Subgroups — #36708008

Order 22784.

548 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#13083549	C₂	1	2	yes	maximal, complemented
2	#26022084	C₂	1	6	yes	maximal, complemented
3	#26022055	C₂	1	3	yes	maximal, complemented
4	C₄×#15667952	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	6	yes	—
7	C₄⋊#13083549	C₂²	1	6	yes	—
8	C₂²⋊#13083549	C₂²	1	6	yes	—
9	#19151374	C₂²	1	3	yes	—
10	#19151374	C₂²	1	1	yes	characteristic
11	C₄×#13083549	C₂²	1	6	yes	—
12	C₄×#41×C₈₉	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic
14	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₈₉	C₄	1	3	yes	—
16	C₈×D₄×C₈₉	C₂²	1	3	yes	—
17	#431×C₈₉	C₄	1	3	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
19	#15667952	—	2	12	no	—
20	C₂×#13083549	C₂³	1	6	yes	—
21	C₂×#13083549	C₂³	1	2	yes	—
22	C₂×#41×C₈₉	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
25	C₄×Q₈×C₈₉	C₈	1	1	yes	characteristic
26	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₈₉	C₈	1	3	yes	—
28	C₄×D₄×C₈₉	C₂×C₄	1	3	yes	—
29	#117×C₈₉	C₈	1	3	yes	—
30	C₂×C₄²×C₈₉	C₂×C₄	1	3	yes	—
31	C₄⋊C₈×C₈₉	C₂×C₄	1	3	yes	—
32	C₄⋊C₈×C₈₉	C₂×C₄	1	3	yes	—
33	C₂²⋊C₈×C₈₉	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₈₉	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₈₉	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₈₉	C₂³	1	3	yes	—
37	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
38	#13083549	—	2	6	no	—
39	#13083549	#41	1	4	yes	complemented
40	#41×C₈₉	—	2	4	no	—
41	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
42	C₂×D₄×C₈₉	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₈₉	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₈₉	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₈₉	—	2	6	no	—
47	C₂×C₈×C₈₉	—	2	6	no	—
48	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄×C₈₉	C₂×C₈	1	3	yes	—
53	C₄⋊C₄×C₈₉	C₂×C₈	1	3	yes	—
54	C₂²⋊C₄×C₈₉	C₂×C₈	1	6	yes	—
55	C₄²×C₈₉	C₂×C₈	1	3	yes	—
56	C₄²×C₈₉	C₂×C₈	1	3	yes	—
57	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
59	C₂³×C₈₉	C₂²×C₈	1	3	yes	—
60	Q₈×C₈₉	M₃₂	1	4	yes	complemented
61	D₄×C₈₉	—	2	6	no	—
62	C₂×C₄×C₈₉	—	2	3	no	—
63	C₂×C₄×C₈₉	—	2	3	no	—
64	C₂×C₄×C₈₉	—	2	6	no	—
65	C₂×C₄×C₈₉	—	4	3	no	—
66	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₈₉	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₈₉	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₈₉	—	2	3	no	—
73	C₈×C₈₉	—	2	3	no	—
74	C₈×C₈₉	#131	1	2	yes	—
75	C₈×C₈₉	C₂×#41	1	2	yes	—
76	C₂²×C₈₉	—	2	3	no	—
77	C₂²×C₈₉	—	2	3	no	—
78	C₂²×C₈₉	—	2	3	no	—
79	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₈₉	—	2	1	no	—
81	C₄×C₈₉	—	2	1	no	—
82	C₄×C₈₉	—	2	3	no	—
83	C₄×C₈₉	—	2	3	no	—
84	C₄×C₈₉	C₂×M₃₂	1	6	yes	—
85	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
86	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
87	#33485	—	89	1	no	maximal, Sylow
88	C₂×C₈₉	—	4	3	no	—
89	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
90	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
91	C₂×C₈₉	#3407	1	1	yes	characteristic
92	C₈×#41	D₈₉	1	1	yes	characteristic
93	Q₈×C₁₆	—	89	2	no	—
94	#2169	—	89	6	no	—
95	#2163	—	89	3	no	—
96	C₄×M₃₂	—	89	3	no	—
97	C₈₉	#33485	1	1	yes	characteristic, complemented, Sylow
98	C₄×#41	Dic₈₉	1	1	yes	characteristic
