Subgroups — #36472954

Order 22784.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 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	C₄×#15667952	C₂	1	1	yes	maximal, characteristic
3	#26018340	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19152155	C₄	1	1	yes	characteristic
6	C₂×#15667952	C₂²	1	2	yes	—
7	#19151728	—	2	1	no	—
8	#19151728	—	2	1	no	—
9	C₈×Dic₁₇₈	—	2	1	no	—
10	C₈×Dic₁₇₈	—	2	1	no	—
11	#19151624	C₄	1	2	yes	—
12	#19151622	C₄	1	1	yes	characteristic
13	#19151553	C₂²	1	2	yes	—
14	#19151374	—	2	1	no	—
15	C₄×#13083549	—	2	1	no	—
16	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₈₉	—	2	1	no	—
19	C₄⋊C₁₆×C₈₉	—	2	1	no	—
20	C₂×#13083558	C₂×C₄	1	2	yes	—
21	#15668012	C₈	1	1	yes	characteristic
22	#15668012	C₈	1	1	yes	characteristic
23	#15668004	C₈	1	2	yes	—
24	#15667991	—	2	1	no	—
25	#15667991	—	2	1	no	—
26	C₄×Dic₁₇₈	—	2	1	no	—
27	C₄×Dic₁₇₈	—	2	1	no	—
28	#15667971	C₂×C₄	1	2	yes	—
29	#15667952	—	2	2	no	—
30	#15667952	Q₈	1	4	yes	complemented
31	C₂×#13083549	—	2	2	no	—
32	C₄⋊#10623494	—	2	1	no	—
33	C₄⋊#10623494	—	2	1	no	—
34	#15667941	—	2	2	no	—
35	#15667941	—	2	2	no	—
36	#15667922	—	2	2	no	—
37	C₈×Dic₈₉	—	2	2	no	—
38	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₈₉	—	2	2	no	—
42	C₂×C₁₆×C₈₉	—	2	4	no	—
43	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂×Dic₁₇₈	—	2	1	no	—
48	C₂×Dic₁₇₈	—	2	1	no	—
49	#13083558	—	2	4	no	—
50	C₂×#10623494	—	2	2	no	—
51	C₄⋊Dic₈₉	—	2	1	no	—
52	C₄⋊Dic₈₉	—	2	1	no	—
53	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
54	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
55	#13083553	—	2	2	no	—
56	#13083553	—	2	2	no	—
57	C₂²⋊Dic₈₉	—	2	2	no	—
58	C₄×Dic₈₉	—	2	2	no	—
59	#13083549	—	2	4	no	—
60	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂²×C₄	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	C₂×D₄	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	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
74	C₁₆×C₈₉	—	4	2	no	—
75	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₉	—	2	2	no	—
77	C₂×C₄×C₈₉	—	4	1	no	—
78	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
84	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#131	1	2	yes	—
86	C₂×Dic₈₉	—	2	2	no	—
87	C₂×Dic₈₉	—	2	2	no	—
88	Dic₁₇₈	—	4	2	no	—
89	Dic₁₇₈	—	4	2	no	—
90	C₈×C₈₉	—	2	1	no	—
91	C₈×C₈₉	—	2	1	no	—
92	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
93	C₈×C₈₉	#124	1	1	yes	characteristic
94	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
95	#10623494	—	4	2	no	—
96	C₂²×C₈₉	—	2	1	no	—
97	C₂²×C₈₉	—	2	1	no	—
98	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₂²×C₈₉	#446	1	2	yes	—
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	#350	1	2	yes	—
104	C₄×C₈₉	#391	1	1	yes	characteristic
105	C₄×C₈₉	#408	1	1	yes	characteristic
106	C₄×C₈₉	C₈×Q₈	1	2	yes	—
107	Dic₈₉	—	4	2	no	—
108	Dic₈₉	—	4	2	no	—
109	#13265	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	—	2	2	no	—
111	C₂×C₈₉	#1984	1	1	yes	characteristic
112	C₂×C₈₉	C₂×#350	1	1	yes	characteristic
113	C₂×C₈₉	#2117	1	1	yes	characteristic
114	#2960	—	89	1	no	—
115	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
116	C₄×M₃₂	—	89	1	no	—
117	#1379	—	89	4	no	—
118	C₈₉	#13265	1	1	yes	characteristic, complemented, Sylow
119	#533	—	89	1	no	—
120	C₂×M₃₂	—	89	2	no	—
121	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
