Subgroups — #13511491

Order 1792.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3383	C₂	1	1	yes	maximal, characteristic, complemented
2	#11737357	C₂	1	1	yes	maximal, characteristic
3	#11729141	C₂	1	1	yes	maximal, characteristic
4	#11729136	C₂	1	1	yes	maximal, characteristic, complemented
5	#11729111	C₂	1	2	yes	maximal, complemented
6	#11729105	C₂	1	2	yes	maximal
7	Q₈×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#11728670	C₂	1	1	yes	maximal, characteristic, complemented
9	#11728669	C₂	1	2	yes	maximal
10	C₄×#6403	C₂	1	2	yes	maximal, complemented
11	C₄×#6402	C₂	1	1	yes	maximal, characteristic
12	C₇×#557	—	2	1	no	—
13	#90275	—	2	1	no	—
14	C₇×#500	C₂²	1	1	yes	characteristic
15	C₇×#499	C₂²	1	2	yes	—
16	C₇×#479	—	2	2	no	—
17	C₇×#475	—	2	1	no	—
18	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
19	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
20	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
21	C₂×#6403	C₂²	1	2	yes	—
22	C₂×#6402	C₂²	1	1	yes	characteristic
23	#89617	—	2	2	no	—
24	#89613	—	2	1	no	—
25	#89471	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₈	C₂²	1	2	yes	—
27	#89468	C₂²	1	2	yes	—
28	C₄×D₅₆	C₂²	1	1	yes	characteristic
29	#89359	C₂²	1	1	yes	characteristic
30	#89287	C₂²	1	2	yes	—
31	#89287	C₂²	1	2	yes	—
32	#89287	C₂²	1	2	yes	—
33	#89286	C₂²	1	1	yes	characteristic
34	#89286	C₂²	1	2	yes	—
35	#89285	C₂²	1	1	yes	characteristic
36	#89284	C₂²	1	2	yes	—
37	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
38	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
39	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
41	#33931	—	7	1	no	maximal, Sylow
42	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
43	C₇×C₄⋊Q₈	—	2	1	no	—
44	C₇×C₄⋊Q₈	—	4	1	no	—
45	C₇×#125	—	2	2	no	—
46	C₇×#125	—	4	2	no	—
47	C₄×C₇×Q₈	—	2	1	no	—
48	C₄×C₇×Q₈	—	2	2	no	—
49	C₄×C₇×Q₈	—	4	1	no	—
50	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
51	#6509	—	4	1	no	—
52	C₂×Dic₂₈	—	2	1	no	—
53	C₂×Dic₂₈	C₂³	1	2	yes	—
54	C₂×D₅₆	C₂³	1	1	yes	characteristic
55	C₂×#1132	—	2	2	no	—
56	#6461	—	4	2	no	—
57	#6455	—	4	1	no	—
58	#6438	—	2	2	no	—
59	C₄×D₂₈	—	2	1	no	—
60	#6432	—	2	1	no	—
61	C₄×Dic₁₄	—	2	2	no	—
62	C₇×#107	—	2	1	no	—
63	C₇×#107	C₂³	1	1	yes	characteristic
64	C₇×#107	C₂³	1	2	yes	—
65	C₇×#106	—	2	2	no	—
66	C₇×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₇×C₄⋊C₈	D₄	1	2	yes	—
68	C₇×#103	—	2	2	no	—
69	C₇×#103	—	2	1	no	—
70	C₄×C₇×C₈	C₂³	1	2	yes	—
71	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
72	#6403	—	2	4	no	—
73	#6402	—	2	2	no	—
74	#6395	—	2	1	no	—
75	#6392	C₂³	1	1	yes	characteristic
76	#6390	—	2	2	no	—
77	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
78	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
80	#3393	—	7	1	no	—
81	#3383	D₇	1	1	yes	characteristic, complemented
82	#2242	—	7	2	no	—
83	#2241	—	7	1	no	—
84	#2229	—	7	2	no	—
85	#2224	—	7	1	no	—
86	#2211	—	7	2	no	—
87	#2209	—	7	1	no	—
88	Q₈×C₁₆	—	7	1	no	—
89	C₄×Q₃₂	—	7	2	no	—
90	C₄×SD₃₂	—	7	1	no	—
91	C₂×C₇×Q₈	—	2	1	no	—
92	C₂×C₇×Q₈	—	4	1	no	—
93	C₂×C₇×Q₈	D₈	1	1	yes	characteristic
94	C₂×D₂₈	—	2	1	no	—
