Subgroups — #13337647

Order 1792.

602 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	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#11739876	C₂	1	2	yes	maximal
3	#11735920	C₂	1	1	yes	maximal, characteristic
4	#11728701	C₂	1	4	yes	maximal, complemented
5	#11728699	C₂	1	2	yes	maximal, complemented
6	#11728696	C₂	1	2	yes	maximal, complemented
7	#11728192	C₂	1	2	yes	maximal
8	#11727485	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
10	#90161	C₂²	1	2	yes	—
11	C₇×C₈×Q₈	—	2	1	no	—
12	C₇×C₈×Q₈	—	2	1	no	—
13	C₇×C₈×Q₈	C₂²	1	2	yes	—
14	C₇×C₈×Q₈	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₈	—	2	1	no	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×C₈	—	2	1	no	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	#89997	—	2	2	no	—
20	#89888	—	2	1	no	—
21	#89888	—	2	1	no	—
22	#89794	—	2	4	no	—
23	#89793	C₂²	1	4	yes	—
24	#89792	C₂²	1	4	yes	—
25	#89736	—	2	2	no	—
26	#89696	C₂²	1	2	yes	—
27	#89471	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₈	—	2	1	no	—
29	C₄×Dic₂₈	—	2	1	no	—
30	#89454	C₂²	1	1	yes	characteristic
31	#89453	C₂²	1	2	yes	—
32	#89277	C₂²	1	4	yes	—
33	#89276	C₂²	1	4	yes	—
34	#89253	C₂²	1	2	yes	—
35	#89253	C₂²	1	2	yes	—
36	#19262	—	7	1	no	maximal, Sylow
37	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	—	2	1	no	—
39	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
41	C₄×C₇×Q₈	—	2	1	no	—
42	C₄×C₇×Q₈	—	2	1	no	—
43	C₄×C₇×Q₈	D₄	1	2	yes	—
44	C₄×C₇×Q₈	C₂³	1	2	yes	—
45	C₂×C₇×C₄⋊C₄	—	2	1	no	—
46	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇	—	2	1	no	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	#6489	—	2	2	no	—
50	#6487	—	2	2	no	—
51	#6487	—	2	2	no	—
52	C₂×#1136	—	2	2	no	—
53	C₂×Dic₂₈	—	2	1	no	—
54	C₂×Dic₂₈	—	2	1	no	—
55	#6433	—	2	2	no	—
56	#6432	C₂³	1	2	yes	—
57	#6432	C₂³	1	2	yes	—
58	C₄×Dic₁₄	—	2	2	no	—
59	C₄×Dic₁₄	—	2	2	no	—
60	C₇×C₄⋊C₈	—	2	2	no	—
61	C₇×C₄⋊C₈	—	2	2	no	—
62	C₇×C₄⋊C₈	—	2	1	no	—
63	C₇×C₄⋊C₈	C₂³	1	2	yes	—
64	C₄×C₇×C₈	—	2	2	no	—
65	C₄×C₇×C₈	—	2	2	no	—
66	C₄×C₇×C₈	D₄	1	1	yes	characteristic
67	C₄×C₇×C₈	D₄	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	#6409	—	2	4	no	—
71	#6409	—	2	4	no	—
72	#6392	—	2	1	no	—
73	#6392	—	2	1	no	—
74	#6390	—	2	2	no	—
75	#6390	—	2	2	no	—
76	#6382	—	2	4	no	—
77	#6381	—	2	4	no	—
78	C₄⋊(C₇⋊C₈)	—	2	2	no	—
79	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
80	#3170	—	7	1	no	—
81	#3078	—	7	2	no	—
82	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
83	#1684	—	7	4	no	—
84	#1676	—	7	2	no	—
85	#1670	—	7	2	no	—
86	#1567	—	7	1	no	—
87	#1550	—	7	2	no	—
88	C₂×C₇×Q₈	—	2	1	no	—
89	C₂×C₇×Q₈	—	2	2	no	—
90	C₂×C₇×Q₈	Q₁₆	1	4	yes	complemented
91	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₇	—	2	1	no	—
93	C₂²×C₄×C₇	—	2	1	no	—
94	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇	#41	1	1	yes	characteristic
97	C₂×Dic₁₄	—	2	2	no	—
98	C₂×Dic₁₄	—	2	2	no	—
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	4	1	no	—
102	C₂×C₇×C₈	—	4	2	no	—
103	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
107	C₇×C₄⋊C₄	—	2	2	no	—
108	C₇×C₄⋊C₄	—	2	1	no	—
109	C₇×C₄⋊C₄	—	2	2	no	—
110	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
