Subgroups — #36464256

Order 22784.

322 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029186	C₂	1	1	yes	maximal, characteristic
2	#26022008	C₂	1	1	yes	maximal, characteristic
3	#26018235	C₂	1	4	yes	maximal, complemented
4	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19152155	C₄	1	1	yes	characteristic
6	C₂×#15667952	—	2	1	no	—
7	#19151728	—	2	1	no	—
8	#19151728	C₂²	1	2	yes	—
9	C₈×Dic₁₇₈	—	2	1	no	—
10	C₈×Dic₁₇₈	C₄	1	2	yes	—
11	#19151624	—	2	1	no	—
12	#19151622	C₄	1	1	yes	characteristic
13	#19151553	—	2	1	no	—
14	C₄⋊#13083549	—	2	1	no	—
15	C₄⋊#13083549	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₈₉	—	2	1	no	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#346×C₈₉	—	2	1	no	—
19	#346×C₈₉	C₂²	1	2	yes	—
20	C₂×#13083558	—	2	1	no	—
21	#15668012	C₈	1	1	yes	characteristic
22	#15668012	C₈	1	1	yes	characteristic
23	#15668004	—	2	1	no	—
24	#15667991	—	2	1	no	—
25	#15667991	C₈	1	2	yes	—
26	C₄×Dic₁₇₈	—	2	1	no	—
27	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
28	#15667971	—	2	1	no	—
29	#15667952	—	4	2	no	—
30	C₂×#13083549	—	2	2	no	—
31	C₄⋊#10623494	—	2	1	no	—
32	C₄⋊#10623494	C₂×C₄	1	2	yes	—
33	#15667941	—	2	2	no	—
34	#15667941	—	2	2	no	—
35	#15667922	—	2	2	no	—
36	C₈×Dic₈₉	—	2	2	no	—
37	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₈₉	—	2	4	no	—
41	C₂×C₁₆×C₈₉	—	2	2	no	—
42	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₇₈	—	2	1	no	—
47	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
48	#13083558	—	4	2	no	—
49	C₂×#10623494	—	2	2	no	—
50	C₄⋊Dic₈₉	—	2	1	no	—
51	C₄⋊Dic₈₉	—	2	1	no	—
52	C₄⋊Dic₈₉	—	2	1	no	—
53	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
54	#13083553	—	2	2	no	—
55	#13083553	—	2	2	no	—
56	C₂²⋊Dic₈₉	—	2	2	no	—
57	C₄×Dic₈₉	—	2	2	no	—
58	#13083549	—	4	2	no	—
59	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
66	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
72	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₁₆×C₈₉	—	2	4	no	—
74	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₈₉	—	4	1	no	—
76	C₂×C₄×C₈₉	—	4	1	no	—
77	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#103	1	2	yes	—
81	C₂×C₄×C₈₉	#104	1	2	yes	—
82	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₈₉	—	2	2	no	—
86	C₂×Dic₈₉	—	2	2	no	—
87	Dic₁₇₈	—	4	2	no	—
88	Dic₁₇₈	—	4	2	no	—
89	C₈×C₈₉	—	2	1	no	—
90	C₈×C₈₉	—	2	1	no	—
91	C₈×C₈₉	#103	1	2	yes	—
92	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
94	#10623494	—	4	2	no	—
95	C₂²×C₈₉	—	2	1	no	—
96	C₂²×C₈₉	—	2	1	no	—
97	C₂²×C₈₉	Q₈⋊C₈	1	2	yes	—
98	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	Q₈⋊C₈	1	2	yes	—
104	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
105	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
106	Dic₈₉	—	4	2	no	—
107	Dic₈₉	—	4	2	no	—
108	#11612	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	—	2	2	no	—
110	C₂×C₈₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
111	C₂×C₈₉	#2115	1	1	yes	characteristic
112	C₂×C₈₉	#2117	1	1	yes	characteristic
113	#2960	—	89	1	no	—
114	#2150	—	89	1	no	—
115	C₂×#346	D₈₉	1	1	yes	characteristic
116	#1338	—	89	4	no	—
117	C₈₉	#11612	1	1	yes	characteristic, complemented, Sylow
118	#533	—	89	1	no	—
119	C₂×M₃₂	—	178	1	no	—
120	C₂²×C₁₆	—	2	1	no	—
121	#446	—	89	2	no	—
122	#446	—	178	1	no	—
