Subgroups — #36445565

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15667941	C₂	1	2	yes	maximal
2	C₄×#15667922	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
4	#26018079	C₂	1	2	yes	maximal
5	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667941	C₄	1	4	yes	—
7	C₂×#15667922	C₄	1	2	yes	—
8	C₂×#15667922	C₄	1	2	yes	—
9	C₂×C₄×#10623494	C₂²	1	2	yes	—
10	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
11	C₄²×Dic₈₉	C₂²	1	1	yes	characteristic
12	#19151352	C₄	1	4	yes	—
13	C₈×#10623494	—	2	2	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#322×C₈₉	—	2	2	no	—
17	C₂²×#10623494	C₂×C₄	1	2	yes	—
18	C₂²×#10623494	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
23	#15667941	—	2	8	no	—
24	#15667922	—	2	4	no	—
25	#15667922	C₈	1	8	yes	complemented
26	C₄×#10623494	—	2	2	no	—
27	C₄×#10623494	—	2	2	no	—
28	C₄×#10623494	—	2	2	no	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₈₉	—	2	2	no	—
33	C₄×C₈×C₈₉	—	2	2	no	—
34	C₄×C₈×C₈₉	D₄	1	2	yes	—
35	C₄×C₈×C₈₉	Q₈	1	2	yes	—
36	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
37	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
38	C₂²×Dic₈₉	C₄²	1	2	yes	—
39	C₂×#10623494	—	2	2	no	—
40	C₂×#10623494	—	2	4	no	—
41	C₂×#10623494	—	2	2	no	—
42	C₂×#10623494	—	2	4	no	—
43	C₂×#10623494	C₂×C₈	1	4	yes	—
44	C₂×#10623494	M₁₆	1	8	yes	—
45	C₄×Dic₈₉	—	2	2	no	—
46	C₄×Dic₈₉	—	2	4	no	—
47	C₄×Dic₈₉	—	2	4	no	—
48	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₈₉	—	2	4	no	—
52	C₂×C₈×C₈₉	—	2	4	no	—
53	C₂×C₈×C₈₉	—	2	4	no	—
54	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
56	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
57	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₈₉	#41	1	2	yes	—
60	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
67	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₈₉	#131	1	4	yes	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	—	2	2	no	—
71	C₂×Dic₈₉	—	2	1	no	—
72	C₂×Dic₈₉	—	2	1	no	—
73	C₂×Dic₈₉	—	2	2	no	—
74	C₂×Dic₈₉	—	2	2	no	—
75	C₂×Dic₈₉	C₄×C₈	1	4	yes	—
76	C₈×C₈₉	—	4	4	no	—
77	C₈×C₈₉	C₄⋊C₈	1	8	yes	complemented
78	#10623494	—	2	4	no	—
79	#10623494	—	2	8	no	—
80	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
82	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
83	C₂²×C₈₉	C₄×M₁₆	1	1	yes	characteristic
84	C₂²×C₈₉	#405	1	2	yes	—
85	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
86	C₄×C₈₉	#431	1	2	yes	—
87	C₄×C₈₉	#435	1	4	yes	—
88	C₄×C₈₉	#446	1	4	yes	—
89	Dic₈₉	—	2	4	no	—
90	Dic₈₉	—	4	1	no	—
91	Dic₈₉	—	4	1	no	—
92	#8262	—	89	1	no	maximal, Sylow
93	C₂×C₈₉	C₈×M₁₆	1	2	yes	—
94	C₂×C₈₉	#1453	1	1	yes	characteristic
95	C₂×C₈₉	#1455	1	1	yes	characteristic
96	C₂×C₈₉	C₄×C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₈₉	#1771	1	2	yes	—
98	C₄×C₄⋊C₈	—	89	2	no	—
99	C₄×#97	—	89	1	no	—
100	C₂×#322	D₈₉	1	1	yes	characteristic
101	C₂×C₈²	—	89	1	no	—
102	#1273	—	89	2	no	—
103	C₈₉	#8262	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₄⋊C₈	—	89	4	no	—
