Subgroups — #13323544

Order 1792.

682 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11736722	C₂	1	2	yes	maximal, complemented
2	C₄×#6467	C₂	1	1	yes	maximal, characteristic
3	C₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	#11728508	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726318	C₂	1	2	yes	maximal, complemented
7	#11726317	C₂	1	2	yes	maximal, complemented
8	#11726277	C₂	1	4	yes	maximal, complemented
9	C₂×#6467	C₄	1	1	yes	characteristic
10	C₄×#1157	C₂²	1	2	yes	—
11	C₇×#431	C₂²	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	#89885	C₄	1	2	yes	—
14	#89878	C₄	1	1	yes	characteristic
15	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
17	#89610	C₂²	1	4	yes	—
18	#89603	C₂²	1	4	yes	—
19	#89502	—	2	4	no	—
20	C₄×Dic₂₈	C₄	1	1	yes	characteristic
21	C₄×Dic₂₈	C₂²	1	1	yes	characteristic
22	C₄×D₅₆	C₄	1	1	yes	characteristic
23	C₄×D₅₆	C₂²	1	1	yes	characteristic
24	C₄×#1132	C₄	1	2	yes	—
25	C₄×#1132	C₂²	1	2	yes	—
26	C₄×#1131	—	2	2	no	—
27	#89457	—	2	2	no	—
28	C₇×#335	C₂²	1	2	yes	—
29	C₇×#334	C₂²	1	2	yes	—
30	#89279	C₂²	1	4	yes	—
31	#89278	C₂²	1	4	yes	—
32	C₈×C₇⋊C₈	C₂²	1	2	yes	—
33	C₈×C₇⋊C₈	C₂²	1	2	yes	—
34	#17886	—	7	1	no	maximal, Sylow
35	C₂×#1157	C₂×C₄	1	2	yes	—
36	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
39	C₄×C₂²⋊D₇	—	2	2	no	—
40	C₄×C₂²⋊D₇	—	2	2	no	—
41	#6489	C₂×C₄	1	2	yes	—
42	C₂²×C₇⋊C₈	—	2	1	no	—
43	C₂²×C₇⋊C₈	D₄	1	2	yes	—
44	C₂×Dic₂₈	C₂×C₄	1	1	yes	characteristic
45	#6467	—	2	4	no	—
46	C₂×D₅₆	C₂×C₄	1	1	yes	characteristic
47	C₂×#1132	C₂×C₄	1	2	yes	—
48	C₂×#1131	—	2	4	no	—
49	C₄×D₂₈	C₂×C₄	1	2	yes	—
50	C₄×D₂₈	C₂³	1	2	yes	—
51	#6435	—	2	2	no	—
52	C₄²×D₇	—	2	2	no	—
53	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₄	C₂³	1	2	yes	—
55	C₇×C₄⋊C₈	C₂³	1	4	yes	—
56	C₇×C₂²⋊C₈	—	2	2	no	—
57	C₄×C₇×C₈	—	2	2	no	—
58	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	#6395	C₂×C₄	1	2	yes	—
63	#6395	C₂×C₄	1	2	yes	—
64	(C₂×D₇)⋊C₈	—	2	4	no	—
65	#6392	C₂×C₄	1	1	yes	characteristic
66	#6392	C₂×C₄	1	1	yes	characteristic
67	#6391	C₂×C₄	1	2	yes	—
68	#6390	C₂×C₄	1	2	yes	—
69	#6390	C₂×C₄	1	2	yes	—
70	#6389	—	2	4	no	—
71	Dic₇⋊C₈	—	2	4	no	—
72	C₄×C₇⋊C₈	—	2	1	no	—
73	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
74	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
75	C₄×C₇⋊C₈	C₂³	1	2	yes	—
76	C₄×C₇⋊C₈	C₂³	1	2	yes	—
77	#2965	—	7	2	no	—
78	C₄×#135	—	7	1	no	—
79	#1591	—	7	2	no	—
80	#1590	—	7	4	no	—
81	#1589	—	7	2	no	—
82	#1564	D₇	1	1	yes	characteristic, complemented
83	#1476	—	7	2	no	—
84	C₂×C₈²	—	7	1	no	—
85	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	#41	1	1	yes	characteristic
88	C₂×C₂²⋊D₇	—	2	2	no	—
89	#1157	—	2	4	no	—
90	#1157	—	2	4	no	—
91	C₂×D₂₈	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₇	—	2	2	no	—
93	C₂×C₄×D₇	—	2	2	no	—
94	C₂×C₄×D₇	—	2	2	no	—
95	C₂×Dic₁₄	C₂²×C₄	1	2	yes	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	4	no	—
98	C₂×C₇×C₈	M₁₆	1	4	yes	complemented
99	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	#41	1	1	yes	characteristic
106	C₄²×C₇	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₇	—	2	2	no	—
108	(C₂×D₇)⋊C₄	—	2	2	no	—
109	(C₂×D₇)⋊C₄	—	2	2	no	—
110	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
111	C₄⋊Dic₇	C₂²×C₄	1	2	yes	—
