Subgroups — #13508547

Order 1792.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736025	C₂	1	1	yes	maximal, characteristic
2	#11735990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729204	C₂	1	1	yes	maximal, characteristic
5	C₂×#89313	C₂	1	1	yes	maximal, characteristic
6	#11729181	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#89308	C₂	1	1	yes	maximal, characteristic, complemented
8	#11727062	C₂	1	4	yes	maximal, complemented
9	#11727061	C₂	1	4	yes	maximal, complemented
10	#90436	—	2	1	no	—
11	C₂×#6467	C₂²	1	1	yes	characteristic
12	#90210	—	2	1	no	—
13	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
15	C₇×#405	C₂²	1	1	yes	characteristic
16	#89893	—	2	1	no	—
17	C₂×#6393	C₂²	1	1	yes	characteristic
18	#89878	C₂²	1	1	yes	characteristic
19	C₂×#6391	C₂²	1	1	yes	characteristic
20	#89777	—	2	1	no	—
21	#89489	—	2	2	no	—
22	#89474	—	2	2	no	—
23	#89472	C₂²	1	4	yes	—
24	#89455	C₂²	1	2	yes	complemented
25	#89452	C₂²	1	2	yes	complemented
26	C₇×#364	C₂²	1	4	yes	complemented
27	#89313	C₂²	1	4	yes	—
28	#89313	C₂²	1	2	yes	—
29	#89313	C₂²	1	2	yes	—
30	#89308	C₂²	1	4	yes	complemented
31	#89308	C₂²	1	2	yes	complemented
32	#89308	C₂²	1	2	yes	complemented
33	#33638	—	7	1	no	maximal, Sylow
34	C₂×#1157	—	2	1	no	—
35	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₇×#117	D₄	1	1	yes	characteristic
38	#6489	—	2	1	no	—
39	C₂×C₄⋊Dic₇	—	2	1	no	—
40	C₂×#1136	—	2	1	no	—
41	#6470	—	4	2	no	—
42	D₇×M₁₆	—	4	2	no	—
43	C₂×Dic₂₈	Q₈	1	1	yes	characteristic, complemented
44	#6467	—	2	1	no	—
45	#6467	—	2	1	no	—
46	#6467	Q₈	1	2	yes	complemented
47	#6467	C₂³	1	2	yes	—
48	C₂×D₅₆	Q₈	1	1	yes	characteristic, complemented
49	C₂×#1132	—	2	1	no	—
50	#6464	—	2	4	no	—
51	#6453	—	4	1	no	—
52	#6451	—	4	1	no	—
53	#6441	—	4	2	no	—
54	#6433	—	2	2	no	—
55	#6432	—	2	2	no	—
56	C₇×M₃₂	—	2	1	no	—
57	C₇×M₃₂	—	2	1	no	—
58	C₇×M₃₂	C₂³	1	2	yes	—
59	C₇×M₃₂	C₂³	1	2	yes	—
60	C₂×C₇×C₁₆	—	2	1	no	—
61	C₂×C₇×C₁₆	—	2	1	no	—
62	C₇×#97	D₄	1	2	yes	—
63	C₄×C₇×C₈	C₂³	1	2	yes	—
64	#6393	—	2	1	no	—
65	#6393	C₂³	1	2	yes	—
66	#6392	—	2	1	no	—
67	#6392	—	2	2	no	—
68	#6391	—	2	1	no	—
69	#6391	—	2	2	no	—
70	#6391	C₂³	1	2	yes	—
71	#6391	C₂³	1	1	yes	characteristic
72	#6391	C₂³	1	1	yes	characteristic
73	#6379	—	2	4	no	—
74	#3163	—	7	1	no	—
75	#2957	—	7	1	no	—
76	#2231	—	7	4	no	—
77	#2230	—	7	4	no	—
78	#2155	—	7	1	no	—
79	C₂×#365	—	7	1	no	—
80	#2153	D₇	1	1	yes	characteristic, complemented
81	#2145	—	7	1	no	—
82	C₂×#360	—	7	1	no	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₇	—	4	1	no	—
