Subgroups — C₂×#22577534 #31789717

Order 17536.

400 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	#22577534	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#11764040	C₂	1	2	yes	maximal
3	C₂×C₄⋊#11764040	C₂	1	1	yes	maximal, characteristic
4	C₂×#17983999	C₂	1	2	yes	maximal
5	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11764040	C₂²	1	1	yes	characteristic
8	C₂²×#11764040	C₂²	1	2	yes	—
9	C₂²×#11764040	C₂²	1	1	yes	characteristic
10	C₄⋊#11764040	C₂²	1	4	yes	—
11	C₄⋊#11764040	C₂²	1	8	yes	—
12	#17983999	C₂²	1	8	yes	—
13	C₄×#11764040	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₃₇	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₁₃₇	C₄	1	4	yes	—
16	C₄×Q₈×C₁₃₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
21	C₂×#11764040	—	2	2	no	—
22	C₂×#11764040	—	2	4	no	—
23	C₂×#11764040	Q₈	1	4	yes	complemented
24	C₂×#11764040	C₂³	1	2	yes	—
25	C₂×#11764040	C₂³	1	4	yes	—
26	C₂×#11764040	C₂³	1	2	yes	—
27	C₂×Q₈×C₁₃₇	—	2	4	no	—
28	C₂×Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
35	C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
36	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
38	C₄²×C₁₃₇	C₂³	1	2	yes	—
39	C₄²×C₁₃₇	C₂³	1	4	yes	—
40	#11764040	—	2	2	no	—
41	#11764040	—	2	4	no	—
42	#11764040	C₂×Q₈	1	4	yes	complemented
43	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₁₃₇	—	2	8	no	—
45	C₂×C₄×C₁₃₇	—	2	4	no	—
46	C₂×C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄×C₁₃₇	—	2	4	no	—
48	C₂×C₄×C₁₃₇	M₁₆	1	4	yes	—
49	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
51	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₁₃₇	#41	1	2	yes	—
56	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
57	C₂²×C₁₃₇	C₄×Q₈	1	2	yes	—
58	C₂²×C₁₃₇	C₂×M₁₆	1	2	yes	—
59	C₂²×C₁₃₇	#131	1	2	yes	—
60	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₁₃₇	—	2	4	no	—
62	C₄×C₁₃₇	—	2	2	no	—
63	C₄×C₁₃₇	—	2	4	no	—
64	C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
65	C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
66	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
67	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
68	C₂×C₁₃₇	#446	1	4	yes	complemented
69	C₂×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
71	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
72	C₁₃₇	C₂×#446	1	1	yes	characteristic, complemented, Sylow
73	C₂×#446	—	137	1	no	maximal, Sylow
74	C₂×C₄×Q₈	D₁₃₇	1	1	yes	characteristic
75	#446	—	137	8	no	—
76	C₂×C₄⋊C₈	—	137	1	no	—
77	C₂×C₄⋊C₈	—	137	2	no	—
78	C₂×#97	—	137	2	no	—
79	C₂×C₄×C₈	—	137	1	no	—
80	C₂²×Q₈	Dic₁₃₇	1	1	yes	characteristic
81	C₂²×C₈	—	137	1	no	—
82	C₂²×C₈	—	137	2	no	—
83	C₂²×C₈	—	137	1	no	—
84	C₄×Q₈	Dic₁₃₇	1	4	yes	—
85	C₄×Q₈	C₂×D₁₃₇	1	4	yes	—
86	C₂×C₄⋊C₄	Dic₁₃₇	1	2	yes	—
87	C₂×C₄⋊C₄	Dic₁₃₇	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
89	C₂×C₄²	C₂×D₁₃₇	1	2	yes	—
90	C₄⋊C₈	—	137	4	no	—
91	C₄⋊C₈	—	137	8	no	—
92	#97	—	137	8	no	—
93	C₄×C₈	—	137	4	no	—
94	C₂×Q₈	—	2	4	no	—
95	C₂×Q₈	C₂×Dic₁₃₇	1	4	yes	—
96	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
98	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
99	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
101	C₂×C₈	—	274	2	no	—
102	C₂×C₈	—	274	4	no	—
103	C₂×C₈	—	137	2	no	—
104	C₂×C₈	—	137	4	no	—
105	C₂×C₈	—	137	4	no	—
106	C₂×C₈	—	137	2	no	—
107	C₄⋊C₄	C₂×Dic₁₃₇	1	8	yes	—
108	C₄⋊C₄	C₂×Dic₁₃₇	1	4	yes	—
109	C₄²	C₂×Dic₁₃₇	1	4	yes	—
110	C₄²	C₂×Dic₁₃₇	1	2	yes	—
111	C₄²	C₂²×D₁₃₇	1	2	yes	—
112	C₄²	C₂²×D₁₃₇	1	4	yes	—
113	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
114	Q₈	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#14555494	1	4	yes	—
119	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
120	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
121	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
122	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
123	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
124	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
125	C₂×C₄	#14555511	1	2	yes	—
126	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
127	C₈	—	274	2	no	—
128	C₈	—	274	4	no	—
129	C₈	—	137	4	no	—
130	C₂²	Q₈×Dic₁₃₇	1	2	yes	—
131	C₂²	C₂×#14555494	1	2	yes	—
132	C₂²	#17984095	1	2	yes	—
133	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	4	no	—
137	C₄	Q₈×Dic₁₃₇	1	2	yes	—
138	C₄	C₂×#14555494	1	4	yes	—
139	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
140	C₄	C₂×#14555511	1	1	yes	characteristic
141	C₂	#22577534	1	4	yes	direct product factor
142	C₂	C₂×Q₈×Dic₁₃₇	1	1	yes	characteristic
143	C₂	C₂²×#14555494	1	1	yes	characteristic
144	C₂	C₂×#17984095	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₇
commutator	C₂²×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×C₄×Q₈×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₃₇
hypocenter	C₁₃₇	C₂×#446
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22577534	C₂	direct (N × H)
23. C₂×#11764040	Q₈	semidirect
42. #11764040	C₂×Q₈	semidirect
68. C₂×C₁₃₇	#446	semidirect
72. C₁₃₇	C₂×#446	semidirect
141. C₂	#22577534	direct (N × H)

Maximal subgroups

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

subgroup	index
#22577534	2
C₂×C₄⋊#11764040	2
C₂×#17983999	2
C₂×C₄×#11764040	2
C₂×C₄×Q₈×C₁₃₇	2
C₂×#446	137

Subgroups — C2×#22577534 #31789717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577534 #31789717