112	Dic₇⋊C₄	—	2	2	no	—
113	Dic₇⋊C₄	—	2	2	no	—
114	C₄×Dic₇	—	2	2	no	—
115	C₄×Dic₇	—	2	2	no	—
116	C₄×Dic₇	—	2	2	no	—
117	C₂×C₇⋊C₈	—	2	4	no	—
118	C₂×C₇⋊C₈	—	2	2	no	—
119	C₂×C₇⋊C₈	—	2	4	no	—
120	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
121	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
122	Dic₂₈	—	2	2	no	—
123	D₅₆	—	2	2	no	—
124	#1132	—	2	4	no	—
125	#1131	—	4	8	no	—
126	C₂×#135	—	7	1	no	—
127	C₄×#41	—	7	2	no	—
128	#446	—	7	4	no	—
129	C₄×Q₁₆	—	7	1	no	—
130	C₄×Q₁₆	—	7	1	no	—
131	C₄×SD₁₆	—	7	2	no	—
132	C₄×SD₁₆	—	7	2	no	—
133	C₄×D₈	—	7	1	no	—
134	C₄×D₈	—	7	1	no	—
135	#436	—	7	4	no	—
136	#435	—	14	4	no	—
137	#431	—	14	2	no	—
138	#431	C₂×D₇	1	2	yes	—
139	#427	—	7	1	no	—
140	#416	—	7	2	no	—
141	C₄×M₁₆	—	14	2	no	—
142	C₂×C₄×C₈	—	7	1	no	—
143	C₂×C₄×C₈	—	7	1	no	—
144	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
145	#335	C₂×D₇	1	2	yes	—
146	#334	C₂×D₇	1	2	yes	—
147	Q₈⋊C₈	—	7	4	no	—
148	D₄⋊C₈	—	7	4	no	—
149	C₈²	—	7	2	no	—
150	C₈²	—	7	2	no	—
151	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
152	C₂²×D₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
159	C₂×C₄×C₇	#131	1	2	yes	—
160	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
163	C₂²⋊D₇	—	4	4	no	—
164	C₂×Dic₇	—	2	2	no	—
165	C₂×Dic₇	—	2	2	no	—
166	C₂×Dic₇	—	2	2	no	—
167	D₂₈	—	2	2	no	—
168	D₂₈	—	2	2	no	—
169	C₄×D₇	—	4	4	no	—
170	C₄×D₇	—	4	4	no	—
171	C₄×D₇	—	4	4	no	—
172	Dic₁₄	—	2	2	no	—
173	Dic₁₄	—	2	2	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	2	1	no	—
176	C₇×C₈	—	4	4	no	—
177	C₇×C₈	C₂×M₁₆	1	2	yes	—
178	C₇×C₈	C₂×M₁₆	1	2	yes	—
179	C₇⋊C₈	—	2	4	no	—
180	C₇⋊C₈	—	2	2	no	—
181	C₇⋊C₈	#135	1	4	yes	complemented
182	C₂×#41	—	7	2	no	—
183	#135	—	14	4	no	—
184	C₂×Q₁₆	—	7	1	no	—
185	C₂×SD₁₆	—	7	2	no	—
186	C₂×D₈	—	7	1	no	—
187	C₂×M₁₆	—	14	4	no	—
188	C₂²×C₈	—	7	2	no	—
189	C₂²×C₈	—	14	1	no	—
190	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
192	C₄×Q₈	—	7	2	no	—
193	C₄×Q₈	—	7	2	no	—
194	C₄×D₄	—	7	2	no	—
195	C₄×D₄	—	7	2	no	—
196	C₄×D₄	—	14	2	no	—
197	C₄×D₄	—	14	2	no	—
198	#117	—	7	2	no	—
199	#117	—	14	2	no	—
200	C₂×C₄²	—	14	2	no	—
201	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
202	#107	—	7	1	no	—
203	#107	—	7	1	no	—
204	#106	—	7	2	no	—
205	C₄⋊C₈	—	14	4	no	—
206	C₄⋊C₈	C₂²×D₇	1	4	yes	—
207	#103	—	7	2	no	—
208	#103	—	7	2	no	—
209	#102	—	7	2	no	—
210	#102	—	7	2	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	14	4	no	—
213	#97	—	14	4	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	—	7	1	no	—
216	C₄×C₈	—	7	1	no	—
217	C₄×C₈	—	7	2	no	—
218	C₄×C₈	—	7	2	no	—
219	C₄×C₈	—	14	1	no	—
220	C₄×C₈	C₄×D₇	1	1	yes	characteristic
221	C₄×C₈	C₄×D₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
224	C₂²×C₇	—	2	1	no	—
225	C₂²×C₇	—	2	1	no	—
226	C₂²×C₇	#436	1	2	yes	—
227	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂×D₇	—	4	2	no	—
229	C₂×D₇	—	4	2	no	—
230	C₂×D₇	—	4	2	no	—
231	C₄×C₇	—	2	1	no	—
232	C₄×C₇	—	2	1	no	—
233	C₄×C₇	#436	1	2	yes	—
234	C₄×C₇	#443	1	2	yes	—
235	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
236	C₄×C₇	C₂×#131	1	1	yes	characteristic
237	C₄×C₇	C₂×#135	1	2	yes	—
238	Dic₇	—	4	2	no	—
239	Dic₇	—	4	2	no	—
240	Dic₇	—	4	2	no	—
241	#41	—	14	4	no	—
242	#41	—	14	4	no	—
243	C₂×Q₈	—	7	2	no	—
244	C₂×D₄	—	7	2	no	—
