Subgroups — C₈×C₄⋊#10623494 #36445549

Order 22784.

550 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10623494	C₂	1	2	yes	maximal
2	C₂×C₈×#10623494	C₂	1	2	yes	maximal
3	#26018079	C₂	1	2	yes	maximal
4	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₄	1	4	yes	—
6	C₂×C₄×#10623494	C₂²	1	4	yes	—
7	#19151353	C₄	1	4	yes	—
8	C₈×#10623494	—	2	4	no	—
9	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄³×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×#10623494	C₂×C₄	1	8	yes	—
15	C₄⋊#10623494	C₈	1	16	yes	direct product factor
16	C₄×#10623494	—	2	4	no	—
17	C₄×#10623494	—	2	4	no	—
18	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₈₉	—	2	4	no	—
26	C₄×C₈×C₈₉	—	2	2	no	—
27	C₄×C₈×C₈₉	C₈	1	8	yes	complemented
28	C₄×C₈×C₈₉	D₄	1	2	yes	—
29	C₄×C₈×C₈₉	Q₈	1	2	yes	—
30	C₂×#10623494	—	2	8	no	—
31	C₂×#10623494	—	2	8	no	—
32	C₂×#10623494	C₂×C₈	1	16	yes	—
33	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
34	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
38	C₂×C₈×C₈₉	—	2	2	no	—
39	C₂×C₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
41	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
44	C₄²×C₈₉	—	2	2	no	—
45	C₄²×C₈₉	—	2	4	no	—
46	C₄²×C₈₉	C₂×C₈	1	8	yes	—
47	C₄²×C₈₉	C₂×C₈	1	4	yes	—
48	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₈₉	#41	1	2	yes	—
51	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	2	2	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	C₄×C₈	1	4	yes	—
56	C₂×C₄×C₈₉	C₄×C₈	1	4	yes	—
57	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
64	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₈₉	#131	1	4	yes	—
67	C₈×C₈₉	—	2	4	no	—
68	C₈×C₈₉	C₄⋊C₈	1	8	yes	complemented
69	#10623494	—	2	16	no	—
70	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₈₉	C₂×C₄×C₈	1	2	yes	—
72	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
73	C₂²×C₈₉	C₄×M₁₆	1	1	yes	characteristic
74	C₂²×C₈₉	#405	1	2	yes	—
75	C₄×C₈₉	—	2	4	no	—
76	C₄×C₈₉	—	2	2	no	—
77	C₄×C₈₉	C₈²	1	4	yes	complemented
78	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
79	C₄×C₈₉	#431	1	2	yes	—
80	C₄×C₈₉	C₈×D₄	1	4	yes	—
81	C₄×C₈₉	C₈×Q₈	1	4	yes	—
82	C₈×C₄⋊C₈	—	89	1	no	maximal, Sylow
83	C₂×C₈₉	C₂×C₈²	1	1	yes	characteristic
84	C₂×C₈₉	C₈×M₁₆	1	2	yes	—
85	C₂×C₈₉	#1450	1	1	yes	characteristic
86	C₂×C₈₉	C₄×C₄⋊C₈	1	1	yes	characteristic
87	C₂×C₈₉	C₈×C₄⋊C₄	1	2	yes	—
88	C₄×C₄⋊C₈	—	89	2	no	—
89	C₄²×C₈	D₈₉	1	1	yes	characteristic
90	C₂×C₈²	—	89	2	no	—
91	#1273	—	89	2	no	—
92	C₈₉	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
93	C₂×C₄⋊C₈	—	89	4	no	—
94	C₂×C₄×C₈	—	89	4	no	—
95	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
96	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
97	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
99	C₄³	C₂×D₈₉	1	1	yes	characteristic
100	#336	—	89	4	no	—
101	C₈²	—	178	4	no	—
102	C₂²×C₈	—	89	8	no	—
103	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
104	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
105	C₂×C₄²	C₄×D₈₉	1	2	yes	—
106	C₂×C₄²	C₄×D₈₉	1	2	yes	—
107	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
109	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
110	C₄⋊C₈	—	89	16	no	—
111	C₄×C₈	—	2	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	178	4	no	—
114	C₄×C₈	—	178	4	no	—
115	C₄×C₈	#10623494	1	8	yes	complemented
116	C₄×C₈	Dic₁₇₈	1	2	yes	—
117	C₄×C₈	D₃₅₆	1	2	yes	—
118	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
119	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
120	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
122	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	89	16	no	—
126	C₂×C₈	—	178	8	no	—
127	C₂×C₈	—	178	8	no	—
128	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
129	C₂×C₈	C₂×#10623494	1	4	yes	—
130	C₂×C₈	C₂×#10623494	1	4	yes	—
131	C₂×C₈	#13083558	1	4	yes	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	4	no	—
134	C₄²	C₈×D₈₉	1	8	yes	—
135	C₄²	C₂×#10623494	1	4	yes	—
136	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
137	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
138	C₄²	#13083574	1	2	yes	—
139	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	C₈×Dic₈₉	1	4	yes	—
144	C₂×C₄	C₈×Dic₈₉	1	4	yes	—
145	C₂×C₄	C₄×#10623494	1	2	yes	—
146	C₂×C₄	C₄×#10623494	1	2	yes	—
147	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
148	C₂×C₄	#15667971	1	1	yes	characteristic
149	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
150	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
151	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
152	C₂×C₄	C₂²×#10623494	1	2	yes	—
153	C₂×C₄	C₂×#13083558	1	2	yes	—
154	C₂×C₄	#15668018	1	4	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	178	16	no	—
157	C₈	C₄⋊#10623494	1	8	yes	direct product factor
158	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
159	C₂²	C₂×C₈×Dic₈₉	1	2	yes	—
160	C₂²	C₂×C₄×#10623494	1	1	yes	characteristic
161	C₂²	C₄×#13083558	1	1	yes	characteristic
162	C₂²	#19151553	1	2	yes	—
163	C₄	—	2	4	no	—
164	C₄	—	2	2	no	—
165	C₄	C₈×#10623494	1	4	yes	complemented
166	C₄	C₂×C₄⋊#10623494	1	2	yes	—
167	C₄	#19151651	1	2	yes	—
168	C₄	C₈×D₃₅₆	1	4	yes	—
169	C₄	C₈×Dic₁₇₈	1	4	yes	—
170	C₂	C₂×C₈×#10623494	1	1	yes	characteristic
171	C₂	C₈×#13083558	1	2	yes	—
172	C₂	#26018535	1	1	yes	characteristic
173	C₂	C₄×C₄⋊#10623494	1	1	yes	characteristic
174	C₂	C₈×C₄⋊Dic₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₈₉
commutator	C₂×C₈₉	C₂×C₈²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
15. C₄⋊#10623494	C₈	direct (N × H)
27. C₄×C₈×C₈₉	C₈	semidirect
68. C₈×C₈₉	C₄⋊C₈	semidirect
77. C₄×C₈₉	C₈²	semidirect
92. C₈₉	C₈×C₄⋊C₈	semidirect
115. C₄×C₈	#10623494	semidirect
157. C₈	C₄⋊#10623494	direct (N × H)
165. C₄	C₈×#10623494	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10623494	2
C₂×C₈×#10623494	2
#26018079	2
C₄²×C₈×C₈₉	2
C₈×C₄⋊C₈	89

Subgroups — C8×C4⋊#10623494 #36445549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₄⋊#10623494 #36445549