Subgroups — #36707923

Order 22784.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022161	C₂	1	4	yes	maximal, complemented
2	Q₈×#13083549	C₂	1	4	yes	maximal, complemented
3	#26022011	C₂	1	2	yes	maximal
4	#26022008	C₂	1	1	yes	maximal, characteristic
5	#26021849	C₂	1	2	yes	maximal
6	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2960	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667952	C₂²	1	2	yes	—
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	C₄⋊#13083549	C₂²	1	4	yes	—
12	C₄⋊#13083549	C₂²	1	8	yes	—
13	#19151374	C₂²	1	4	yes	—
14	C₄×#13083549	C₂²	1	2	yes	—
15	C₄×#13083549	C₂²	1	4	yes	—
16	C₄×#13083549	C₂²	1	2	yes	—
17	#533×C₈₉	C₄	1	1	yes	characteristic
18	#446×C₈₉	C₄	1	4	yes	—
19	C₈×Q₈×C₈₉	C₂²	1	4	yes	—
20	#424×C₈₉	C₄	1	2	yes	—
21	#423×C₈₉	C₄	1	1	yes	characteristic
22	#405×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#15667952	—	2	4	no	—
25	C₂×#13083549	—	2	2	no	—
26	C₂×#13083549	Q₈	1	4	yes	complemented
27	C₂×#13083549	C₂³	1	2	yes	—
28	C₂×#13083549	C₂³	1	4	yes	—
29	C₂×#13083549	C₂³	1	1	yes	characteristic
30	C₂×#13083549	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₈₉	C₈	1	2	yes	—
35	#125×C₈₉	C₈	1	2	yes	—
36	C₂²⋊Q₈×C₈₉	C₈	1	4	yes	—
37	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
38	#117×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
41	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
42	#97×C₈₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₈₉	C₂³	1	2	yes	—
45	C₄×C₈×C₈₉	C₂³	1	4	yes	—
46	#13083549	—	2	2	no	—
47	#13083549	—	2	4	no	—
48	#13083549	C₂×Q₈	1	2	yes	—
49	#13083549	C₂×Q₈	1	2	yes	—
50	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₈₉	—	2	4	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₈₉	#41	1	2	yes	—
60	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
62	C₄⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
63	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
64	C₂²⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
65	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
68	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
69	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
70	Q₈×C₈₉	—	2	8	no	—
71	C₂×C₄×C₈₉	—	2	4	no	—
72	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₈₉	#131	1	2	yes	—
78	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₈₉	—	2	2	no	—
80	C₈×C₈₉	—	2	4	no	—
81	C₈×C₈₉	C₄×Q₈	1	2	yes	—
82	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	C₈×Q₈	1	2	yes	—
86	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₈₉	—	2	2	no	—
88	C₄×C₈₉	—	2	8	no	—
89	C₄×C₈₉	C₈×Q₈	1	2	yes	—
90	C₄×C₈₉	#504	1	4	yes	—
91	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
93	#33474	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	—	2	2	no	—
95	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
96	C₂×C₈₉	C₂×#504	1	2	yes	—
97	#2960	D₈₉	1	1	yes	characteristic
98	#2183	—	89	4	no	—
99	Q₈×C₁₆	—	89	4	no	—
100	#2151	—	89	2	no	—
101	#2150	—	89	1	no	—
102	#2108	—	89	2	no	—
103	C₂×C₄×C₁₆	—	89	1	no	—
104	C₈₉	#33474	1	1	yes	characteristic, complemented, Sylow
105	#533	Dic₈₉	1	1	yes	characteristic
106	C₂×M₃₂	—	89	2	no	—
107	C₂²×C₁₆	—	89	1	no	—
108	C₂²×C₁₆	—	89	1	no	—
109	#446	Dic₈₉	1	4	yes	—
110	C₈×Q₈	C₂×D₈₉	1	4	yes	—
111	#424	Dic₈₉	1	2	yes	—
112	#423	Dic₈₉	1	1	yes	characteristic
113	#405	C₂×D₈₉	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	C₄⋊C₁₆	—	89	4	no	—
116	C₄⋊C₁₆	—	89	8	no	—
117	#346	—	89	4	no	—
118	C₄×C₁₆	—	89	2	no	—
119	C₄×C₁₆	—	89	2	no	—
120	C₄×C₁₆	—	89	4	no	—
121	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
122	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
124	C₄⋊Q₈	#10623494	1	2	yes	—
125	#125	#10623494	1	2	yes	—
126	C₂²⋊Q₈	#10623494	1	4	yes	—
127	C₄×Q₈	C₂×Dic₈₉	1	4	yes	—
128	#117	C₂×Dic₈₉	1	2	yes	—
129	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
130	M₃₂	—	178	4	no	—
131	C₂×C₁₆	—	89	4	no	—
132	C₂×C₁₆	—	89	1	no	—
133	C₂×C₁₆	—	89	1	no	—
134	C₂×C₁₆	—	89	2	no	—
135	C₂×C₁₆	—	89	4	no	—
136	C₂×C₁₆	—	178	2	no	—
137	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
138	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
139	#97	C₂×Dic₈₉	1	4	yes	—
140	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
141	C₄×C₈	C₂²×D₈₉	1	2	yes	—
142	C₄×C₈	C₂²×D₈₉	1	4	yes	—
143	C₂×Q₈	C₂×#10623494	1	4	yes	—
144	C₂²×C₄	C₂×#10623494	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
149	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
150	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₈₉	1	2	yes	—
152	C₂×C₈	#13083575	1	2	yes	—
153	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
154	C₄⋊C₄	C₂×#10623494	1	4	yes	—
155	C₄⋊C₄	C₂×#10623494	1	8	yes	—
156	C₄⋊C₄	C₂×#10623494	1	4	yes	—
157	C₂²⋊C₄	C₂×#10623494	1	4	yes	—
158	C₄²	C₂×#10623494	1	1	yes	characteristic
159	C₄²	C₂×#10623494	1	1	yes	characteristic
160	C₄²	C₂²×Dic₈₉	1	2	yes	—
161	C₄²	C₂²×Dic₈₉	1	4	yes	—
162	C₁₆	—	89	2	no	—
163	C₁₆	—	89	2	no	—
164	C₁₆	—	178	2	no	—
165	C₁₆	—	178	4	no	—
166	C₂³	C₂²×#10623494	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	Q₈×Dic₈₉	1	2	yes	—
170	C₂×C₄	C₂²×#10623494	1	2	yes	—
171	C₂×C₄	C₂²×#10623494	1	8	yes	—
172	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
173	C₂×C₄	C₂²×#10623494	1	2	yes	—
174	C₂×C₄	#15668019	1	2	yes	—
175	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	Q₈×Dic₈₉	1	2	yes	—
179	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
180	C₈	C₂×#13083575	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×#10623494	1	2	yes	—
183	C₂²	C₂³×#10623494	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×#10623494	1	2	yes	—
187	C₄	#19152015	1	4	yes	—
188	C₄	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
189	C₄	C₂×#15668019	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×#10623494	1	1	yes	characteristic
192	C₂	C₂×#19152015	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	#33474
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
2. Q₈×#13083549	C₂	semidirect
26. C₂×#13083549	Q₈	semidirect
104. C₈₉	#33474	semidirect

Maximal subgroups

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

subgroup	index
#26022161	2
Q₈×#13083549	2
#26022011	2
#26022008	2
#26021849	2
C₂×C₄×#13083549	2
C₈₉×#2960	2
#33474	89