Subgroups — #36472996

Order 22784.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 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	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic
3	#26018349	C₂	1	4	yes	maximal
4	C₄×#108×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19152155	—	2	1	no	—
6	C₂²×#13083549	C₂²	1	2	yes	—
7	#19152003	C₄	1	1	yes	characteristic
8	#19152003	C₄	1	1	yes	characteristic
9	#19152000	—	2	1	no	—
10	#19152000	C₄	1	2	yes	—
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×#108×C₈₉	C₂²	1	2	yes	—
17	C₄×M₁₆×C₈₉	—	2	1	no	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#341×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	C₂×#13083549	—	2	2	no	—
29	C₂×#13083549	—	2	4	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	C₂×M₁₆×C₈₉	—	2	2	no	—
37	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	#108×C₈₉	—	2	2	no	—
41	#108×C₈₉	Q₈	1	4	yes	complemented
42	#97×C₈₉	—	4	1	no	—
43	C₄×C₈×C₈₉	—	4	1	no	—
44	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
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	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	M₁₆×C₈₉	—	2	4	no	—
63	M₁₆×C₈₉	—	4	2	no	—
64	C₂×C₈×C₈₉	—	2	2	no	—
65	C₂×C₈×C₈₉	—	4	2	no	—
66	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₈₉	#41	1	2	yes	—
72	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₈₉	D₈	1	1	yes	characteristic
75	C₄²×C₈₉	D₈	1	1	yes	characteristic
76	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₉	—	2	2	no	—
78	C₂×C₄×C₈₉	—	4	1	no	—
79	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	#137	1	2	yes	—
87	C₂×Dic₈₉	—	4	2	no	—
88	C₂×Dic₈₉	—	4	2	no	—
89	Dic₁₇₈	—	8	2	no	—
90	Dic₁₇₈	—	8	2	no	—
91	C₈×C₈₉	—	2	1	no	—
92	C₈×C₈₉	—	2	1	no	—
93	C₈×C₈₉	—	4	4	no	—
94	C₈×C₈₉	C₄×Q₈	1	2	yes	—
95	C₈×C₈₉	#124	1	1	yes	characteristic
96	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
97	C₂²×C₈₉	—	2	1	no	—
98	C₂²×C₈₉	—	2	1	no	—
99	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
100	C₂²×C₈₉	#475	1	2	yes	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	—	2	1	no	—
104	C₄×C₈₉	#359	1	2	yes	—
105	C₄×C₈₉	#391	1	1	yes	characteristic
106	C₄×C₈₉	#419	1	1	yes	characteristic
107	C₄×C₈₉	#474	1	2	yes	—
108	Dic₈₉	—	8	2	no	—
109	Dic₈₉	—	8	2	no	—
110	#13274	—	89	1	no	maximal, Sylow
111	C₂×C₈₉	—	2	2	no	—
112	C₂×C₈₉	#1988	1	1	yes	characteristic
113	C₂×C₈₉	C₂×#359	1	1	yes	characteristic
114	C₂×C₈₉	#2143	1	1	yes	characteristic
115	#3160	—	89	1	no	—
116	C₂×C₄⋊C₁₆	—	89	1	no	—
117	C₄×#108	D₈₉	1	1	yes	characteristic
118	#1382	—	89	4	no	—
119	C₈₉	#13274	1	1	yes	characteristic, complemented, Sylow
120	#533	—	178	1	no	—
121	C₂²×C₁₆	—	89	2	no	—
122	#500	—	89	1	no	—
123	#500	—	89	1	no	—
124	#499	—	89	2	no	—
125	#499	—	178	1	no	—
126	#498	—	178	1	no	—
127	C₂×#108	C₂×D₈₉	1	2	yes	—
128	#427	—	89	2	no	—
129	#427	—	178	1	no	—
130	C₄×M₁₆	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
132	C₄⋊C₁₆	—	178	1	no	—
133	C₄⋊C₁₆	—	178	1	no	—
134	#341	—	2	1	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
138	C₄⋊Q₈	—	178	1	no	—
