Subgroups — #26033168

Order 11392.

360 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	#19152325	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	#19152008	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152005	C₂	1	1	yes	maximal, characteristic, complemented
5	#19152002	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151896	C₂	1	1	yes	maximal, characteristic
7	#19151894	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151892	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151882	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151876	C₂	1	1	yes	maximal, characteristic
11	#19151727	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151698	C₂	1	1	yes	maximal, characteristic
13	#19151695	C₂	1	1	yes	maximal, characteristic
14	#19151689	C₂	1	1	yes	maximal, characteristic
15	#478×C₈₉	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₁₇₈	—	2	1	no	—
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	#15668012	C₂²	1	1	yes	characteristic
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	#15668003	C₂²	1	1	yes	characteristic
24	#15668002	—	2	1	no	—
25	#15668002	—	2	1	no	—
26	#15667991	—	2	1	no	—
27	#15667989	—	2	1	no	—
28	#15667989	—	2	1	no	—
29	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
31	#15667977	—	2	1	no	—
32	#15667977	C₂²	1	1	yes	characteristic
33	#15667977	C₂²	1	1	yes	characteristic
34	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
35	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
36	#15667946	C₂²	1	1	yes	characteristic
37	#15667946	C₂²	1	1	yes	characteristic
38	#15667945	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667945	C₂²	1	1	yes	characteristic
41	#15667943	C₂²	1	1	yes	characteristic
42	#15667943	C₂²	1	1	yes	characteristic
43	#15667941	C₂²	1	1	yes	characteristic
44	#15667941	C₂²	1	1	yes	characteristic
45	#15667937	C₂²	1	1	yes	characteristic
46	#15667937	C₂²	1	1	yes	characteristic
47	#15667935	C₂²	1	1	yes	characteristic
48	#15667933	C₂²	1	1	yes	characteristic
49	#15667923	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×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	—	4	4	no	—
64	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	#13083562	Q₈	1	4	yes	complemented
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₈₉	—	2	1	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	D₄	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	—	2	1	no	—
84	#13083553	D₄	1	1	yes	characteristic
85	#13083553	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₄×Dic₈₉	—	2	1	no	—
89	C₄×Dic₈₉	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	—	2	1	no	—
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	—	2	1	no	—
99	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₄²×C₈₉	C₂³	1	1	yes	characteristic
104	C₂³×C₈₉	—	2	1	no	—
105	D₄×C₈₉	—	2	1	no	—
106	D₄×C₈₉	C₂×Q₈	1	2	yes	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	4	2	no	—
111	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₈₉	—	2	1	no	—
115	C₂×Dic₈₉	—	2	1	no	—
116	C₂×Dic₈₉	—	2	1	no	—
117	C₂×Dic₈₉	—	2	1	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	C₂×Dic₈₉	—	4	2	no	—
120	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
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	Dic₁₇₈	—	2	1	no	—
125	Dic₁₇₈	—	2	2	no	—
126	Dic₁₇₈	—	4	2	no	—
127	Dic₁₇₈	—	4	2	no	—
128	Dic₁₇₈	—	4	2	no	—
129	Dic₁₇₈	—	4	2	no	—
130	Dic₁₇₈	C₂×Q₈	1	2	yes	complemented
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	—	2	1	no	—
133	#10623494	—	2	1	no	—
134	#10623494	C₂×Q₈	1	2	yes	complemented
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	—	4	1	no	—
137	C₂²×C₈₉	—	4	1	no	—
138	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
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	—	4	1	no	—
144	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
145	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
146	Dic₈₉	—	2	2	no	—
147	Dic₈₉	—	2	1	no	—
148	Dic₈₉	—	2	1	no	—
149	Dic₈₉	—	2	1	no	—
150	Dic₈₉	—	4	1	no	—
151	Dic₈₉	—	4	1	no	—
152	Dic₈₉	—	4	1	no	—
153	Dic₈₉	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₈₉	—	4	2	no	—
155	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
156	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₈₉	#578	1	1	yes	characteristic