245	C₂×D₄	—	14	2	no	—
246	C₂²×C₄	—	14	2	no	—
247	C₂²×C₄	—	14	2	no	—
248	C₂²×C₄	—	14	2	no	—
249	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
250	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
251	C₂²×C₄	#1161	1	1	yes	characteristic
252	Q₁₆	—	14	2	no	—
253	SD₁₆	—	14	4	no	—
254	D₈	—	14	2	no	—
255	M₁₆	—	28	8	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	7	2	no	—
259	C₂×C₈	—	7	2	no	—
260	C₂×C₈	—	14	4	no	—
261	C₂×C₈	—	14	4	no	—
262	C₂×C₈	—	14	2	no	—
263	C₂×C₈	#1131	1	4	yes	complemented
264	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
268	C₄⋊C₄	—	7	2	no	—
269	C₄⋊C₄	—	7	2	no	—
270	C₄⋊C₄	—	14	2	no	—
271	C₄⋊C₄	—	14	2	no	—
272	C₂²⋊C₄	—	14	2	no	—
273	C₂²⋊C₄	—	14	2	no	—
274	C₂²⋊C₄	—	14	2	no	—
275	C₄²	—	14	2	no	—
276	C₄²	—	14	2	no	—
277	C₄²	—	14	2	no	—
278	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
279	C₄²	D₄×D₇	1	1	yes	characteristic
280	C₄²	#1161	1	1	yes	characteristic
281	C₄²	C₂³×D₇	1	1	yes	characteristic
282	C₂×C₇	—	2	2	no	—
283	C₂×C₇	C₂×#436	1	1	yes	characteristic
284	C₂×C₇	C₄×#135	1	1	yes	characteristic
285	C₂×C₇	C₂×#443	1	1	yes	characteristic
286	D₇	—	8	2	no	—
287	C₂³	—	14	2	no	—
288	C₂³	#6456	1	1	yes	characteristic
289	Q₈	—	14	2	no	—
290	Q₈	—	14	2	no	—
291	D₄	—	14	2	no	—
292	D₄	—	14	2	no	—
293	D₄	—	28	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	14	2	no	—
297	C₂×C₄	—	14	2	no	—
298	C₂×C₄	—	14	2	no	—
299	C₂×C₄	—	28	4	no	—
300	C₂×C₄	—	28	4	no	—
301	C₂×C₄	—	28	4	no	—
302	C₂×C₄	#6456	1	1	yes	characteristic
303	C₂×C₄	#6456	1	1	yes	characteristic
304	C₂×C₄	#6456	1	1	yes	characteristic
305	C₂×C₄	C₂×#1131	1	2	yes	—
306	C₂×C₄	#6470	1	2	yes	—
307	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
309	C₂×C₄	C₂×#1161	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	7	4	no	—
314	C₈	—	14	4	no	—
315	C₈	—	14	2	no	—
316	C₈	C₂×#1131	1	2	yes	—
317	C₈	C₂×#1131	1	2	yes	—
318	C₇	#17886	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	28	2	no	—
322	C₂²	—	28	2	no	—
323	C₂²	—	28	2	no	—
324	C₂²	#89610	1	2	yes	—
325	C₂²	C₂×#6456	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	28	2	no	—
329	C₄	—	28	2	no	—
330	C₄	—	28	2	no	—
331	C₄	#89610	1	2	yes	—
332	C₄	#89667	1	2	yes	—
333	C₄	C₂²×#1131	1	1	yes	characteristic
334	C₄	C₂×#6470	1	1	yes	characteristic
335	C₄	D₇×#135	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	56	2	no	—
338	C₂	C₂×#89610	1	1	yes	characteristic
339	C₂	#11736610	1	1	yes	characteristic
340	C₂	C₂×#89667	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#1564	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂×#6467	C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#1564	D₇
hypocenter	C₇	#17886
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
1. #11736722	C₂	semidirect
3. C₇×#1564	C₂	semidirect
4. #11728508	C₂	semidirect
5. C₂×C₈×C₇⋊C₈	C₂	semidirect
6. #11726318	C₂	semidirect
7. #11726317	C₂	semidirect
8. #11726277	C₂	semidirect
82. #1564	D₇	semidirect
98. C₂×C₇×C₈	M₁₆	semidirect
181. C₇⋊C₈	#135	semidirect
263. C₂×C₈	#1131	semidirect
318. C₇	#17886	semidirect

Maximal subgroups

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

subgroup	index
#11736722	2
C₄×#6467	2
C₇×#1564	2
#11728508	2
C₂×C₈×C₇⋊C₈	2
#11726318	2
#11726317	2
#11726277	2
#17886	7