85	C₂²×Dic₇	—	4	1	no	—
86	#1157	—	2	2	no	—
87	#1157	—	2	2	no	—
88	#1157	—	4	1	no	—
89	#1157	—	4	1	no	—
90	C₂×D₂₈	—	2	1	no	—
91	C₂×C₄×D₇	—	4	1	no	—
92	C₂×Dic₁₄	—	2	1	no	—
93	C₂×Dic₁₄	—	4	2	no	—
94	C₇×M₁₆	SD₁₆	1	2	yes	—
95	C₇×M₁₆	SD₁₆	1	2	yes	—
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
98	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₇×C₈	#41	1	1	yes	characteristic
101	C₂×C₇×C₈	#41	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
103	C₇×C₄⋊C₄	—	2	1	no	—
104	C₇×C₂²⋊C₄	—	2	1	no	—
105	C₄²×C₇	C₂×D₄	1	2	yes	—
106	C₂²⋊Dic₇	—	4	1	no	—
107	C₄⋊Dic₇	—	2	1	no	—
108	C₄⋊Dic₇	—	2	2	no	—
109	C₄⋊Dic₇	—	2	1	no	—
110	C₄⋊Dic₇	—	4	2	no	—
111	C₄⋊Dic₇	—	4	1	no	—
112	Dic₇⋊C₄	—	4	2	no	—
113	Dic₇⋊C₄	—	4	2	no	—
114	C₄×Dic₇	—	4	1	no	—
115	#1136	—	2	2	no	—
116	#1136	—	4	1	no	—
117	C₂×C₇⋊C₈	—	4	1	no	—
118	Dic₂₈	—	2	1	no	—
119	Dic₂₈	C₂×Q₈	1	2	yes	—
120	D₅₆	—	2	1	no	—
121	D₅₆	C₂×Q₈	1	2	yes	—
122	#1132	—	2	2	no	—
123	#1132	—	2	2	no	—
124	#1131	—	4	4	no	—
125	C₈×D₇	—	4	4	no	—
126	C₇×C₁₆	—	2	2	no	—
127	C₇×C₁₆	—	2	2	no	—
128	C₂×#135	—	7	1	no	—
129	#568	—	14	1	no	—
130	#544	—	14	1	no	—
131	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
132	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
133	#501	—	14	2	no	—
134	#499	—	7	2	no	—
135	#498	—	7	2	no	—
136	#444	—	14	2	no	—
137	#443	—	7	4	no	—
138	C₂×#108	—	7	1	no	—
139	#428	—	14	1	no	—
140	#427	—	7	1	no	—
141	C₂×#106	—	7	1	no	—
142	#421	—	14	1	no	—
143	#405	C₂×D₇	1	1	yes	characteristic
144	#365	—	7	4	no	—
145	#365	—	7	2	no	—
146	#365	—	7	2	no	—
147	#364	C₂×D₇	1	4	yes	complemented
148	#360	—	7	4	no	—
149	#360	—	7	2	no	—
150	#360	—	7	2	no	—
151	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂²×D₇	—	4	1	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
158	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
160	C₂²⋊D₇	—	4	2	no	—
161	C₂²⋊D₇	—	4	2	no	—
162	C₂×Dic₇	—	4	1	no	—
163	C₂×Dic₇	—	4	1	no	—
164	C₂×Dic₇	—	4	1	no	—
165	C₂×Dic₇	—	4	2	no	—
166	C₂×Dic₇	—	16	1	no	—
167	D₂₈	—	2	2	no	—
168	D₂₈	—	4	1	no	—
169	C₄×D₇	—	4	2	no	—
170	C₄×D₇	—	4	2	no	—
171	Dic₁₄	—	2	2	no	—
172	Dic₁₄	—	4	1	no	—
173	Dic₁₄	—	16	2	no	—
174	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
175	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
176	C₇×C₈	C₂×SD₁₆	1	4	yes	—
177	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
178	C₇×C₈	C₂×#41	1	1	yes	characteristic
179	C₇⋊C₈	—	4	2	no	—
180	C₂×#41	—	14	1	no	—
181	#135	—	7	2	no	—
182	#135	—	7	2	no	—
183	#135	—	14	1	no	—
184	#135	—	14	1	no	—
185	C₂×Q₁₆	—	7	1	no	—