95	C₂×C₄×D₇	—	4	1	no	—
96	C₂×Dic₁₄	—	2	2	no	—
97	C₇×Q₁₆	—	2	1	no	—
98	C₇×Q₁₆	#41	1	2	yes	—
99	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
102	C₇×C₄⋊C₄	—	2	1	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	4	1	no	—
105	C₇×C₄⋊C₄	—	4	2	no	—
106	C₇×C₄⋊C₄	—	4	2	no	—
107	C₇×C₄⋊C₄	—	4	2	no	—
108	C₇×C₄⋊C₄	—	4	1	no	—
109	C₇×C₄⋊C₄	D₈	1	1	yes	characteristic
110	C₇×C₄⋊C₄	D₈	1	2	yes	—
111	C₄²×C₇	—	4	2	no	—
112	C₄²×C₇	—	4	1	no	—
113	C₄²×C₇	C₂×D₄	1	2	yes	—
114	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
115	(C₂×D₇)⋊C₄	—	4	1	no	—
116	(C₂×D₇)⋊C₄	—	4	2	no	—
117	(C₂×D₇)⋊C₄	—	4	2	no	—
118	C₄⋊Dic₇	—	2	1	no	—
119	C₄⋊Dic₇	—	4	1	no	—
120	C₄⋊Dic₇	—	4	1	no	—
121	Dic₇⋊C₄	—	4	2	no	—
122	C₄×Dic₇	—	4	2	no	—
123	Dic₂₈	—	2	2	no	—
124	Dic₂₈	—	2	2	no	—
125	Dic₂₈	—	8	1	no	—
126	D₅₆	—	2	1	no	—
127	D₅₆	—	2	1	no	—
128	#1132	—	8	2	no	—
129	C₇⋊C₁₆	—	2	1	no	—
130	C₇⋊C₁₆	—	2	2	no	—
131	C₇⋊C₁₆	—	2	1	no	—
132	#557	—	2	1	no	—
133	#556	—	14	1	no	—
134	C₂×Q₃₂	—	7	2	no	—
135	C₂×SD₃₂	—	7	1	no	—
136	#500	C₂×D₇	1	1	yes	characteristic
137	#499	C₂×D₇	1	2	yes	—
138	#495	—	7	2	no	—
139	#494	—	7	1	no	—
140	#479	—	2	2	no	—
141	#475	—	2	1	no	—
142	#464	—	14	2	no	—
143	#461	—	14	1	no	—
144	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
145	C₄×Q₁₆	—	7	2	no	—
146	C₄×Q₁₆	C₂×D₇	1	1	yes	characteristic
147	C₄×Q₁₆	C₂×D₇	1	2	yes	—
148	C₄×D₈	—	7	1	no	—
149	#367	—	7	1	no	—
150	#366	—	7	2	no	—
151	C₄⋊C₁₆	—	7	1	no	—
152	C₄⋊C₁₆	—	7	2	no	—
153	#358	—	7	1	no	—
154	#358	—	7	2	no	—
155	#358	—	7	2	no	—
156	#358	—	7	2	no	—
157	#357	—	7	1	no	—
158	#357	—	7	2	no	—
159	C₄×C₁₆	—	7	1	no	—
160	C₄×C₁₆	—	7	2	no	—
161	C₂²×D₇	—	4	1	no	—
162	C₇×Q₈	—	2	2	no	—
163	C₇×Q₈	—	2	2	no	—
164	C₇×Q₈	—	4	1	no	—
165	C₇×Q₈	—	8	2	no	—
166	C₂×C₄×C₇	—	4	1	no	—
167	C₂×C₄×C₇	—	4	1	no	—
168	C₂×C₄×C₇	—	4	2	no	—
169	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
170	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
171	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×Dic₇	—	4	1	no	—
175	C₂×Dic₇	—	4	2	no	—
176	D₂₈	—	4	1	no	—
177	D₂₈	—	4	1	no	—
178	C₄×D₇	—	8	2	no	—
179	Dic₁₄	—	4	2	no	—
180	Dic₁₄	—	4	2	no	—
181	C₇×C₈	—	2	2	no	—
182	C₇×C₈	—	2	1	no	—
183	C₇×C₈	C₂×#41	1	1	yes	characteristic
184	C₇×C₈	2₋¹⁺⁴	1	1	yes	characteristic
185	C₂×Q₁₆	—	7	2	no	—
186	C₂×Q₁₆	—	14	1	no	—
187	C₂×Q₁₆	C₂²×D₇	1	1	yes	characteristic
188	C₂×SD₁₆	—	14	2	no	—
189	C₂×D₈	—	7	1	no	—
190	C₄⋊Q₈	—	2	1	no	—
191	C₄⋊Q₈	—	4	1	no	—
192	C₄⋊Q₈	—	14	1	no	—
193	#126	—	28	2	no	—
194	#125	—	2	2	no	—
195	#125	—	4	2	no	—
196	#124	—	14	2	no	—
197	C₂²⋊Q₈	—	28	1	no	—
198	C₄×Q₈	—	2	1	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	4	1	no	—
201	C₄×Q₈	—	14	2	no	—
202	C₄×Q₈	C₂²⋊D₇	1	1	yes	characteristic
203	C₄×D₄	—	14	1	no	—
204	#117	—	28	1	no	—
205	Q₃₂	—	14	4	no	—
206	SD₃₂	—	14	2	no	—
207	C₂×C₁₆	—	7	1	no	—
208	C₂×C₁₆	—	7	2	no	—
209	C₂×C₁₆	—	7	1	no	—
210	#107	—	2	1	no	—
211	#107	—	7	1	no	—
212	#107	C₂²×D₇	1	2	yes	—
213	#107	C₂²×D₇	1	1	yes	characteristic
214	#106	—	2	2	no	—
215	C₄⋊C₈	C₂²⋊D₇	1	1	yes	characteristic
216	C₄⋊C₈	C₂²⋊D₇	1	2	yes	—
217	#103	—	2	2	no	—
218	#103	—	2	1	no	—
219	#103	—	14	2	no	—
220	#102	—	14	1	no	—
