Subgroups — C₂×C₈×Dic₁₇₈ #26029173

Order 11392.

440 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₁₇₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667941	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₈₉	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	C₄	1	1	yes	characteristic
8	C₂²×#10623494	C₂²	1	2	yes	—
9	C₄×Dic₁₇₈	C₄	1	4	yes	—
10	C₄×Dic₁₇₈	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
12	C₂×#13083553	C₄	1	2	yes	—
13	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
14	C₄⋊#10623494	C₂²	1	4	yes	—
15	#15667941	C₂²	1	8	yes	—
16	C₈×Dic₈₉	C₂²	1	8	yes	—
17	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₈₉	C₂²	1	4	yes	—
21	C₂×Dic₁₇₈	C₈	1	8	yes	direct product factor
22	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	—
23	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
25	C₂×#10623494	—	2	4	no	—
26	C₂×#10623494	C₂³	1	4	yes	—
27	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
28	#13083553	C₂×C₄	1	8	yes	—
29	C₄×Dic₈₉	C₂×C₄	1	4	yes	—
30	C₄×Dic₈₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	Q₈	1	4	yes	complemented
36	C₂×C₈×C₈₉	C₂³	1	2	yes	—
37	C₂×C₈×C₈₉	C₂³	1	2	yes	—
38	C₄²×C₈₉	C₂×C₄	1	2	yes	—
39	C₄²×C₈₉	C₂³	1	2	yes	—
40	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₈₉	#41	1	2	yes	—
48	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	C₂×C₈	1	8	yes	—
51	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
52	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
53	Dic₁₇₈	C₂×C₈	1	16	yes	direct product factor
54	C₈×C₈₉	—	2	2	no	—
55	C₈×C₈₉	C₂×Q₈	1	4	yes	complemented
56	#10623494	—	2	4	no	—
57	C₂²×C₈₉	C₄×Q₈	1	2	yes	—
58	C₂²×C₈₉	C₂²×C₈	1	2	yes	—
59	C₂²×C₈₉	#131	1	2	yes	—
60	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	C₄×Q₈	1	2	yes	—
63	C₄×C₈₉	C₂²×C₈	1	4	yes	—
64	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
66	Dic₈₉	—	2	4	no	—
67	Dic₈₉	C₂²×C₈	1	8	yes	—
68	C₂×C₈₉	C₈×Q₈	1	4	yes	complemented
69	C₂×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₈₉	C₂³×C₈	1	1	yes	characteristic
71	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
72	C₂×C₈×Q₈	—	89	1	no	maximal, Sylow
73	C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×Q₈	—	89	1	no	—
75	C₈×Q₈	—	89	8	no	—
76	C₂×C₄⋊C₈	—	89	2	no	—
77	C₂×C₄⋊C₈	—	89	1	no	—
78	C₂×C₄×C₈	—	89	2	no	—
79	C₂×C₄×C₈	D₈₉	1	1	yes	characteristic
80	C₂²×Q₈	—	89	1	no	—
81	C₂²×C₈	—	89	2	no	—
82	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
84	C₄×Q₈	—	89	4	no	—
85	C₄×Q₈	—	89	4	no	—
86	C₂×C₄⋊C₄	—	89	2	no	—
87	C₂×C₄⋊C₄	—	89	1	no	—
88	C₂×C₄²	—	89	2	no	—
89	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
90	C₄⋊C₈	—	89	8	no	—
91	C₄⋊C₈	—	89	4	no	—
92	C₄×C₈	—	89	8	no	—
93	C₄×C₈	C₂×D₈₉	1	4	yes	—
94	C₂×Q₈	—	89	8	no	—
95	C₂×Q₈	—	89	4	no	—
96	C₂²×C₄	—	89	2	no	—
97	C₂²×C₄	—	89	2	no	—
98	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	89	4	no	—
103	C₂×C₈	—	178	4	no	—
104	C₂×C₈	Dic₁₇₈	1	4	yes	direct product factor
105	C₂×C₈	C₂²×D₈₉	1	2	yes	—
106	C₂×C₈	C₂²×D₈₉	1	2	yes	—
107	C₄⋊C₄	—	89	8	no	—
108	C₄⋊C₄	—	89	4	no	—
109	C₄²	—	89	4	no	—
110	C₄²	—	89	4	no	—
111	C₄²	C₄×D₈₉	1	2	yes	—
112	C₄²	C₂²×D₈₉	1	2	yes	—
113	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
114	Q₈	—	89	16	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	89	8	no	—
117	C₂×C₄	—	89	4	no	—
118	C₂×C₄	—	89	4	no	—
119	C₂×C₄	—	178	4	no	—
120	C₂×C₄	C₈×D₈₉	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
124	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
125	C₂×C₄	#13083574	1	2	yes	—
126	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	178	4	no	—
129	C₈	C₂×Dic₁₇₈	1	4	yes	direct product factor
130	C₂²	C₄×Dic₁₇₈	1	2	yes	—
131	C₂²	C₂×C₈×D₈₉	1	2	yes	—
132	C₂²	#15668018	1	2	yes	—
133	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	89	8	no	—
136	C₄	—	178	4	no	—
137	C₄	C₄×Dic₁₇₈	1	2	yes	—
138	C₄	C₂×C₈×D₈₉	1	4	yes	—
139	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
140	C₄	C₂×#13083574	1	1	yes	characteristic
141	C₂	C₈×Dic₁₇₈	1	4	yes	direct product factor
142	C₂	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
143	C₂	C₂²×C₈×D₈₉	1	1	yes	characteristic
144	C₂	C₂×#15668018	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×C₄×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₈₉
hypocenter	C₈₉	C₂×C₈×Q₈
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₁₇₈	C₂	direct (N × H)
21. C₂×Dic₁₇₈	C₈	direct (N × H)
35. C₂×C₈×C₈₉	Q₈	semidirect
53. Dic₁₇₈	C₂×C₈	direct (N × H)
55. C₈×C₈₉	C₂×Q₈	semidirect
68. C₂×C₈₉	C₈×Q₈	semidirect
73. C₈₉	C₂×C₈×Q₈	semidirect
104. C₂×C₈	Dic₁₇₈	direct (N × H)
129. C₈	C₂×Dic₁₇₈	direct (N × H)
141. C₂	C₈×Dic₁₇₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₇₈	2
C₈×Dic₁₇₈	2
C₂×C₄⋊#10623494	2
C₂×#15667941	2
C₂×C₈×Dic₈₉	2
C₂×C₄×C₈×C₈₉	2
C₂×C₈×Q₈	89

Subgroups — C2×C8×Dic178 #26029173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Dic₁₇₈ #26029173