Subgroups — Q₈×#15667952 #36707942

Order 22784.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	Q₈×M₃₂
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

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
20. #15667952	Q₈	direct (N × H)
40. #13083549	C₂×Q₈	semidirect
63. Q₈×C₈₉	M₃₂	semidirect
100. C₈₉	Q₈×M₃₂	semidirect
158. Q₈	#15667952	direct (N × H)

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
#26022008	2
C₄×#15667952	2
C₂×C₈×Q₈×C₈₉	2
Q₈×M₃₂	89

Subgroups — Q8×#15667952 #36707942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#15667952 #36707942