221	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₇	1	2	yes	—
223	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
224	C₂×D₇	—	8	1	no	—
225	C₂×D₇	—	8	1	no	—
226	C₂×D₇	—	8	1	no	—
227	C₄×C₇	—	2	2	no	—
228	C₄×C₇	—	2	2	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	4	2	no	—
231	C₄×C₇	—	8	2	no	—
232	C₄×C₇	—	8	1	no	—
233	C₄×C₇	#508	1	2	yes	—
234	C₄×C₇	#548	1	1	yes	characteristic
235	C₄×C₇	#577	1	1	yes	characteristic
236	Dic₇	—	8	2	no	—
237	Dic₇	—	8	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	14	2	no	—
241	C₂×Q₈	#1141	1	1	yes	characteristic
242	C₂×D₄	—	14	1	no	—
243	C₂²×C₄	—	28	1	no	—
244	Q₁₆	—	2	1	no	—
245	Q₁₆	—	14	2	no	—
246	Q₁₆	—	14	2	no	—
247	Q₁₆	—	56	1	no	—
248	Q₁₆	#1161	1	2	yes	—
249	SD₁₆	—	56	2	no	—
250	D₈	—	14	1	no	—
251	D₈	—	14	1	no	—
252	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
253	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
254	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	14	1	no	—
263	C₄⋊C₄	—	28	2	no	—
264	C₄⋊C₄	—	28	1	no	—
265	C₄⋊C₄	—	28	1	no	—
266	C₄⋊C₄	#1141	1	1	yes	characteristic
267	C₄⋊C₄	#1141	1	2	yes	—
268	C₂²⋊C₄	—	28	2	no	—
269	C₂²⋊C₄	—	28	1	no	—
270	C₂²⋊C₄	—	28	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	28	2	no	—
274	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
276	C₁₆	—	14	1	no	—
277	C₁₆	—	14	2	no	—
278	C₁₆	—	14	1	no	—
279	C₂×C₇	#3358	1	1	yes	characteristic
280	C₂×C₇	C₂×#508	1	1	yes	characteristic
281	C₂×C₇	#3417	1	1	yes	characteristic
282	D₇	—	16	1	no	—
283	C₂³	—	28	1	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	4	1	no	—
287	Q₈	—	8	2	no	—
288	Q₈	—	28	2	no	—
289	Q₈	—	28	2	no	—
290	D₄	—	28	1	no	—
291	D₄	—	28	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	28	1	no	—
296	C₂×C₄	—	28	2	no	—
297	C₂×C₄	—	56	2	no	—
298	C₂×C₄	C₂×#1141	1	2	yes	—
299	C₂×C₄	C₂×#1141	1	1	yes	characteristic
300	C₂×C₄	C₂×#1141	1	2	yes	—
301	C₂×C₄	C₂×#1141	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	2	1	no	—
305	C₈	C₂×#1161	1	1	yes	characteristic
306	C₈	#6551	1	1	yes	characteristic
307	C₇	#33931	1	1	yes	characteristic, complemented, Sylow
308	C₂²	—	56	1	no	—
309	C₂²	—	56	1	no	—
310	C₂²	—	56	1	no	—
311	C₂²	C₂²×#1141	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	56	2	no	—
319	C₄	—	56	1	no	—
320	C₄	#89966	1	2	yes	—
321	C₄	#90508	1	1	yes	characteristic
322	C₄	#90528	1	1	yes	characteristic
323	C₂	—	112	1	no	—
324	C₂	#11741007	1	1	yes	characteristic
325	C₂	C₂×#89966	1	1	yes	characteristic
326	C₂	#11741462	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#3383	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂×D₅₆	C₂³
hypercenter	#3383	D₇
hypocenter	C₇	#33931
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
1. C₇×#3383	C₂	semidirect
4. #11729136	C₂	semidirect
5. #11729111	C₂	semidirect
7. Q₈×C₇⋊C₁₆	C₂	semidirect
8. #11728670	C₂	semidirect
10. C₄×#6403	C₂	semidirect
81. #3383	D₇	semidirect
307. C₇	#33931	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3383	2
#11737357	2
#11729141	2
#11729136	2
#11729111	2
#11729105	2
Q₈×C₇⋊C₁₆	2
#11728670	2
#11728669	2
C₄×#6403	2
C₄×#6402	2
#33931	7