Subgroups — #36707929

Order 22784.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022161	C₂	1	8	yes	maximal, complemented
2	#26022011	C₂	1	2	yes	maximal
3	#26022008	C₂	1	1	yes	maximal, characteristic
4	#26021849	C₂	1	2	yes	maximal
5	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2960	C₂	1	1	yes	maximal, characteristic
7	C₂×#15667952	C₂²	1	2	yes	—
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₄⋊#13083549	C₂²	1	4	yes	—
11	C₄⋊#13083549	C₂²	1	8	yes	—
12	#19151374	C₂²	1	4	yes	—
13	#19151374	C₂²	1	4	yes	—
14	C₄×#13083549	C₂²	1	4	yes	—
15	#533×C₈₉	C₄	1	1	yes	characteristic
16	#446×C₈₉	C₄	1	4	yes	—
17	C₈×Q₈×C₈₉	C₂²	1	4	yes	—
18	#424×C₈₉	C₄	1	2	yes	—
19	#423×C₈₉	C₄	1	1	yes	characteristic
20	#405×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#15667952	—	2	4	no	—
23	C₂×#13083549	—	2	2	no	—
24	C₂×#13083549	Q₈	1	4	yes	complemented
25	C₂×#13083549	C₂³	1	4	yes	—
26	C₂×#13083549	C₂³	1	2	yes	—
27	C₂×#13083549	C₂³	1	2	yes	—
28	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₈₉	C₈	1	2	yes	—
32	#125×C₈₉	C₈	1	2	yes	—
33	C₂²⋊Q₈×C₈₉	C₈	1	4	yes	—
34	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
35	#117×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
38	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
39	#97×C₈₉	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₈₉	C₂³	1	4	yes	—
42	C₄×C₈×C₈₉	C₂³	1	2	yes	—
43	#13083549	—	2	2	no	—
44	#13083549	—	2	4	no	—
45	#13083549	C₂×Q₈	1	4	yes	—
46	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₈₉	—	2	4	no	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₈₉	#41	1	2	yes	—
56	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
58	C₄⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
60	C₂²⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
61	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
64	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
65	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₈₉	—	4	4	no	—
67	C₂×C₄×C₈₉	—	4	2	no	—
68	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₈₉	C₂²×C₈	1	8	yes	—
72	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₈₉	#131	1	2	yes	—
74	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
75	C₈×C₈₉	—	2	2	no	—
76	C₈×C₈₉	—	2	4	no	—
77	C₈×C₈₉	C₄×Q₈	1	2	yes	—
78	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
80	C₂²×C₈₉	—	2	2	no	—
81	C₂²×C₈₉	C₈×Q₈	1	2	yes	—
82	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₈₉	—	2	2	no	—
84	C₄×C₈₉	—	2	2	no	—
85	C₄×C₈₉	—	2	8	no	—
86	C₄×C₈₉	C₈×Q₈	1	2	yes	—
87	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
88	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
89	#33475	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	—	2	2	no	—
91	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₈₉	#3407	1	2	yes	—
93	#2960	D₈₉	1	1	yes	characteristic
94	#2183	—	89	8	no	—
95	#2151	—	89	2	no	—
96	#2150	—	89	1	no	—
97	#2108	—	89	2	no	—
98	C₂×#346	—	89	1	no	—
99	C₈₉	#33475	1	1	yes	characteristic, complemented, Sylow
100	#533	Dic₈₉	1	1	yes	characteristic
101	C₂×M₃₂	—	89	2	no	—
102	C₂²×C₁₆	—	89	1	no	—
103	C₂²×C₁₆	—	89	1	no	—
104	#446	Dic₈₉	1	4	yes	—
105	C₈×Q₈	C₂×D₈₉	1	4	yes	—
106	#424	Dic₈₉	1	2	yes	—
107	#423	Dic₈₉	1	1	yes	characteristic
108	#405	C₂×D₈₉	1	2	yes	—
109	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
110	C₄⋊C₁₆	—	89	4	no	—
111	C₄⋊C₁₆	—	89	8	no	—
112	#346	—	89	4	no	—
113	#346	—	89	4	no	—
114	C₄×C₁₆	—	89	4	no	—
115	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
116	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
118	C₄⋊Q₈	#10623494	1	2	yes	—
119	#125	#10623494	1	2	yes	—
120	C₂²⋊Q₈	#10623494	1	4	yes	—
121	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
122	#117	C₂×Dic₈₉	1	2	yes	—
123	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
124	M₃₂	—	178	4	no	—
125	C₂×C₁₆	—	89	4	no	—
126	C₂×C₁₆	—	89	2	no	—
127	C₂×C₁₆	—	89	4	no	—
128	C₂×C₁₆	—	89	2	no	—
129	C₂×C₁₆	—	178	2	no	—
130	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
131	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
132	#97	C₂×Dic₈₉	1	4	yes	—
133	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
134	C₄×C₈	C₂²×D₈₉	1	4	yes	—
135	C₄×C₈	C₂²×D₈₉	1	2	yes	—
136	C₂×Q₈	C₂×#10623494	1	4	yes	—
137	C₂²×C₄	C₂×#10623494	1	2	yes	—
138	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
139	M₁₆	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
142	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
143	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₈₉	1	2	yes	—
145	C₂×C₈	#13083575	1	2	yes	—
146	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
147	C₄⋊C₄	C₂×#10623494	1	4	yes	—
148	C₄⋊C₄	C₂×#10623494	1	8	yes	—
149	C₄⋊C₄	C₂×#10623494	1	4	yes	—
150	C₂²⋊C₄	C₂×#10623494	1	4	yes	—
151	C₄²	C₂×#10623494	1	1	yes	characteristic
152	C₄²	C₂×#10623494	1	1	yes	characteristic
153	C₄²	C₂²×Dic₈₉	1	2	yes	—
154	C₄²	C₂²×Dic₈₉	1	4	yes	—
155	C₁₆	—	89	4	no	—
156	C₁₆	—	178	2	no	—
157	C₁₆	—	178	4	no	—
158	C₂³	C₂²×#10623494	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
162	C₂×C₄	C₂²×#10623494	1	2	yes	—
163	C₂×C₄	C₂²×#10623494	1	2	yes	—
164	C₂×C₄	C₂²×#10623494	1	8	yes	—
165	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
166	C₂×C₄	#15668019	1	2	yes	—
167	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	4	no	—
170	C₈	Q₈×Dic₈₉	1	2	yes	—
171	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
172	C₈	C₂×#13083575	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×#10623494	1	2	yes	—
175	C₂²	C₂³×#10623494	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×#10623494	1	2	yes	—
180	C₄	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
181	C₄	C₂×#15668019	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×#10623494	1	1	yes	characteristic
184	C₂	#26033355	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2960	C₂
prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2960	D₈₉
hypocenter	C₈₉	#33475
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #26022161	C₂	semidirect
24. C₂×#13083549	Q₈	semidirect
99. C₈₉	#33475	semidirect

Maximal subgroups

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

subgroup	index
#26022161	2
#26022011	2
#26022008	2
#26021849	2
C₂×#19151374	2
C₈₉×#2960	2
#33475	89