Subgroups — #26033163

Order 11392.

340 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152307	C₂	1	1	yes	maximal, characteristic
3	#19152007	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152006	C₂	1	1	yes	maximal, characteristic, complemented
5	#19152002	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151903	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151891	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151886	C₂	1	1	yes	maximal, characteristic
10	#19151878	C₂	1	1	yes	maximal, characteristic
11	#19151876	C₂	1	1	yes	maximal, characteristic
12	#19151726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151699	C₂	1	1	yes	maximal, characteristic
14	SD₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
15	#440×C₈₉	C₂	1	1	yes	maximal, characteristic
16	#15668012	—	2	1	no	—
17	#15668012	C₂²	1	1	yes	characteristic
18	#15668011	—	2	1	no	—
19	#15668011	—	2	1	no	—
20	#15668011	C₂²	1	1	yes	characteristic
21	#15668011	C₂²	1	1	yes	characteristic
22	#15668011	C₂²	1	1	yes	characteristic
23	#15668004	C₂²	1	1	yes	characteristic
24	#15668003	C₂²	1	1	yes	characteristic
25	#15667991	—	2	1	no	—
26	#15667989	—	2	1	no	—
27	#15667989	—	2	1	no	—
28	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
29	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₇₈	—	2	1	no	—
31	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
34	C₂×#13083553	—	2	1	no	—
35	#15667946	C₂²	1	1	yes	characteristic
36	#15667946	C₂²	1	1	yes	characteristic
37	#15667945	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667944	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	C₄⋊#10623494	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667941	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667933	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	#107×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂×#10623494	C₂³	1	1	yes	characteristic
67	C₂×#10623494	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₈₉	—	2	1	no	—
69	C₄⋊Dic₈₉	—	2	1	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	#13083553	—	2	1	no	—
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	D₄	1	1	yes	characteristic
84	#13083553	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₄×Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
90	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	SD₁₆×C₈₉	Q₈	1	4	yes	complemented
95	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂²⋊C₄×C₈₉	—	2	1	no	—
101	C₄²×C₈₉	—	2	1	no	—
102	C₄²×C₈₉	C₂³	1	1	yes	characteristic
103	C₂³×C₈₉	—	2	1	no	—
104	Q₈×C₈₉	—	2	1	no	—
105	Q₈×C₈₉	C₂×Q₈	1	2	yes	—
106	D₄×C₈₉	—	2	1	no	—
107	D₄×C₈₉	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	8	1	no	—
112	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₈₉	—	2	1	no	—
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	4	2	no	—
120	C₂×Dic₈₉	—	8	1	no	—
121	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
125	Dic₁₇₈	—	2	2	no	—
126	Dic₁₇₈	—	8	1	no	—
127	Dic₁₇₈	—	8	1	no	—
128	Dic₁₇₈	—	8	1	no	—
129	Dic₁₇₈	—	8	1	no	—
130	C₈×C₈₉	—	2	1	no	—
131	C₈×C₈₉	C₂×Q₈	1	2	yes	—
132	#10623494	—	2	1	no	—
133	#10623494	—	2	1	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	—	4	1	no	—
136	C₂²×C₈₉	—	4	1	no	—
137	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	—	4	1	no	—
142	C₄×C₈₉	—	4	1	no	—
143	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
144	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₈₉	—	2	1	no	—
146	Dic₈₉	—	2	1	no	—
147	Dic₈₉	—	2	1	no	—
148	Dic₈₉	—	4	1	no	—
149	Dic₈₉	—	4	1	no	—
150	Dic₈₉	—	4	1	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	#137	1	2	yes	—
153	C₂×C₈₉	—	4	2	no	—
154	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
155	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
156	C₂×C₈₉	#576	1	1	yes	characteristic
157	#3385	—	89	1	no	maximal, Sylow
158	C₈₉	#3385	1	1	yes	characteristic, complemented, Sylow
159	Q₈²	—	89	1	no	—
160	#554	—	89	1	no	—
161	#501	—	89	1	no	—
162	#501	—	89	1	no	—
163	#500	—	89	1	no	—
164	#479	—	89	1	no	—
165	#479	—	89	1	no	—
166	#477	—	89	1	no	—
167	#476	—	89	1	no	—
168	#474	—	89	1	no	—
169	#474	—	89	1	no	—
170	#446	—	89	1	no	—
