Subgroups — C₂×Q₈×#11719910 #28328418

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719910	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#11719910	C₂	1	3	yes	maximal
3	C₂×C₄×#11719910	C₂	1	3	yes	maximal
4	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	3	yes	—
6	C₂²×#11719910	C₂²	1	1	yes	characteristic
7	C₄⋊#11719910	C₂²	1	12	yes	—
8	C₄×#11719910	C₂²	1	12	yes	—
9	Q₈×#88322	C₄	1	4	yes	—
10	Q₈×#88322	C₂²	1	4	yes	—
11	C₂×C₄⋊#88322	C₄	1	3	yes	—
12	C₂×C₄×#88322	C₂²	1	3	yes	—
13	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#11719910	—	2	6	no	—
15	C₂×#11719910	Q₈	1	4	yes	direct product factor
16	C₂×#11719910	C₂³	1	6	yes	—
17	C₂×#11719910	C₂³	1	2	yes	—
18	C₂²×#88322	C₂×C₄	1	3	yes	—
19	C₂²×#88322	C₂³	1	1	yes	characteristic
20	C₄⋊#88322	C₂×C₄	1	12	yes	—
21	C₄×#88322	C₂×C₄	1	6	yes	—
22	C₄×#88322	C₂³	1	6	yes	—
23	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₅₃	C₈	1	4	yes	—
25	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₅₃	C₈	1	3	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
28	C₂×Q₈×C₅₃	C₁₆	1	8	yes	complemented
29	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	C₂×C₈	1	12	yes	—
34	C₄²×C₅₃	C₂×C₈	1	6	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
36	C₂×#88322	—	2	6	no	—
37	C₂×#88322	C₂²×C₄	1	6	yes	—
38	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
39	C₂×#88322	C₂×Q₈	1	2	yes	—
40	C₂×#88322	#41	1	2	yes	—
41	C₂×#88322	C₂⁴	1	1	yes	characteristic
42	#11719910	—	2	6	no	—
43	#11719910	C₂×Q₈	1	4	yes	direct product factor
44	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
45	Q₈×C₅₃	C₂×C₁₆	1	16	yes	complemented
46	C₂×C₄×C₅₃	—	2	6	no	—
47	C₂×C₄×C₅₃	C₂×C₁₆	1	12	yes	—
48	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
50	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
52	C₂×C₄×C₅₃	#131	1	2	yes	—
53	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	#88322	—	2	6	no	—
55	#88322	C₄×Q₈	1	2	yes	—
56	#88322	C₂²×Q₈	1	1	yes	characteristic
57	#88322	C₂×#41	1	1	yes	characteristic
58	C₂×Q₈×C₁₆	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
60	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
61	C₂²×C₅₃	#504	1	2	yes	—
62	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
63	C₄×C₅₃	—	2	6	no	—
64	C₄×C₅₃	C₈×Q₈	1	2	yes	—
65	C₄×C₅₃	C₂²×C₁₆	1	12	yes	—
66	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
67	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
68	C₂×C₈×Q₈	—	53	1	no	—
69	Q₈×C₁₆	—	53	8	no	—
70	C₂×C₄⋊C₁₆	—	53	3	no	—
71	C₂×C₄×C₁₆	—	53	3	no	—
72	C₂×C₅₃	Q₈×C₁₆	1	4	yes	complemented
73	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
74	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
75	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
76	C₂×C₄×Q₈	#6157	1	1	yes	characteristic