122	#446	—	178	1	no	—
123	#446	—	178	1	no	—
124	C₈×Q₈	—	178	1	no	—
125	C₈×Q₈	—	178	1	no	—
126	#424	—	89	2	no	—
127	#423	—	89	1	no	—
128	#405	—	89	2	no	—
129	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
130	C₄⋊C₁₆	—	2	1	no	—
131	C₄⋊C₁₆	—	2	1	no	—
132	#346	—	178	1	no	—
133	C₄×C₁₆	—	178	1	no	—
134	C₂×M₁₆	—	89	2	no	—
135	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
137	C₄⋊Q₈	—	89	1	no	—
138	C₄⋊Q₈	—	89	1	no	—
139	#125	—	89	2	no	—
140	C₂²⋊Q₈	—	178	1	no	—
141	C₂²⋊Q₈	—	178	1	no	—
142	C₄×Q₈	—	178	1	no	—
143	C₄×Q₈	—	178	1	no	—
144	#117	—	89	2	no	—
145	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
146	M₃₂	—	89	4	no	—
147	M₃₂	—	178	2	no	—
148	C₂×C₁₆	—	2	2	no	—
149	C₂×C₁₆	—	2	4	no	—
150	C₂×C₁₆	—	178	2	no	—
151	C₄⋊C₈	—	178	2	no	—
152	C₄⋊C₈	—	178	2	no	—
153	C₄⋊C₈	—	178	1	no	—
154	C₄⋊C₈	—	178	1	no	—
155	#97	—	178	2	no	—
156	C₄×C₈	—	178	2	no	—
157	C₄×C₈	D₃₅₆	1	1	yes	characteristic
158	C₄×C₈	D₃₅₆	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
161	C₂×Q₈	—	178	1	no	—
162	C₂×Q₈	—	178	1	no	—
163	C₂²×C₄	#13083557	1	1	yes	characteristic
164	C₂²×C₄	#13083557	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
166	M₁₆	—	178	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	178	2	no	—
169	C₂×C₈	#13083551	1	1	yes	characteristic
170	C₂×C₈	#13083551	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
172	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
173	C₂×C₈	Q₈×D₈₉	1	2	yes	—
174	C₂×C₈	#13083573	1	2	yes	—
175	C₄⋊C₄	—	89	2	no	—
176	C₄⋊C₄	—	89	2	no	—
177	C₄⋊C₄	—	178	2	no	—
178	C₄⋊C₄	—	178	2	no	—
179	C₄⋊C₄	—	178	1	no	—
180	C₄⋊C₄	—	178	1	no	—
181	C₂²⋊C₄	—	178	2	no	—
182	C₄²	—	178	2	no	—
183	C₄²	#13083551	1	1	yes	characteristic
184	C₄²	#13083551	1	1	yes	characteristic
185	C₄²	C₈×D₈₉	1	1	yes	characteristic
186	C₄²	C₈×D₈₉	1	1	yes	characteristic
187	C₁₆	—	4	2	no	—
188	C₁₆	—	178	4	no	—
189	C₂³	C₂×#13083557	1	1	yes	characteristic
190	Q₈	—	356	2	no	—
191	Q₈	—	356	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	178	2	no	—
195	C₂×C₄	—	178	2	no	—
196	C₂×C₄	#15667924	1	1	yes	characteristic
197	C₂×C₄	#15667924	1	1	yes	characteristic
198	C₂×C₄	#15667924	1	1	yes	characteristic
199	C₂×C₄	#15667924	1	1	yes	characteristic
200	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
201	C₂×C₄	#15667977	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
203	C₂×C₄	#15668020	1	2	yes	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	356	2	no	—
207	C₈	#15667992	1	2	yes	—
208	C₈	#15667999	1	1	yes	characteristic
209	C₈	#15668010	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×#15667924	1	1	yes	characteristic
213	C₂²	#19151725	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	356	2	no	—
218	C₄	—	356	2	no	—
219	C₄	#19151381	1	2	yes	—
220	C₄	#19151503	1	1	yes	characteristic
221	C₄	#19151566	1	1	yes	characteristic
222	C₄	#19151722	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	#26021187	1	1	yes	characteristic
225	C₂	C₂×#19151381	1	1	yes	characteristic
226	C₂	#26021879	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#13265
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
30. #15667952	Q₈	semidirect
118. C₈₉	#13265	semidirect

Maximal subgroups

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

subgroup	index
#26029186	2
C₄×#15667952	2
#26018340	2
C₂×C₄⋊C₁₆×C₈₉	2
#13265	89