186	C₂×SD₁₆	—	14	1	no	—
187	C₂×D₈	—	7	1	no	—
188	#131	—	14	4	no	—
189	#131	—	28	2	no	—
190	C₂×M₁₆	—	14	1	no	—
191	C₂×M₁₆	—	28	2	no	—
192	C₂×M₁₆	D₂₈	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
194	C₄⋊Q₈	—	14	2	no	—
195	C₄⋊Q₈	—	28	1	no	—
196	#125	—	14	2	no	—
197	#125	—	28	1	no	—
198	C₂²⋊Q₈	—	28	2	no	—
199	#117	—	14	1	no	—
200	#117	D₂₈	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	14	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	2	1	no	—
204	M₃₂	C₂²×D₇	1	2	yes	—
205	M₃₂	C₂²×D₇	1	2	yes	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	#108	—	7	2	no	—
209	#108	—	14	1	no	—
210	#107	—	14	1	no	—
211	#107	—	14	2	no	—
212	#106	—	7	2	no	—
213	#106	—	7	1	no	—
214	#106	—	7	1	no	—
215	#106	—	14	1	no	—
216	#106	—	14	2	no	—
217	#104	—	14	4	no	—
218	#97	D₂₈	1	2	yes	—
219	C₄×C₈	C₂²×D₇	1	2	yes	—
220	C₂²×C₇	—	4	1	no	—
221	C₂²×C₇	#476	1	1	yes	characteristic
222	C₂²×C₇	#476	1	1	yes	characteristic
223	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂×D₇	—	4	2	no	—
225	C₂×D₇	—	16	1	no	—
226	C₄×C₇	—	4	2	no	—
227	C₄×C₇	#476	1	1	yes	characteristic
228	C₄×C₇	#476	1	1	yes	characteristic
229	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
230	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
231	Dic₇	—	4	2	no	—
232	Dic₇	—	8	1	no	—
233	Dic₇	—	8	2	no	—
234	Dic₇	—	8	1	no	—
235	#41	—	14	2	no	—
236	#41	—	14	2	no	—
237	#41	—	28	1	no	—
238	#41	—	28	1	no	—
239	C₂×Q₈	—	14	1	no	—
240	C₂×Q₈	—	28	2	no	—
241	C₂×D₄	—	14	1	no	—
242	C₂×D₄	—	28	1	no	—
243	C₂²×C₄	—	28	1	no	—
244	C₂²×C₄	—	28	1	no	—
245	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
246	Q₁₆	—	7	2	no	—
247	Q₁₆	—	14	1	no	—
248	SD₁₆	—	14	2	no	—
249	SD₁₆	—	14	2	no	—
250	D₈	—	7	2	no	—
251	D₈	—	14	1	no	—
252	M₁₆	—	14	2	no	—
253	M₁₆	—	28	1	no	—
254	M₁₆	—	28	4	no	—
255	M₁₆	#1132	1	2	yes	—
256	M₁₆	#1132	1	2	yes	—
257	C₂×C₈	—	28	1	no	—
258	C₂×C₈	—	28	4	no	—
259	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
260	C₂×C₈	C₂×D₂₈	1	2	yes	—
261	C₂×C₈	#1159	1	1	yes	characteristic
262	C₂×C₈	#1159	1	1	yes	characteristic
263	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
264	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	14	2	no	—
268	C₄⋊C₄	—	14	1	no	—
269	C₄⋊C₄	—	14	1	no	—
270	C₄⋊C₄	—	28	2	no	—
271	C₄⋊C₄	—	28	2	no	—
272	C₄⋊C₄	—	28	2	no	—
273	C₄⋊C₄	—	28	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	28	1	no	—
276	C₄²	—	28	1	no	—
277	C₄²	C₂×D₂₈	1	2	yes	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	2	no	—
280	C₂×C₇	—	2	1	no	—
281	C₂×C₇	—	2	1	no	—