105	C₂×#97	—	89	2	no	—
106	C₂×#97	—	89	2	no	—
107	C₂×C₄×C₈	—	89	2	no	—
108	C₂×C₄×C₈	—	89	1	no	—
109	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
111	C₄³	—	89	1	no	—
112	#336	—	89	4	no	—
113	#322	—	2	2	no	—
114	C₈²	—	178	2	no	—
115	C₂²×C₈	—	89	4	no	—
116	C₂²×C₈	—	89	2	no	—
117	C₂²×C₈	C₄×D₈₉	1	4	yes	—
118	C₂²×C₈	C₄×D₈₉	1	2	yes	—
119	C₂×C₄²	—	89	1	no	—
120	C₂×C₄²	—	89	2	no	—
121	C₂×C₄²	—	89	1	no	—
122	C₂×C₄²	—	89	2	no	—
123	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
124	C₄⋊C₈	—	178	8	no	—
125	#97	—	89	8	no	—
126	#97	—	178	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	178	2	no	—
130	C₄×C₈	—	178	2	no	—
131	C₄×C₈	—	178	2	no	—
132	C₄×C₈	Dic₁₇₈	1	2	yes	—
133	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
134	C₂²×C₄	—	89	2	no	—
135	C₂²×C₄	—	89	1	no	—
136	C₂²×C₄	—	89	1	no	—
137	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	89	4	no	—
143	C₂×C₈	—	89	8	no	—
144	C₂×C₈	—	178	2	no	—
145	C₂×C₈	—	178	4	no	—
146	C₂×C₈	—	178	2	no	—
147	C₂×C₈	—	178	4	no	—
148	C₂×C₈	#13083553	1	4	yes	—
149	C₂×C₈	C₈×D₈₉	1	4	yes	—
150	C₂×C₈	#13083557	1	4	yes	—
151	C₄²	—	89	4	no	—
152	C₄²	—	178	4	no	—
153	C₄²	—	178	2	no	—
154	C₄²	—	178	4	no	—
155	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
156	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
157	C₄²	#13083574	1	2	yes	—
158	C₂³	C₄²×D₈₉	1	1	yes	characteristic
159	C₂×C₄	—	89	4	no	—
160	C₂×C₄	—	178	1	no	—
161	C₂×C₄	—	178	1	no	—
162	C₂×C₄	—	178	2	no	—
163	C₂×C₄	—	178	2	no	—
164	C₂×C₄	—	178	2	no	—
165	C₂×C₄	—	178	2	no	—
166	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
167	C₂×C₄	#15667969	1	1	yes	characteristic
168	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
169	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
170	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
171	C₂×C₄	M₁₆×D₈₉	1	4	yes	—
172	C₂×C₄	C₂×#13083557	1	2	yes	—
173	C₂×C₄	#15668020	1	4	yes	—
174	C₈	—	4	4	no	—
175	C₈	—	178	4	no	—
176	C₈	—	178	8	no	—
177	C₈	#15667941	1	8	yes	complemented
178	C₂²	C₄×#13083553	1	1	yes	characteristic
179	C₂²	C₄×C₈×D₈₉	1	1	yes	characteristic
180	C₂²	C₄×#13083557	1	1	yes	characteristic
181	C₂²	#19151550	1	2	yes	—
182	C₂²	#19151555	1	2	yes	—
183	C₄	—	356	1	no	—
184	C₄	—	356	1	no	—
185	C₄	—	178	4	no	—
186	C₄	C₂×#15667941	1	2	yes	—
187	C₄	#19151649	1	2	yes	—
188	C₄	#19151667	1	4	yes	—
189	C₄	#19151726	1	4	yes	—
190	C₂	#26018531	1	2	yes	—
191	C₂	#26018550	1	1	yes	characteristic
192	C₂	#26018556	1	1	yes	characteristic
193	C₂	C₄×#15667941	1	1	yes	characteristic
194	C₂	#26020128	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄×C₈
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#8262
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
25. #15667922	C₈	semidirect
77. C₈×C₈₉	C₄⋊C₈	semidirect
103. C₈₉	#8262	semidirect
177. C₈	#15667941	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15667941	2
C₄×#15667922	2
C₂×C₈×#10623494	2
#26018079	2
C₂×#322×C₈₉	2
#8262	89