99	C₂×M₃₂	—	89	6	no	—
100	C₈×Q₈	Dic₈₉	1	1	yes	characteristic
101	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
102	C₈×D₄	Dic₈₉	1	3	yes	—
103	C₈×D₄	C₂×D₈₉	1	3	yes	—
104	#431	Dic₈₉	1	3	yes	—
105	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
106	C₄⋊C₁₆	—	89	6	no	—
107	C₂²⋊C₁₆	—	89	6	no	—
108	#346	—	89	1	no	—
109	#346	—	89	3	no	—
110	C₄×C₁₆	—	89	6	no	—
111	C₂×#41	#10623494	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₈₉	1	3	yes	—
113	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
114	C₄×Q₈	#10623494	1	1	yes	characteristic
115	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
116	C₄×D₄	#10623494	1	3	yes	—
117	C₄×D₄	C₂×Dic₈₉	1	3	yes	—
118	#117	#10623494	1	3	yes	—
119	C₂×C₄²	C₂×Dic₈₉	1	3	yes	—
120	M₃₂	—	178	12	no	—
121	C₂×C₁₆	—	89	2	no	—
122	C₂×C₁₆	—	89	6	no	—
123	C₄⋊C₈	C₂×Dic₈₉	1	3	yes	—
124	C₄⋊C₈	C₂×Dic₈₉	1	3	yes	—
125	C₂²⋊C₈	C₂×Dic₈₉	1	6	yes	—
126	C₄×C₈	C₂×Dic₈₉	1	3	yes	—
127	C₄×C₈	C₂×Dic₈₉	1	3	yes	—
128	C₄×C₈	C₂²×D₈₉	1	3	yes	—
129	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
130	#41	—	2	4	no	—
131	C₂×Q₈	C₂×#10623494	1	1	yes	characteristic
132	C₂×D₄	C₂×#10623494	1	3	yes	—
133	C₂²×C₄	C₂×#10623494	1	3	yes	—
134	C₂²×C₄	C₂×#10623494	1	3	yes	—
135	C₂²×C₄	C₂²×Dic₈₉	1	3	yes	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	2	6	no	—
138	C₂×C₈	C₂²×Dic₈₉	1	3	yes	—
139	C₂×C₈	C₂²×Dic₈₉	1	3	yes	—
140	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
141	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
142	C₄⋊C₄	C₂×#10623494	1	3	yes	—
143	C₄⋊C₄	C₂×#10623494	1	3	yes	—
144	C₂²⋊C₄	C₂×#10623494	1	6	yes	—
145	C₄²	C₂×#10623494	1	3	yes	—
146	C₄²	C₂×#10623494	1	3	yes	—
147	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
148	C₄²	C₂²×Dic₈₉	1	3	yes	—
149	C₁₆	—	89	4	no	—
150	C₁₆	—	178	6	no	—
151	C₂³	C₂²×#10623494	1	3	yes	—
152	Q₈	#15667952	1	4	yes	complemented
153	D₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	—	4	3	no	—
158	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
159	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
160	C₂×C₄	C₂²×#10623494	1	3	yes	—
161	C₂×C₄	C₂²×#10623494	1	3	yes	—
162	C₂×C₄	C₂²×#10623494	1	3	yes	—
163	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	2	3	no	—
166	C₈	#15668019	1	2	yes	—
167	C₈	#41×D₈₉	1	2	yes	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	—	2	3	no	—
171	C₂²	C₂³×#10623494	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	3	no	—
176	C₄	C₂×#15667952	1	6	yes	—
177	C₄	#41×Dic₈₉	1	1	yes	characteristic
178	C₄	C₂×#15668019	1	1	yes	characteristic
179	C₂	—	4	3	no	—
180	C₂	#41×#10623494	1	1	yes	characteristic
181	C₂	C₂²×#15667952	1	1	yes	characteristic
182	C₂	#26033355	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈×#41×C₈₉	C₂
prime radical	C₂×#41×C₈₉	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₈×#41	D₈₉
hypocenter	C₈₉	#33485
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#13083549	C₂	semidirect
2. #26022084	C₂	semidirect
3. #26022055	C₂	semidirect
4. C₄×#15667952	C₂	semidirect
39. #13083549	#41	semidirect
60. Q₈×C₈₉	M₃₂	semidirect
97. C₈₉	#33485	semidirect
152. Q₈	#15667952	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#13083549	2
#26022084	2
#26022055	2
C₄×#15667952	2
C₈×#41×C₈₉	2
#33485	89