Subgroups — #13508455

Order 1792.

534 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₈	C₂	1	1	yes	maximal, characteristic
3	#11729144	C₂	1	2	yes	maximal, complemented
4	#11729143	C₂	1	2	yes	maximal, complemented
5	#11729111	C₂	1	4	yes	maximal, complemented
6	#11728408	C₂	1	2	yes	maximal
7	#11728403	C₂	1	2	yes	maximal
8	#11727497	C₂	1	1	yes	maximal, characteristic
9	C₇×#533	—	2	1	no	—
10	C₂²×Dic₂₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₄	—	2	1	no	—
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	1	yes	characteristic
14	C₇×#499	—	2	1	no	—
15	C₇×#499	C₂²	1	2	yes	—
16	C₇×#498	—	2	1	no	—
17	C₇×#427	—	2	1	no	—
18	C₇×#427	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×#6392	C₂²	1	1	yes	characteristic
21	C₂×#6390	—	2	1	no	—
22	C₂×#6386	C₂²	1	2	yes	—
23	C₄×Dic₂₈	—	2	1	no	—
24	C₄×Dic₂₈	—	2	1	no	—
25	C₄×Dic₂₈	C₂²	1	2	yes	—
26	C₄×Dic₂₈	C₂²	1	2	yes	—
27	#89287	C₂²	1	4	yes	—
28	#89287	C₂²	1	4	yes	—
29	#89285	C₂²	1	4	yes	—
30	#89284	C₂²	1	4	yes	—
31	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
32	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
33	#33628	—	7	1	no	maximal, Sylow
34	C₂²×Dic₁₄	—	2	1	no	—
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₇×C₄⋊Q₈	—	2	1	no	—
38	C₇×C₄⋊Q₈	—	2	1	no	—
39	C₇×#125	—	2	2	no	—
40	C₇×C₂²⋊Q₈	—	4	1	no	—
41	C₇×C₂²⋊Q₈	—	4	1	no	—
42	C₄×C₇×Q₈	—	4	1	no	—
43	C₄×C₇×Q₈	—	4	1	no	—
44	C₇×#117	—	2	2	no	—
45	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
46	C₂×C₄⋊Dic₇	—	2	1	no	—
47	C₂×Dic₇⋊C₄	—	4	1	no	—
48	C₂×C₄×Dic₇	—	4	1	no	—
49	C₂×Dic₂₈	—	2	1	no	—
50	C₂×Dic₂₈	—	2	2	no	—
51	C₂×Dic₂₈	Q₈	1	4	yes	complemented
52	C₂×Dic₂₈	C₂³	1	2	yes	—
53	C₄×Dic₁₄	—	2	2	no	—
54	C₄×Dic₁₄	—	2	2	no	—
55	C₄×Dic₁₄	—	4	1	no	—
56	C₄×Dic₁₄	—	4	1	no	—
57	C₇×#107	—	2	1	no	—
58	C₇×#107	—	2	1	no	—
59	C₇×#107	C₂³	1	2	yes	—
60	C₇×#107	C₂³	1	2	yes	—
61	C₇×#106	—	2	2	no	—
62	C₇×#106	—	2	2	no	—
63	C₄×C₇×C₈	D₄	1	1	yes	characteristic
64	C₄×C₇×C₈	D₄	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	#6392	—	2	1	no	—
68	#6392	C₂³	1	2	yes	—
69	#6390	—	2	2	no	—
70	#6390	—	2	2	no	—
71	#6386	—	2	4	no	—
72	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
73	#3160	D₇	1	1	yes	characteristic
74	C₂×C₄×Q₁₆	—	7	1	no	—
75	#2229	—	7	4	no	—
76	#2227	—	7	2	no	—
77	#2225	—	7	2	no	—
78	#2159	—	7	2	no	—
79	#2150	—	7	1	no	—
80	#2141	—	7	2	no	—
81	C₂×C₇×Q₈	—	4	1	no	—
82	C₂×C₇×Q₈	—	4	1	no	—
83	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₇	—	4	1	no	—
87	C₂²×Dic₇	—	4	1	no	—
88	C₂×Dic₁₄	—	2	4	no	—
89	C₂×Dic₁₄	—	2	2	no	—
90	C₂×Dic₁₄	—	4	1	no	—
91	C₂×Dic₁₄	—	4	2	no	—
92	C₂×C₇×C₈	—	2	2	no	—
93	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
97	C₂×C₇×C₈	#41	1	2	yes	—
98	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
99	C₇×C₄⋊C₄	—	2	2	no	—
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	4	2	no	—
102	C₇×C₄⋊C₄	—	4	2	no	—
103	C₇×C₄⋊C₄	—	4	1	no	—
104	C₇×C₄⋊C₄	—	4	1	no	—