158	#3386	—	89	1	no	maximal, Sylow
159	C₈₉	#3386	1	1	yes	characteristic, complemented, Sylow
160	#557	—	89	1	no	—
161	D₄×Q₈	—	89	1	no	—
162	#501	—	89	1	no	—
163	#501	—	89	1	no	—
164	#500	—	89	1	no	—
165	#478	—	89	1	no	—
166	#478	D₈₉	1	1	yes	characteristic
167	#477	—	89	1	no	—
168	#477	—	89	1	no	—
169	#475	—	89	1	no	—
170	#474	—	89	1	no	—
171	#446	—	89	1	no	—
172	#440	—	89	1	no	—
173	#440	—	89	1	no	—
174	C₄×SD₁₆	—	89	1	no	—
175	C₂²×Q₈	—	178	1	no	—
176	C₂×SD₁₆	—	89	1	no	—
177	C₄⋊Q₈	—	89	1	no	—
178	C₄⋊Q₈	—	89	1	no	—
179	C₄⋊Q₈	—	89	1	no	—
180	C₄⋊Q₈	—	89	1	no	—
181	C₄⋊Q₈	—	178	1	no	—
182	#125	—	89	1	no	—
183	#125	—	178	1	no	—
184	#125	—	178	1	no	—
185	#125	C₂×D₈₉	1	1	yes	characteristic
186	C₂²⋊Q₈	—	178	1	no	—
187	C₂²⋊Q₈	—	178	1	no	—
188	C₂²⋊Q₈	—	178	1	no	—
189	C₄×Q₈	—	89	1	no	—
190	C₄×Q₈	—	89	1	no	—
191	C₄×Q₈	—	89	1	no	—
192	C₄×Q₈	—	89	1	no	—
193	C₄×Q₈	—	178	1	no	—
194	C₄×D₄	—	89	1	no	—
195	C₄×D₄	—	89	1	no	—
196	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
197	#107	—	89	1	no	—
198	#107	—	89	1	no	—
199	#107	—	89	1	no	—
200	#107	—	89	1	no	—
201	#107	—	89	1	no	—
202	#107	C₂×D₈₉	1	1	yes	characteristic
203	#106	—	89	1	no	—
204	#106	—	89	1	no	—
205	#106	C₂×D₈₉	1	1	yes	characteristic
206	C₄⋊C₈	—	89	1	no	—
207	C₄⋊C₈	—	89	1	no	—
208	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
209	#103	—	89	1	no	—
210	#103	—	89	1	no	—
211	#103	—	89	1	no	—
212	#102	—	89	1	no	—
213	#102	C₂×D₈₉	1	1	yes	characteristic
214	#102	C₂×D₈₉	1	1	yes	characteristic
215	#97	—	89	1	no	—
216	#97	—	89	1	no	—
217	C₄×C₈	—	89	1	no	—
218	C₂×Q₈	—	89	1	no	—
219	C₂×Q₈	—	89	1	no	—
220	C₂×Q₈	—	356	4	no	—
221	C₂×Q₈	—	178	1	no	—
222	C₂×Q₈	—	178	1	no	—
223	C₂×Q₈	—	178	1	no	—
224	C₂×Q₈	—	178	1	no	—
225	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	178	1	no	—
228	C₂²×C₄	—	178	1	no	—
229	SD₁₆	—	89	4	no	—
230	C₂×C₈	—	89	1	no	—
231	C₂×C₈	—	89	1	no	—
232	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	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	—	89	1	no	—
241	C₄⋊C₄	—	89	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	—	178	1	no	—
251	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	178	1	no	—
256	C₂²⋊C₄	—	178	1	no	—
257	C₄²	—	89	1	no	—
258	C₄²	—	89	1	no	—
259	C₄²	—	178	1	no	—
260	C₄²	—	178	1	no	—
261	C₄²	—	178	1	no	—
262	C₄²	C₂²×D₈₉	1	1	yes	characteristic
263	C₂³	—	2	1	no	—
264	Q₈	—	89	2	no	—
265	Q₈	—	356	2	no	—
266	Q₈	—	356	2	no	—
267	Q₈	—	356	2	no	—
268	Q₈	—	356	2	no	—
269	Q₈	—	178	2	no	—
270	Q₈	—	178	1	no	—
271	D₄	—	2	1	no	—
272	D₄	Q₈×D₈₉	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	89	1	no	—
278	C₂×C₄	—	89	1	no	—
279	C₂×C₄	—	89	1	no	—
280	C₂×C₄	—	89	1	no	—
281	C₂×C₄	—	356	2	no	—
282	C₂×C₄	—	356	2	no	—
283	C₂×C₄	—	178	1	no	—
284	C₂×C₄	—	178	1	no	—
285	C₂×C₄	—	178	1	no	—
286	C₂×C₄	—	178	1	no	—
287	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
288	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	89	2	no	—
293	C₈	—	178	1	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	89	2	no	—
304	C₄	—	356	1	no	—
305	C₄	—	356	1	no	—
306	C₄	—	356	1	no	—
307	C₄	—	178	1	no	—
308	C₄	—	178	1	no	—
309	C₄	—	178	2	no	—
310	C₄	—	178	1	no	—
311	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
312	C₄	#15668059	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	#19152261	1	1	yes	characteristic
315	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
316	C₂	#19152412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#478×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₄×C₈₉	C₂³
hypercenter	#478	D₈₉
hypocenter	C₈₉	#3386
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #19152325	C₂	semidirect
3. #19152008	C₂	semidirect
4. #19152005	C₂	semidirect
5. #19152002	C₂	semidirect
7. #19151894	C₂	semidirect
8. #19151892	C₂	semidirect
9. #19151882	C₂	semidirect
11. #19151727	C₂	semidirect
68. #13083562	Q₈	semidirect
130. Dic₁₇₈	C₂×Q₈	semidirect
134. #10623494	C₂×Q₈	semidirect
159. C₈₉	#3386	semidirect

Maximal subgroups

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

subgroup	index
#19152325	2
D₄×Dic₁₇₈	2
#19152008	2
#19152005	2
#19152002	2
#19151896	2
#19151894	2
#19151892	2
#19151882	2
#19151876	2
#19151727	2
#19151698	2
#19151695	2
#19151689	2
#478×C₈₉	2
#3386	89