171	#440	—	89	1	no	—
172	#440	D₈₉	1	1	yes	characteristic
173	C₄×SD₁₆	—	89	1	no	—
174	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
175	C₄⋊Q₈	—	89	1	no	—
176	C₄⋊Q₈	—	89	1	no	—
177	C₄⋊Q₈	—	89	1	no	—
178	C₄⋊Q₈	—	89	1	no	—
179	C₄⋊Q₈	—	178	1	no	—
180	C₄⋊Q₈	—	178	1	no	—
181	C₄⋊Q₈	—	178	1	no	—
182	#125	—	89	1	no	—
183	#125	—	89	1	no	—
184	C₂²⋊Q₈	—	178	1	no	—
185	C₂²⋊Q₈	—	178	1	no	—
186	C₂²⋊Q₈	—	178	1	no	—
187	C₄×Q₈	—	89	1	no	—
188	C₄×Q₈	—	89	1	no	—
189	C₄×Q₈	—	89	1	no	—
190	C₄×Q₈	—	178	1	no	—
191	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
192	C₄×D₄	—	89	1	no	—
193	C₄×D₄	—	89	1	no	—
194	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	178	1	no	—
196	#107	—	89	1	no	—
197	#107	—	89	1	no	—
198	#107	—	89	1	no	—
199	#107	—	89	1	no	—
200	#107	—	89	1	no	—
201	#107	C₂×D₈₉	1	1	yes	characteristic
202	#106	—	89	1	no	—
203	#106	—	89	1	no	—
204	#106	—	89	1	no	—
205	C₄⋊C₈	—	89	1	no	—
206	C₄⋊C₈	—	89	1	no	—
207	C₄⋊C₈	—	89	1	no	—
208	#103	—	89	1	no	—
209	#103	—	89	1	no	—
210	#103	C₂×D₈₉	1	1	yes	characteristic
211	#102	—	89	1	no	—
212	#102	—	89	1	no	—
213	#102	C₂×D₈₉	1	1	yes	characteristic
214	#97	—	89	1	no	—
215	#97	C₂×D₈₉	1	1	yes	characteristic
216	C₄×C₈	—	89	1	no	—
217	C₂×Q₈	—	89	1	no	—
218	C₂×Q₈	—	178	1	no	—
219	C₂×Q₈	—	178	1	no	—
220	C₂×Q₈	—	178	1	no	—
221	C₂×Q₈	—	178	1	no	—
222	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
223	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	178	1	no	—
226	C₂²×C₄	—	178	1	no	—
227	SD₁₆	Dic₁₇₈	1	4	yes	complemented
228	C₂×C₈	—	89	1	no	—
229	C₂×C₈	—	89	1	no	—
230	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	89	1	no	—
234	C₄⋊C₄	—	89	1	no	—
235	C₄⋊C₄	—	89	1	no	—
236	C₄⋊C₄	—	89	1	no	—
237	C₄⋊C₄	—	89	1	no	—
238	C₄⋊C₄	—	89	1	no	—
239	C₄⋊C₄	—	89	1	no	—
240	C₄⋊C₄	—	178	1	no	—
241	C₄⋊C₄	—	178	1	no	—
242	C₄⋊C₄	—	178	1	no	—
243	C₄⋊C₄	—	178	1	no	—
244	C₄⋊C₄	—	178	1	no	—
245	C₄⋊C₄	—	178	1	no	—
246	C₄⋊C₄	—	178	1	no	—
247	C₄⋊C₄	—	178	1	no	—
248	C₄⋊C₄	—	178	1	no	—
249	C₄⋊C₄	—	178	1	no	—
250	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	178	1	no	—
254	C₂²⋊C₄	—	178	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	89	1	no	—
257	C₄²	—	89	1	no	—
258	C₄²	—	178	1	no	—
259	C₄²	—	178	1	no	—
260	C₄²	C₂²×D₈₉	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	Q₈	—	2	1	no	—
263	Q₈	—	178	2	no	—
264	Q₈	—	712	1	no	—
265	Q₈	—	712	1	no	—
266	Q₈	—	712	1	no	—
267	Q₈	—	712	1	no	—
268	Q₈	C₂×Dic₁₇₈	1	2	yes	—
269	D₄	—	2	1	no	—
270	D₄	C₂×Dic₁₇₈	1	2	yes	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	89	1	no	—
276	C₂×C₄	—	89	1	no	—
277	C₂×C₄	—	89	1	no	—
278	C₂×C₄	—	89	1	no	—
279	C₂×C₄	—	356	2	no	—
280	C₂×C₄	—	178	1	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	—	178	1	no	—
283	C₂×C₄	—	178	1	no	—
284	C₂×C₄	—	712	1	no	—
285	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
286	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
287	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	178	1	no	—
290	C₈	—	178	1	no	—
291	C₈	C₂×Dic₁₇₈	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	89	2	no	—
302	C₄	—	356	1	no	—
303	C₄	—	356	1	no	—
304	C₄	—	356	1	no	—
305	C₄	—	356	1	no	—
306	C₄	—	178	1	no	—
307	C₄	—	178	1	no	—
308	C₄	—	178	1	no	—
309	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
310	C₄	#15668059	1	1	yes	characteristic
311	C₂	—	4	2	no	—
312	C₂	D₄×Dic₁₇₈	1	1	yes	characteristic
313	C₂	#137×D₈₉	1	1	yes	characteristic
314	C₂	#19152402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#440×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₄×C₈₉	C₂³
hypercenter	#440	D₈₉
hypocenter	C₈₉	#3385
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₇₈	C₂	semidirect
3. #19152007	C₂	semidirect
4. #19152006	C₂	semidirect
5. #19152002	C₂	semidirect
6. #19151905	C₂	semidirect
7. #19151903	C₂	semidirect
8. #19151891	C₂	semidirect
12. #19151726	C₂	semidirect
94. SD₁₆×C₈₉	Q₈	semidirect
158. C₈₉	#3385	semidirect
227. SD₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₁₇₈	2
#19152307	2
#19152007	2
#19152006	2
#19152002	2
#19151905	2
#19151903	2
#19151891	2
#19151886	2
#19151878	2
#19151876	2
#19151726	2
#19151699	2
SD₁₆×Dic₈₉	2
#440×C₈₉	2
#3385	89