105	C₇×C₂²⋊C₄	—	4	2	no	—
106	C₄²×C₇	—	4	2	no	—
107	C₄²×C₇	D₈	1	1	yes	characteristic
108	C₄²×C₇	D₈	1	1	yes	characteristic
109	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
111	C₄⋊Dic₇	—	2	2	no	—
112	C₄⋊Dic₇	—	4	1	no	—
113	Dic₇⋊C₄	—	4	2	no	—
114	Dic₇⋊C₄	—	4	2	no	—
115	C₄×Dic₇	—	4	2	no	—
116	C₄×Dic₇	—	4	2	no	—
117	Dic₂₈	—	2	4	no	—
118	Dic₂₈	—	2	2	no	—
119	Dic₂₈	C₂×Q₈	1	4	yes	—
120	C₇⋊C₁₆	—	2	4	no	—
121	C₂²×Q₁₆	—	7	1	no	—
122	#533	—	2	1	no	—
123	C₂×C₄×Q₈	—	14	1	no	—
124	C₂×M₃₂	—	7	2	no	—
125	#500	C₂×D₇	1	1	yes	characteristic
126	#500	C₂×D₇	1	1	yes	characteristic
127	#499	—	2	1	no	—
128	#499	C₂×D₇	1	2	yes	—
129	#498	—	2	1	no	—
130	C₄×Q₁₆	—	7	2	no	—
131	C₄×Q₁₆	—	7	2	no	—
132	C₄×Q₁₆	—	14	1	no	—
133	C₄×Q₁₆	—	14	1	no	—
134	#427	—	2	1	no	—
135	#427	C₂×D₇	1	2	yes	—
136	C₂×#107	—	7	1	no	—
137	C₂×#103	—	14	1	no	—
138	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
139	#367	—	7	4	no	—
140	#366	—	7	4	no	—
141	C₄⋊C₁₆	—	7	2	no	—
142	C₄⋊C₁₆	—	7	2	no	—
143	#358	—	7	4	no	—
144	#358	—	7	4	no	—
145	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
146	C₇×Q₈	—	8	2	no	—
147	C₇×Q₈	—	8	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	4	1	no	—
150	C₂×C₄×C₇	—	4	2	no	—
151	C₂×C₄×C₇	—	4	2	no	—
152	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
153	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₇	#137	1	2	yes	—
159	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×Dic₇	—	4	2	no	—
161	C₂×Dic₇	—	4	4	no	—
162	C₂×Dic₇	—	4	2	no	—
163	C₂×Dic₇	—	8	2	no	—
164	Dic₁₄	—	2	4	no	—
165	Dic₁₄	—	4	2	no	—
166	Dic₁₄	—	4	4	no	—
167	C₇×C₈	—	2	1	no	—
168	C₇×C₈	—	2	1	no	—
169	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
170	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
171	C₇×C₈	C₂×#41	1	1	yes	characteristic
172	C₂²×Q₈	—	14	1	no	—
173	C₂×Q₁₆	—	7	4	no	—
174	C₂×Q₁₆	—	7	2	no	—
175	C₂×Q₁₆	—	14	2	no	—
176	C₂×Q₁₆	—	14	1	no	—
177	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	1	no	—
180	C₄⋊Q₈	—	2	1	no	—
181	#125	—	2	2	no	—
182	C₂²⋊Q₈	—	4	1	no	—
183	C₂²⋊Q₈	—	4	1	no	—
184	C₄×Q₈	—	4	1	no	—
185	C₄×Q₈	—	4	1	no	—
186	C₄×Q₈	—	14	2	no	—
187	C₄×Q₈	—	14	2	no	—
188	C₄×Q₈	—	28	1	no	—
189	C₄×Q₈	—	28	1	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	—	14	1	no	—
192	C₂×C₄⋊C₄	—	28	1	no	—
193	C₂×C₄²	—	28	1	no	—
194	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
195	M₃₂	—	14	4	no	—
196	C₂×C₁₆	—	7	4	no	—
197	#107	—	2	1	no	—
198	#107	—	2	1	no	—
199	#107	—	7	2	no	—
200	#107	—	14	1	no	—
201	#107	C₂²×D₇	1	2	yes	—
202	#107	C₂²×D₇	1	2	yes	—
203	#106	—	2	2	no	—
204	#106	—	2	2	no	—
205	#103	—	14	2	no	—
206	#103	—	14	2	no	—
207	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
208	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
211	C₂²×C₇	—	2	1	no	—
212	C₂²×C₇	—	2	1	no	—
213	C₂²×C₇	#474	1	2	yes	—
214	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
215	C₄×C₇	—	2	1	no	—
216	C₄×C₇	—	2	1	no	—
217	C₄×C₇	—	2	1	no	—