282	C₂×C₇	—	2	1	no	—
283	C₂×C₇	C₂×#476	1	1	yes	characteristic
284	D₇	—	8	2	no	—
285	C₂³	—	28	1	no	—
286	C₂³	#6471	1	1	yes	characteristic
287	Q₈	—	14	2	no	—
288	Q₈	—	28	1	no	—
289	Q₈	—	112	2	no	—
290	D₄	—	14	2	no	—
291	D₄	—	28	2	no	—
292	D₄	—	28	2	no	—
293	D₄	—	28	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	28	1	no	—
296	C₂×C₄	—	28	2	no	—
297	C₂×C₄	—	28	2	no	—
298	C₂×C₄	—	28	1	no	—
299	C₂×C₄	—	28	1	no	—
300	C₂×C₄	—	28	2	no	—
301	C₂×C₄	—	112	1	no	—
302	C₂×C₄	#6460	1	1	yes	characteristic
303	C₂×C₄	#6460	1	1	yes	characteristic
304	C₂×C₄	C₂×#1132	1	1	yes	characteristic
305	C₂×C₄	C₂×#1132	1	1	yes	characteristic
306	C₂×C₄	#6471	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
308	C₈	—	28	2	no	—
309	C₈	#6460	1	1	yes	characteristic
310	C₈	#6460	1	1	yes	characteristic
311	C₈	C₂×#1132	1	4	yes	—
312	C₈	C₂×#1159	1	1	yes	characteristic
313	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
314	C₇	#33638	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	4	1	no	—
316	C₂²	—	28	2	no	—
317	C₂²	—	112	1	no	—
318	C₂²	#89614	1	1	yes	characteristic
319	C₂²	#89614	1	1	yes	characteristic
320	C₂²	C₂×#6471	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	28	2	no	—
323	C₄	—	56	2	no	—
324	C₄	—	56	1	no	—
325	C₄	—	56	1	no	—
326	C₄	#89614	1	1	yes	characteristic
327	C₄	#89614	1	1	yes	characteristic
328	C₄	C₂×#6460	1	1	yes	characteristic
329	C₄	C₂²×#1132	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	56	2	no	—
334	C₂	C₂×#89614	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#89614
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2153	C₂
socle	C₂×C₇	C₂×#476
prime radical	C₂×#6467	C₂²
hypercenter	#2153	D₇
hypocenter	C₇	#33638
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2153

Complements

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

N	H ≅ G/N	type
2. #11735990	C₂	semidirect
3. C₇×#2153	C₂	semidirect
6. #11729181	C₂	semidirect
7. C₂×#89308	C₂	semidirect
8. #11727062	C₂	semidirect
9. #11727061	C₂	semidirect
24. #89455	C₂²	semidirect
25. #89452	C₂²	semidirect
26. C₇×#364	C₂²	semidirect
30. #89308	C₂²	semidirect
31. #89308	C₂²	semidirect
32. #89308	C₂²	semidirect
43. C₂×Dic₂₈	Q₈	semidirect
46. #6467	Q₈	semidirect
48. C₂×D₅₆	Q₈	semidirect
80. #2153	D₇	semidirect
147. #364	C₂×D₇	semidirect
314. C₇	#33638	semidirect

Maximal subgroups

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

subgroup	index
#11736025	2
#11735990	2
C₇×#2153	2
#11729204	2
C₂×#89313	2
#11729181	2
C₂×#89308	2
#11727062	2
#11727061	2
#33638	7