Subgroups — #36473013

Order 22784.

358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic
2	#26022008	C₂	1	1	yes	maximal, characteristic
3	#26018359	C₂	1	4	yes	maximal
4	C₄×#106×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19152155	—	2	1	no	—
6	C₂×#15667952	C₂²	1	2	yes	—
7	#19152003	C₄	1	1	yes	characteristic, complemented
8	#19152003	C₄	1	1	yes	characteristic, complemented
9	#19152000	—	2	1	no	—
10	#19152000	C₄	1	2	yes	complemented
11	#19151996	—	2	1	no	—
12	#19151636	—	2	1	no	—
13	#19151636	C₂²	1	2	yes	—
14	C₄⋊#13083549	—	2	1	no	—
15	C₄⋊#13083549	—	2	1	no	—
16	C₂×#106×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
19	#340×C₈₉	—	2	1	no	—
20	#15668012	—	2	1	no	—
21	#15668012	—	2	1	no	—
22	#15668004	—	2	2	no	—
23	#15667991	—	4	1	no	—
24	#15667991	—	4	1	no	—
25	C₄×Dic₁₇₈	—	4	1	no	—
26	C₄×Dic₁₇₈	—	4	1	no	—
27	#15667971	—	2	2	no	—
28	#15667952	—	2	4	no	—
29	C₂×#13083549	—	2	2	no	—
30	#15667946	—	2	1	no	—
31	#15667946	—	2	1	no	—
32	#15667946	C₂×C₄	1	2	yes	—
33	#15667946	C₂×C₄	1	2	yes	—
34	#15667944	—	2	2	no	—
35	#15667944	—	2	2	no	—
36	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
39	C₂×C₄²×C₈₉	—	4	1	no	—
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	#106×C₈₉	—	2	2	no	—
42	#106×C₈₉	Q₈	1	4	yes	complemented
43	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
47	#95×C₈₉	—	4	1	no	—
48	C₂×Dic₁₇₈	—	4	1	no	—
49	C₂×Dic₁₇₈	—	4	1	no	—
50	C₄⋊Dic₈₉	—	2	2	no	—
51	C₄⋊Dic₈₉	—	2	2	no	—
52	C₄⋊Dic₈₉	—	4	1	no	—
53	C₄⋊Dic₈₉	—	4	1	no	—
54	#13083553	—	4	2	no	—
55	#13083553	—	4	2	no	—
56	C₂²⋊Dic₈₉	—	4	2	no	—
57	C₄×Dic₈₉	—	4	2	no	—
58	#13083549	—	4	2	no	—
59	C₂²×C₄×C₈₉	—	4	2	no	—
60	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₈₉	#41	1	2	yes	—
70	C₄⋊C₄×C₈₉	—	2	4	no	—
71	C₄⋊C₄×C₈₉	—	4	2	no	—
72	C₄²×C₈₉	—	8	2	no	—
73	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
76	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
77	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	4	1	no	—
80	C₂×C₄×C₈₉	—	4	4	no	—
81	C₂×C₄×C₈₉	—	8	2	no	—
82	C₂×C₄×C₈₉	—	8	2	no	—
83	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#137	1	2	yes	—
91	C₂×Dic₈₉	—	4	2	no	—
92	C₂×Dic₈₉	—	4	2	no	—
93	Dic₁₇₈	—	8	2	no	—
94	Dic₁₇₈	—	8	2	no	—
95	C₈×C₈₉	—	2	1	no	—
96	C₈×C₈₉	—	2	1	no	—
97	C₈×C₈₉	C₄×Q₈	1	2	yes	—
98	C₈×C₈₉	#124	1	1	yes	characteristic
99	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
100	C₂²×C₈₉	—	2	1	no	—
101	C₂²×C₈₉	—	2	1	no	—
102	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
103	C₂²×C₈₉	#474	1	2	yes	—
104	C₄×C₈₉	—	2	1	no	—
105	C₄×C₈₉	—	2	1	no	—
106	C₄×C₈₉	—	2	1	no	—
107	C₄×C₈₉	—	8	4	no	—
108	C₄×C₈₉	#360	1	2	yes	—
109	C₄×C₈₉	#391	1	1	yes	characteristic
110	C₄×C₈₉	#419	1	1	yes	characteristic
111	C₄×C₈₉	#475	1	2	yes	—
112	Dic₈₉	—	8	2	no	—
113	Dic₈₉	—	8	2	no	—
114	#13278	—	89	1	no	maximal, Sylow
115	C₂×C₈₉	—	2	2	no	—
116	C₂×C₈₉	#1988	1	1	yes	characteristic
117	C₂×C₈₉	C₂×#360	1	1	yes	characteristic
118	C₂×C₈₉	#2145	1	1	yes	characteristic
119	#3160	—	89	1	no	—
120	#2150	—	89	1	no	—
121	C₄×#106	D₈₉	1	1	yes	characteristic
122	#1386	—	89	4	no	—
123	C₈₉	#13278	1	1	yes	characteristic, complemented, Sylow
124	#533	—	178	1	no	—
125	C₂×M₃₂	—	89	2	no	—
126	#500	—	89	1	no	—
127	#500	—	89	1	no	—
128	#499	—	89	2	no	—
129	#499	—	178	1	no	—
130	#498	—	178	1	no	—
131	#427	—	89	2	no	—
132	#427	—	178	1	no	—
133	C₂×#106	C₂×D₈₉	1	2	yes	—
134	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₁₆	—	178	1	no	—
137	C₄⋊C₁₆	—	178	1	no	—
138	#340	—	2	1	no	—