139	C₄⋊Q₈	—	178	1	no	—
140	#125	—	178	2	no	—
141	C₂²⋊Q₈	—	356	1	no	—
142	C₂²⋊Q₈	—	356	1	no	—
143	C₄×Q₈	—	356	1	no	—
144	C₄×Q₈	—	356	1	no	—
145	#117	—	178	2	no	—
146	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
147	C₂×C₁₆	—	178	2	no	—
148	C₂×C₁₆	—	178	4	no	—
149	#108	—	2	2	no	—
150	#108	Dic₁₇₈	1	4	yes	complemented
151	#107	—	89	2	no	—
152	#107	—	89	2	no	—
153	#107	—	178	1	no	—
154	#107	—	178	1	no	—
155	#106	—	178	2	no	—
156	#106	—	178	2	no	—
157	#97	—	4	1	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
160	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
161	C₄×C₈	D₃₅₆	1	1	yes	characteristic
162	C₄×C₈	D₃₅₆	1	1	yes	characteristic
163	C₂×Q₈	—	356	1	no	—
164	C₂×Q₈	—	356	1	no	—
165	C₂²×C₄	#13083561	1	1	yes	characteristic
166	C₂²×C₄	#13083561	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
168	M₁₆	—	2	4	no	—
169	M₁₆	—	4	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	#13083551	1	1	yes	characteristic
173	C₂×C₈	#13083551	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
175	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
176	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
177	C₂×C₈	#13083574	1	2	yes	—
178	C₄⋊C₄	—	356	2	no	—
179	C₄⋊C₄	—	356	2	no	—
180	C₄⋊C₄	—	356	1	no	—
181	C₄⋊C₄	—	356	1	no	—
182	C₄⋊C₄	—	178	2	no	—
183	C₄⋊C₄	—	178	2	no	—
184	C₂²⋊C₄	—	356	2	no	—
185	C₄²	—	356	2	no	—
186	C₄²	#13083551	1	1	yes	characteristic
187	C₄²	#13083551	1	1	yes	characteristic
188	C₄²	#13083559	1	1	yes	characteristic
189	C₄²	#13083559	1	1	yes	characteristic
190	C₁₆	—	356	2	no	—
191	C₂³	C₂×#13083561	1	1	yes	characteristic
192	Q₈	—	712	2	no	—
193	Q₈	—	712	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	356	2	no	—
197	C₂×C₄	—	356	2	no	—
198	C₂×C₄	#15667932	1	1	yes	characteristic
199	C₂×C₄	#15667932	1	1	yes	characteristic
200	C₂×C₄	#15667932	1	1	yes	characteristic
201	C₂×C₄	#15667932	1	1	yes	characteristic
202	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
203	C₂×C₄	#15667991	1	2	yes	—
204	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
205	C₂×C₄	#15668042	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	4	no	—
209	C₈	C₄×Dic₁₇₈	1	2	yes	—
210	C₈	#15668000	1	1	yes	characteristic
211	C₈	#15668012	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#15667932	1	1	yes	characteristic
215	C₂²	#19151883	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	712	2	no	—
220	C₄	—	712	2	no	—
221	C₄	#19151404	1	2	yes	—
222	C₄	#19151504	1	1	yes	characteristic
223	C₄	#19151606	1	1	yes	characteristic
224	C₄	#19151878	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#26021204	1	1	yes	characteristic
227	C₂	C₂×#19151404	1	1	yes	characteristic
228	C₂	#26021979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×#108×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×#108	D₈₉
hypocenter	C₈₉	#13274
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
41. #108×C₈₉	Q₈	semidirect
119. C₈₉	#13274	semidirect
150. #108	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
C₂×C₄⋊#13083549	2
#26018349	2
C₄×#108×C₈₉	2
#13274	89