	C₂³	1	2	yes	—
63	C₇×#97	—	2	2	no	—
64	C₄×C₇×C₈	—	2	2	no	—
65	C₄×C₇×C₈	D₄	1	1	yes	characteristic
66	C₄×C₇×C₈	D₄	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
70	#6383	—	2	8	no	—
71	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
72	C₄×C₇⋊C₈	—	2	2	no	—
73	#3208	—	7	2	no	—
74	C₂×#493	—	7	1	no	—
75	#2960	D₇	1	1	yes	characteristic, complemented
76	#1688	—	7	4	no	—
77	#1668	—	7	4	no	—
78	#1564	—	7	1	no	—
79	#1545	—	7	2	no	—
80	C₂³×D₇	—	4	2	no	—
81	C₂×C₇×Q₈	—	2	1	no	—
82	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	#41	1	1	yes	characteristic
86	C₂×D₂₈	—	2	4	no	—
87	C₂×D₂₈	—	2	4	no	—
88	C₂×D₂₈	—	2	4	no	—
89	C₂×D₂₈	—	4	4	no	—
90	C₂×D₂₈	—	4	4	no	—
91	C₂×D₂₈	—	4	2	no	—
92	C₂×D₂₈	—	4	2	no	—
93	C₂×D₂₈	—	4	2	no	—
94	C₂×D₂₈	—	8	4	no	—
95	C₂×D₂₈	—	8	4	no	—
96	C₇×M₁₆	—	4	2	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
100	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
104	C₇×C₄⋊C₄	—	2	1	no	—
105	C₇×C₄⋊C₄	—	2	1	no	—
106	C₇×C₄⋊C₄	—	2	1	no	—
107	C₇×C₄⋊C₄	—	2	2	no	—
108	C₇×C₄⋊C₄	—	2	2	no	—
109	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
110	C₇×C₂²⋊C₄	—	2	2	no	—
111	C₄²×C₇	—	2	2	no	—
112	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇	#41	1	1	yes	characteristic
115	C₄²×C₇	C₂⁴	1	1	yes	characteristic
116	#1143	—	8	8	no	—
117	C₂×C₇⋊C₈	—	2	4	no	—
118	D₅₆	—	4	4	no	—
119	D₅₆	—	4	4	no	—
120	D₅₆	—	4	4	no	—
121	D₅₆	—	4	4	no	—
122	C₂²×D₈	—	14	1	no	—
123	C₂²×D₈	—	14	1	no	—
124	#533	C₂×D₇	1	1	yes	characteristic
125	C₂×#127	—	7	2	no	—
126	#493	—	7	2	no	—
127	#493	—	7	2	no	—
128	#493	—	14	1	no	—
129	#493	—	14	1	no	—
130	#492	—	14	2	no	—
131	#486	—	14	2	no	—
132	#460	—	7	8	no	—
133	#450	—	14	4	no	—
134	#446	—	2	1	no	—
135	#446	C₂×D₇	1	2	yes	complemented
136	C₈×Q₈	—	2	1	no	—
137	C₈×Q₈	C₂×D₇	1	2	yes	complemented
138	#431	—	7	2	no	—
139	#424	—	2	1	no	—
140	#423	C₂×D₇	1	1	yes	characteristic
141	#405	—	2	1	no	—
142	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
143	#335	—	7	2	no	—
144	#334	—	7	2	no	—
145	#331	—	7	8	no	—
146	C₂³×C₇	#121	1	1	yes	characteristic
147	C₂²×D₇	—	4	4	no	—
148	C₂²×D₇	—	8	4	no	—
149	C₂²×D₇	—	8	4	no	—
150	C₂²×D₇	—	8	4	no	—
151	C₇×Q₈	—	2	4	no	—
152	C₇×Q₈	—	4	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	#121	1	1	yes	characteristic
158	C₂×C₄×C₇	#121	1	1	yes	characteristic
159	C₂×C₄×C₇	#121	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
161	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
165	D₂₈	—	4	4	no	—
166	D₂₈	—	4	4	no	—
167	D₂₈	—	4	8	no	—
168	D₂₈	—	8	4	no	—
169	D₂₈	—	8	4	no	—
170	D₂₈	—	8	4	no	—
171	D₂₈	—	8	4	no	—
172	C₇×C₈	—	2	1	no	—
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	—	4	2	no	—
175	C₇×C₈	C₂×SD₁₆	1	2	yes	—
176	C₇×C₈	C₂×SD₁₆	1	2	yes	—
177	C₇⋊C₈	—	4	4	no	—
178	C₂²×D₄	—	14	2	no	—
179	C₂²×D₄	—	14	2	no	—
180	C₂²×D₄	—	14	2	no	—
181	C₂×SD₁₆	—	14	8	no	—
182	C₂×D₈	—	14	2	no	—
183	C₂×D₈	—	14	2	no	—
184	C₂×D₈	—	14	2	no	—
185	C₂×D₈	—	14	2	no	—
186	C₂×D₈	—	28	4	no	—
187	C₂×D₈	—	28	4	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
191	C₄⋊Q₈	D₂₈	1	1	yes	characteristic
192	C₄⋊Q₈	D₂₈	1	1	yes	characteristic
193	#127	—	7	4	no	—
194	#127	—	14	2	no	—
195	#127	—	14	2	no	—
196	#127	—	14	2	no	—
197	#125	—	2	1	no	—
198	C₂²⋊Q₈	—	2	1	no	—
199	C₂²⋊Q₈	D₂₈	1	2	yes	—
200	C₄×Q₈	—	2	1	no	—
201	C₄×Q₈	C₂²×D₇	1	2	yes	—
202	#117	—	2	1	no	—
203	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	7	4	no	—
208	C₄⋊C₈	C₂²×D₇	1	2	yes	—
209	#102	—	14	8	no	—
210	C₂²⋊C₈	—	14	2	no	—
211	#97	—	2	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	14	2	no	—
214	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
215	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	#460	1	2	yes	—