111	C₄²×C₇	—	2	2	no	—
112	C₄²×C₇	—	2	2	no	—
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇	#41	1	1	yes	characteristic
116	C₄²×C₇	C₂⁴	1	1	yes	characteristic
117	C₂²⋊Dic₇	—	4	2	no	—
118	C₄⋊Dic₇	—	2	2	no	—
119	C₄⋊Dic₇	—	2	2	no	—
120	C₄⋊Dic₇	—	2	2	no	—
121	C₄⋊Dic₇	—	2	2	no	—
122	Dic₇⋊C₄	—	4	2	no	—
123	Dic₇⋊C₄	—	4	2	no	—
124	C₄×Dic₇	—	4	2	no	—
125	#1136	—	4	4	no	—
126	C₂×C₇⋊C₈	—	2	4	no	—
127	Dic₂₈	—	4	2	no	—
128	Dic₂₈	—	4	2	no	—
129	#533	—	7	2	no	—
130	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
131	#500	—	7	1	no	—
132	#494	—	7	1	no	—
133	#487	—	7	2	no	—
134	#479	—	14	4	no	—
135	#477	—	7	4	no	—
136	#475	—	7	4	no	—
137	#467	—	14	1	no	—
138	#467	—	14	1	no	—
139	C₈×Q₈	—	2	1	no	—
140	C₈×Q₈	—	2	1	no	—
141	C₈×Q₈	C₂×D₇	1	2	yes	—
142	C₈×Q₈	C₂×D₇	1	2	yes	—
143	C₄×Q₁₆	—	14	1	no	—
144	C₄×Q₁₆	—	14	1	no	—
145	#428	—	14	2	no	—
146	#423	—	7	2	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
149	#419	—	14	2	no	—
150	C₂×C₄×C₈	—	2	1	no	—
151	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
152	#335	—	7	2	no	—
153	#335	—	7	2	no	—
154	#333	—	7	4	no	—
155	#332	—	7	4	no	—
156	C₂³×C₇	#121	1	1	yes	characteristic
157	C₇×Q₈	—	2	2	no	—
158	C₇×Q₈	—	2	4	no	—
159	C₇×Q₈	C₂×Q₁₆	1	4	yes	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	4	no	—
162	C₂×C₄×C₇	—	2	1	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	2	no	—
165	C₂×C₄×C₇	—	4	1	no	—
166	C₂×C₄×C₇	—	4	2	no	—
167	C₂×C₄×C₇	#121	1	1	yes	characteristic
168	C₂×C₄×C₇	#121	1	1	yes	characteristic
169	C₂×C₄×C₇	#121	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
171	C₂×C₄×C₇	#135	1	2	yes	—
172	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
175	C₂×Dic₇	—	4	2	no	—
176	C₂×Dic₇	—	4	2	no	—
177	Dic₁₄	—	4	2	no	—
178	Dic₁₄	—	4	2	no	—
179	Dic₁₄	—	8	2	no	—
180	C₇×C₈	—	2	1	no	—
181	C₇×C₈	—	2	1	no	—
182	C₇×C₈	—	2	1	no	—
183	C₇×C₈	—	4	2	no	—
184	C₇×C₈	#137	1	2	yes	—
185	C₇⋊C₈	—	4	4	no	—
186	C₂²×Q₈	D₂₈	1	1	yes	characteristic
187	C₂×Q₁₆	—	14	1	no	—
188	C₂×Q₁₆	—	14	1	no	—
189	C₂×M₁₆	—	14	2	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
193	C₄⋊Q₈	—	7	2	no	—
194	C₄⋊Q₈	—	7	2	no	—
195	#125	—	14	2	no	—
196	C₂²⋊Q₈	—	14	2	no	—
197	C₂²⋊Q₈	—	14	2	no	—
198	C₄×Q₈	—	2	1	no	—
199	C₄×Q₈	—	2	1	no	—
200	C₄×Q₈	—	14	2	no	—
201	C₄×Q₈	—	14	2	no	—
202	C₄×Q₈	D₂₈	1	2	yes	—
203	C₄×Q₈	C₂²×D₇	1	2	yes	—
204	#117	—	14	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	D₂₈	1	1	yes	characteristic
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
209	#107	—	14	1	no	—
210	#107	—	14	1	no	—
211	#107	—	14	4	no	—
212	#106	—	14	4	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	7	4	no	—
217	C₄⋊C₈	—	14	2	no	—
218	C₄⋊C₈	C₂²×D₇	1	2	yes	—
219	#103	—	14	4	no	—
220	#103	—	14	4	no	—
221	#103	—	14	2	no	—
222	#103	—	14	2	no	—
223	C₄×C₈	—	2	2	no	—
224	C₄×C₈	—	2	2	no	—
225	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
226	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
227	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
229	C₂²×C₇	—	2	1	no	—
230	C₂²×C₇	—	2	1	no	—
231	C₂²×C₇	#467	1	2	yes	—
232	C₂²×C₇	C₂×#121	1	1	yes	characteristic
233	C₄×C₇	—	2	1	no	—