218	C₄×C₇	—	8	2	no	—
219	C₄×C₇	—	8	2	no	—
220	C₄×C₇	#474	1	2	yes	—
221	C₄×C₇	#510	1	2	yes	—
222	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
223	C₄×C₇	C₂×#137	1	1	yes	characteristic
224	Dic₇	—	4	4	no	—
225	Dic₇	—	8	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	14	4	no	—
229	C₂×Q₈	—	14	2	no	—
230	C₂×Q₈	—	28	1	no	—
231	C₂×Q₈	—	28	2	no	—
232	C₂²×C₄	—	28	1	no	—
233	C₂²×C₄	—	28	1	no	—
234	C₂²×C₄	#1141	1	1	yes	characteristic
235	C₂²×C₄	#1141	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
237	Q₁₆	—	7	4	no	—
238	Q₁₆	—	14	4	no	—
239	Q₁₆	—	14	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	Q₈×D₇	1	2	yes	—
242	C₂×C₈	#1161	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
246	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	14	2	no	—
254	C₄⋊C₄	—	28	2	no	—
255	C₄⋊C₄	—	28	2	no	—
256	C₄⋊C₄	—	28	1	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	28	2	no	—
260	C₄²	—	28	2	no	—
261	C₄²	#1141	1	1	yes	characteristic
262	C₄²	#1141	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
264	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
265	C₁₆	—	14	4	no	—
266	C₂×C₇	—	2	2	no	—
267	C₂×C₇	C₂×#474	1	1	yes	characteristic
268	C₂×C₇	C₂×#510	1	1	yes	characteristic
269	C₂×C₇	#3414	1	1	yes	characteristic
270	C₂³	C₂×#1141	1	1	yes	characteristic
271	Q₈	—	8	2	no	—
272	Q₈	—	8	2	no	—
273	Q₈	—	14	4	no	—
274	Q₈	—	28	4	no	—
275	Q₈	—	28	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	28	4	no	—
281	C₂×C₄	—	28	2	no	—
282	C₂×C₄	—	28	2	no	—
283	C₂×C₄	—	56	2	no	—
284	C₂×C₄	C₂×#1141	1	1	yes	characteristic
285	C₂×C₄	C₂×#1141	1	1	yes	characteristic
286	C₂×C₄	C₂×#1141	1	1	yes	characteristic
287	C₂×C₄	C₂×#1141	1	1	yes	characteristic
288	C₂×C₄	C₂×#1141	1	1	yes	characteristic
289	C₂×C₄	#6505	1	2	yes	—
290	C₂×C₄	#6509	1	2	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	#6509	1	2	yes	—
295	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
296	C₈	C₂×#1161	1	1	yes	characteristic
297	C₇	#33628	1	1	yes	characteristic, complemented, Sylow
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#89859	1	2	yes	—
301	C₂²	C₂²×#1141	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	28	4	no	—
308	C₄	—	56	2	no	—
309	C₄	#89859	1	2	yes	—
310	C₄	#89967	1	2	yes	—
311	C₄	C₂×#6505	1	1	yes	characteristic
312	C₄	C₂×#6509	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	C₂×#89859	1	1	yes	characteristic
315	C₂	#11741339	1	1	yes	characteristic
316	C₂	C₂×#89967	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#3160	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	#3160	D₇
hypocenter	C₇	#33628
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #11729144	C₂	semidirect
4. #11729143	C₂	semidirect
5. #11729111	C₂	semidirect
51. C₂×Dic₂₈	Q₈	semidirect
297. C₇	#33628	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3160	2
C₂×C₄×Dic₂₈	2
#11729144	2
#11729143	2
#11729111	2
#11728408	2
#11728403	2
#11727497	2
#33628	7