221	C₂²×C₇	C₂×#121	1	1	yes	characteristic
222	C₂×D₇	—	8	4	no	—
223	C₂×D₇	—	8	4	no	—
224	C₂×D₇	—	8	4	no	—
225	C₂×D₇	—	16	8	no	—
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	4	2	no	—
229	C₄×C₇	—	4	2	no	—
230	C₄×C₇	#460	1	2	yes	—
231	C₄×C₇	#472	1	2	yes	—
232	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
233	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₄×C₇	#576	1	2	yes	—
235	C₂⁴	—	28	2	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	C₂×D₂₈	1	2	yes	—
238	C₂×D₄	—	14	4	no	—
239	C₂×D₄	—	14	4	no	—
240	C₂×D₄	—	14	4	no	—
241	C₂×D₄	—	28	2	no	—
242	C₂×D₄	—	28	2	no	—
243	C₂×D₄	—	28	2	no	—
244	C₂×D₄	—	28	4	no	—
245	C₂×D₄	—	28	4	no	—
246	C₂×D₄	—	56	4	no	—
247	C₂×D₄	—	56	4	no	—
248	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
249	C₂²×C₄	#1157	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
251	SD₁₆	—	56	8	no	—
252	D₈	—	28	4	no	—
253	D₈	—	28	4	no	—
254	D₈	—	28	4	no	—
255	D₈	—	28	4	no	—
256	M₁₆	—	4	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	14	4	no	—
260	C₂×C₈	#1143	1	4	yes	complemented
261	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
264	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	C₂×D₂₈	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	C₂×D₂₈	1	1	yes	characteristic
274	C₄²	#1157	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
276	C₄²	C₂³×D₇	1	1	yes	characteristic
277	C₂×C₇	—	2	2	no	—
278	C₂×C₇	C₂×#460	1	1	yes	characteristic
279	C₂×C₇	#3058	1	1	yes	characteristic
280	C₂×C₇	C₂×#472	1	1	yes	characteristic
281	D₇	—	16	4	no	—
282	C₂³	—	28	4	no	—
283	C₂³	—	56	4	no	—
284	C₂³	—	56	4	no	—
285	C₂³	—	56	4	no	—
286	C₂³	#6493	1	1	yes	characteristic
287	Q₈	—	2	4	no	—
288	Q₈	—	4	2	no	—
289	D₄	—	28	8	no	—
290	D₄	—	28	4	no	—
291	D₄	—	28	4	no	—
292	D₄	—	56	4	no	—
293	D₄	—	56	4	no	—
294	D₄	—	56	4	no	—
295	D₄	—	56	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	#6493	1	1	yes	characteristic
301	C₂×C₄	#6493	1	1	yes	characteristic
302	C₂×C₄	#6493	1	1	yes	characteristic
303	C₂×C₄	C₂×#1143	1	2	yes	—
304	C₂×C₄	#6512	1	2	yes	—
305	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
306	C₂×C₄	C₂×#1157	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	—	28	4	no	—
312	C₈	C₂×#1143	1	2	yes	—
313	C₈	C₂×#1143	1	2	yes	—
314	C₇	#19203	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	56	4	no	—
318	C₂²	—	56	4	no	—
319	C₂²	—	56	4	no	—
320	C₂²	—	112	8	no	—
321	C₂²	#89799	1	2	yes	—
322	C₂²	C₂×#6493	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	#89799	1	2	yes	—
328	C₄	#89914	1	2	yes	—
329	C₄	#90443	1	2	yes	—
330	C₄	C₂²×#1143	1	1	yes	characteristic
331	C₄	C₂×#6512	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	112	4	no	—
334	C₂	C₂×#89799	1	1	yes	characteristic
335	C₂	#11741244	1	1	yes	characteristic
336	C₂	C₂×#89914	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6493
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#2960	C₂
socle	C₂²×C₇	C₂×#121
prime radical	C₂×#89467	C₂
hypercenter	#2960	D₇
hypocenter	C₇	#19203
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
1. C₇×#2960	C₂	semidirect
2. #11740800	C₂	semidirect
4. #11728713	C₂	semidirect
5. #11728693	C₂	semidirect
6. #11728256	C₂	semidirect
7. #11727483	C₂	semidirect
13. C₇×#446	C₂²	semidirect
15. C₇×C₈×Q₈	C₂²	semidirect
23. #89799	C₂²	semidirect
29. #89280	C₂²	semidirect
30. #89253	C₂²	semidirect
31. #89252	C₂²	semidirect
75. #2960	D₇	semidirect
99. C₂×C₇×C₈	SD₁₆	semidirect
135. #446	C₂×D₇	semidirect
137. C₈×Q₈	C₂×D₇	semidirect
260. C₂×C₈	#1143	semidirect
314. C₇	#19203	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2960	2
#11740800	2
C₂×#89467	2
#11728713	2
#11728693	2
#11728256	2
#11727483	2
#19203	7