139	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
141	C₄⋊Q₈	—	178	1	no	—
142	C₄⋊Q₈	—	178	1	no	—
143	#125	—	178	2	no	—
144	C₂²⋊Q₈	—	356	1	no	—
145	C₂²⋊Q₈	—	356	1	no	—
146	C₄×Q₈	—	356	1	no	—
147	C₄×Q₈	—	356	1	no	—
148	#117	—	178	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄²	—	4	1	no	—
151	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
152	M₃₂	—	178	4	no	—
153	C₂×C₁₆	—	178	2	no	—
154	#107	—	89	2	no	—
155	#107	—	89	2	no	—
156	#107	—	178	1	no	—
157	#107	—	178	1	no	—
158	#106	—	2	2	no	—
159	#106	—	178	2	no	—
160	#106	—	178	2	no	—
161	#106	Dic₁₇₈	1	4	yes	complemented
162	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
163	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
164	C₄×C₈	D₃₅₆	1	1	yes	characteristic
165	C₄×C₈	D₃₅₆	1	1	yes	characteristic
166	#95	—	4	1	no	—
167	C₂×Q₈	—	356	1	no	—
168	C₂×Q₈	—	356	1	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	#13083559	1	1	yes	characteristic
171	C₂²×C₄	#13083559	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	#13083551	1	1	yes	characteristic
175	C₂×C₈	#13083551	1	1	yes	characteristic
176	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
177	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
178	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
179	C₂×C₈	#13083574	1	2	yes	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	356	2	no	—
183	C₄⋊C₄	—	356	2	no	—
184	C₄⋊C₄	—	356	1	no	—
185	C₄⋊C₄	—	356	1	no	—
186	C₄⋊C₄	—	178	2	no	—
187	C₄⋊C₄	—	178	2	no	—
188	C₂²⋊C₄	—	356	2	no	—
189	C₄²	—	8	2	no	—
190	C₄²	—	356	2	no	—
191	C₄²	#13083551	1	1	yes	characteristic
192	C₄²	#13083551	1	1	yes	characteristic
193	C₄²	#13083561	1	1	yes	characteristic
194	C₄²	#13083561	1	1	yes	characteristic
195	C₁₆	—	356	2	no	—
196	C₂³	C₂×#13083559	1	1	yes	characteristic
197	Q₈	—	712	2	no	—
198	Q₈	—	712	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	356	2	no	—
205	C₂×C₄	—	356	2	no	—
206	C₂×C₄	#15667932	1	1	yes	characteristic
207	C₂×C₄	#15667932	1	1	yes	characteristic
208	C₂×C₄	#15667932	1	1	yes	characteristic
209	C₂×C₄	#15667932	1	1	yes	characteristic
210	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
211	C₂×C₄	#15667991	1	2	yes	—
212	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
213	C₂×C₄	#15668042	1	2	yes	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	C₄×Dic₁₇₈	1	2	yes	—
217	C₈	#15668000	1	1	yes	characteristic
218	C₈	#15668012	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×#15667932	1	1	yes	characteristic
222	C₂²	#19151878	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	4	no	—
227	C₄	—	712	2	no	—
228	C₄	—	712	2	no	—
229	C₄	#19151407	1	2	yes	—
230	C₄	#19151504	1	1	yes	characteristic
231	C₄	#19151606	1	1	yes	characteristic
232	C₄	#19151883	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	#26021204	1	1	yes	characteristic
235	C₂	C₂×#19151407	1	1	yes	characteristic
236	C₂	#26021987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667932
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×#106×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	C₂×D₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×#106	D₈₉
hypocenter	C₈₉	#13278
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
7. #19152003	C₄	semidirect
8. #19152003	C₄	semidirect
10. #19152000	C₄	semidirect
42. #106×C₈₉	Q₈	semidirect
123. C₈₉	#13278	semidirect
161. #106	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
#26022008	2
#26018359	2
C₄×#106×C₈₉	2
#13278	89