234	C₄×C₇	—	2	4	no	—
235	C₄×C₇	—	2	1	no	—
236	C₄×C₇	—	2	1	no	—
237	C₄×C₇	—	4	2	no	—
238	C₄×C₇	#467	1	2	yes	—
239	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
240	C₄×C₇	C₂×#135	1	1	yes	characteristic
241	C₄×C₇	C₂×#137	1	1	yes	characteristic
242	C₄×C₇	#575	1	1	yes	characteristic
243	Dic₇	—	8	2	no	—
244	Dic₇	—	8	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	14	2	no	—
248	C₂×Q₈	—	14	2	no	—
249	C₂×Q₈	Dic₂₈	1	4	yes	complemented
250	C₂×Q₈	C₂×D₂₈	1	2	yes	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
254	C₂²×C₄	#1157	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
256	Q₁₆	—	28	2	no	—
257	Q₁₆	—	28	2	no	—
258	M₁₆	—	28	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	14	4	no	—
264	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
266	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
267	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	14	2	no	—
272	C₄⋊C₄	—	14	2	no	—
273	C₄⋊C₄	—	14	2	no	—
274	C₄⋊C₄	—	14	2	no	—
275	C₄⋊C₄	—	28	2	no	—
276	C₄⋊C₄	—	28	2	no	—
277	C₄⋊C₄	C₂×D₂₈	1	2	yes	—
278	C₂²⋊C₄	—	28	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	28	2	no	—
282	C₄²	C₂×D₂₈	1	1	yes	characteristic
283	C₄²	#1157	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
285	C₄²	C₂³×D₇	1	1	yes	characteristic
286	C₂×C₇	—	2	2	no	—
287	C₂×C₇	#3037	1	1	yes	characteristic
288	C₂×C₇	C₂×#467	1	1	yes	characteristic
289	C₂×C₇	#3067	1	1	yes	characteristic
290	C₂³	#6493	1	1	yes	characteristic
291	Q₈	—	2	4	no	—
292	Q₈	—	2	2	no	—
293	Q₈	—	28	2	no	—
294	Q₈	—	28	2	no	—
295	Q₈	—	56	2	no	—
296	Q₈	C₂×Dic₂₈	1	4	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	28	2	no	—
305	C₂×C₄	—	28	2	no	—
306	C₂×C₄	#6467	1	2	yes	—
307	C₂×C₄	C₂×Dic₂₈	1	2	yes	—
308	C₂×C₄	#6493	1	1	yes	characteristic
309	C₂×C₄	#6493	1	1	yes	characteristic
310	C₂×C₄	#6493	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
312	C₂×C₄	C₂×#1157	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	4	2	no	—
318	C₈	—	28	4	no	—
319	C₈	#6505	1	2	yes	—
320	C₇	#19262	1	1	yes	characteristic, complemented, Sylow
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	#89888	1	2	yes	—
324	C₂²	C₂×#6493	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	—	56	2	no	—
331	C₄	—	56	2	no	—
332	C₄	#89888	1	2	yes	—
333	C₄	C₂×#6467	1	1	yes	characteristic
334	C₄	C₂²×Dic₂₈	1	1	yes	characteristic
335	C₄	C₂×#6505	1	1	yes	characteristic
336	C₄	#90513	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	#11739886	1	1	yes	characteristic
339	C₂	C₂×#89888	1	1	yes	characteristic
340	C₂	#11741250	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6493
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×C₈×Q₈	C₂
socle	C₂²×C₇	C₂×#121
prime radical	C₂³×C₇	#121
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	#19262
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
4. #11728701	C₂	semidirect
5. #11728699	C₂	semidirect
6. #11728696	C₂	semidirect
90. C₂×C₇×Q₈	Q₁₆	semidirect
249. C₂×Q₈	Dic₂₈	semidirect
320. C₇	#19262	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×Q₈	2
#11739876	2
#11735920	2
#11728701	2
#11728699	2
#11728696	2
#11728192	2
#11727485	2
#19262	7