123	C₈×Q₈	—	89	2	no	—
124	C₈×Q₈	—	178	1	no	—
125	#424	—	178	1	no	—
126	#423	—	89	1	no	—
127	#405	—	178	1	no	—
128	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
129	C₄⋊C₁₆	—	89	2	no	—
130	C₄⋊C₁₆	—	178	1	no	—
131	#346	—	2	1	no	—
132	#346	C₂×D₈₉	1	2	yes	—
133	C₂×M₁₆	—	178	1	no	—
134	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	89	1	no	—
137	C₄⋊Q₈	—	89	1	no	—
138	#125	—	178	1	no	—
139	C₂²⋊Q₈	—	89	2	no	—
140	C₂²⋊Q₈	—	178	1	no	—
141	C₄×Q₈	—	89	2	no	—
142	C₄×Q₈	—	178	1	no	—
143	#117	—	178	1	no	—
144	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
145	M₃₂	—	356	2	no	—
146	C₂×C₁₆	—	2	4	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	178	2	no	—
149	C₄⋊C₈	—	89	2	no	—
150	C₄⋊C₈	—	178	2	no	—
151	C₄⋊C₈	—	178	2	no	—
152	C₄⋊C₈	—	178	1	no	—
153	#97	—	178	2	no	—
154	C₄×C₈	—	178	2	no	—
155	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
156	C₄×C₈	D₃₅₆	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
159	C₂×Q₈	—	89	2	no	—
160	C₂×Q₈	—	178	1	no	—
161	C₂²×C₄	#13083551	1	1	yes	characteristic
162	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
163	C₂²×C₄	#13083557	1	1	yes	characteristic
164	M₁₆	—	356	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	178	2	no	—
167	C₂×C₈	#13083551	1	1	yes	characteristic
168	C₂×C₈	#13083551	1	1	yes	characteristic
169	C₂×C₈	#13083563	1	2	yes	—
170	C₂×C₈	Dic₃₅₆	1	2	yes	—
171	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
172	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
173	C₄⋊C₄	—	89	2	no	—
174	C₄⋊C₄	—	178	2	no	—
175	C₄⋊C₄	—	178	1	no	—
176	C₄⋊C₄	—	178	1	no	—
177	C₄⋊C₄	—	178	2	no	—
178	C₄⋊C₄	—	178	1	no	—
179	C₂²⋊C₄	—	178	2	no	—
180	C₄²	—	178	2	no	—
181	C₄²	#13083551	1	1	yes	characteristic
182	C₄²	C₈×D₈₉	1	1	yes	characteristic
183	C₄²	#13083557	1	1	yes	characteristic
184	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₁₆	—	356	2	no	—
187	C₂³	#15667924	1	1	yes	characteristic
188	Q₈	—	356	2	no	—
189	Q₈	—	356	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	178	2	no	—
193	C₂×C₄	—	178	2	no	—
194	C₂×C₄	#15667924	1	1	yes	characteristic
195	C₂×C₄	#15667924	1	1	yes	characteristic
196	C₂×C₄	#15667924	1	1	yes	characteristic
197	C₂×C₄	#15667937	1	2	yes	—
198	C₂×C₄	#15667940	1	2	yes	—
199	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
201	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	356	2	no	—
205	C₈	#15667937	1	2	yes	—
206	C₈	C₂×#13083563	1	1	yes	characteristic
207	C₈	C₂×Dic₃₅₆	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#19151330	1	2	yes	—
211	C₂²	C₂×#15667924	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	356	2	no	—
217	C₄	—	356	2	no	—
218	C₄	#19151330	1	2	yes	—
219	C₄	C₂×#15667937	1	1	yes	characteristic
220	C₄	C₂×#15667940	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#19151330	1	1	yes	characteristic
223	C₂	#26021872	1	1	yes	characteristic
224	C₂	#26021881	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₉	C₂²⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#11612
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #26018235	C₂	semidirect
117. C₈₉	#11612	semidirect

Maximal subgroups

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

subgroup	index
#26029186	2
#26022008	2
#26018235	2
C₂×#346×C₈₉	2
#11612	89