77	C₂²×C₁₆	—	53	1	no	—
78	C₂²×C₁₆	—	53	3	no	—
79	C₈×Q₈	—	53	4	no	—
80	C₈×Q₈	—	53	4	no	—
81	C₂×C₄⋊C₈	—	53	3	no	—
82	C₂×C₄×C₈	—	53	3	no	—
83	C₄⋊C₁₆	—	53	12	no	—
84	C₄×C₁₆	—	53	12	no	—
85	C₅₃	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	#88324	1	1	yes	characteristic
87	C₂²×C₈	—	53	1	no	—
88	C₂²×C₈	—	53	3	no	—
89	C₄×Q₈	#88324	1	4	yes	—
90	C₄×Q₈	C₂×#6157	1	4	yes	—
91	C₂×C₄⋊C₄	#88324	1	3	yes	—
92	C₂×C₄²	C₂×#6157	1	3	yes	—
93	C₂×C₁₆	—	53	2	no	—
94	C₂×C₁₆	—	53	4	no	—
95	C₂×C₁₆	—	53	6	no	—
96	C₂×C₁₆	—	106	6	no	—
97	C₄⋊C₈	—	53	12	no	—
98	C₄×C₈	—	53	6	no	—
99	C₄×C₈	—	53	6	no	—
100	C₂×Q₈	#11719910	1	8	yes	direct product factor
101	C₂×Q₈	C₂×#88324	1	4	yes	—
102	C₂²×C₄	C₂×#88324	1	3	yes	—
103	C₂²×C₄	C₂×#88324	1	3	yes	—
104	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
105	C₂×C₈	—	53	2	no	—
106	C₂×C₈	—	53	2	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	53	1	no	—
109	C₂×C₈	—	53	6	no	—
110	C₂×C₈	—	106	6	no	—
111	C₄⋊C₄	C₂×#88324	1	12	yes	—
112	C₄²	C₂×#88324	1	6	yes	—
113	C₄²	C₂²×#6157	1	6	yes	—
114	C₁₆	—	53	4	no	—
115	C₁₆	—	106	6	no	—
116	C₂³	C₂²×#88324	1	1	yes	characteristic
117	Q₈	C₂×#11719910	1	16	yes	direct product factor
118	C₂×C₄	—	2	6	no	—
119	C₂×C₄	C₂×#11719910	1	12	yes	—
120	C₂×C₄	Q₈×#6157	1	2	yes	—
121	C₂×C₄	C₂²×#88324	1	6	yes	—
122	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
123	C₂×C₄	C₂²×#88324	1	6	yes	—
124	C₂×C₄	#13133305	1	2	yes	—
125	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
126	C₈	—	53	2	no	—
127	C₈	—	53	1	no	—
128	C₈	—	53	1	no	—
129	C₈	—	106	6	no	—
130	C₂²	Q₈×#88324	1	2	yes	—
131	C₂²	C₂²×#11719910	1	2	yes	—
132	C₂²	#16811681	1	2	yes	—
133	C₂²	C₂³×#88324	1	1	yes	characteristic
134	C₄	—	2	6	no	—
135	C₄	Q₈×#88324	1	2	yes	—
136	C₄	C₂²×#11719910	1	12	yes	—
137	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
138	C₄	C₂×#13133305	1	1	yes	characteristic
139	C₂	Q₈×#11719910	1	4	yes	direct product factor
140	C₂	C₂×Q₈×#88324	1	1	yes	characteristic
141	C₂	C₂³×#11719910	1	1	yes	characteristic
142	C₂	C₂×#16811681	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×Q₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×Q₈	#6157
hypocenter	C₅₃	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11719910	C₂	direct (N × H)
15. C₂×#11719910	Q₈	direct (N × H)
28. C₂×Q₈×C₅₃	C₁₆	semidirect
43. #11719910	C₂×Q₈	direct (N × H)
45. Q₈×C₅₃	C₂×C₁₆	semidirect
72. C₂×C₅₃	Q₈×C₁₆	semidirect
85. C₅₃	C₂×Q₈×C₁₆	semidirect
100. C₂×Q₈	#11719910	direct (N × H)
117. Q₈	C₂×#11719910	direct (N × H)
139. C₂	Q₈×#11719910	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#11719910	2
C₂×C₄⋊#11719910	2
C₂×C₄×#11719910	2
C₂×Q₈×#88322	2
C₂×Q₈×C₁₆	53

Subgroups — C2×Q8×#11719910 #28328418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×#11719910 #28328418