Subgroups — C₂×#22577960 #31794080

Order 17536.

1168 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577960	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984088	C₂	1	2	yes	maximal, complemented
3	C₂×#17984080	C₂	1	2	yes	maximal, complemented
4	C₂×#17984067	C₂	1	4	yes	maximal
5	C₂×C₄×Dic₂₇₄	C₂	1	4	yes	maximal, complemented
6	C₂×#17984047	C₂	1	2	yes	maximal
7	C₂²×C₄²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₇₄	C₂²	1	4	yes	—
9	#17984088	C₂²	1	16	yes	complemented
10	#17984080	C₂²	1	16	yes	complemented
11	#17984067	C₂²	1	32	yes	—
12	C₄×Dic₂₇₄	C₂²	1	32	yes	complemented
13	#17984047	C₂²	1	16	yes	—
14	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
16	C₂×#14555489	C₂²	1	8	yes	—
17	C₂×C₂²⋊Dic₁₃₇	C₂²	1	4	yes	—
18	C₂×C₄×Dic₁₃₇	C₂²	1	4	yes	—
19	C₂³×C₄×C₁₃₇	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₃₇	C₂²	1	8	yes	—
24	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
25	C₂×Dic₂₇₄	—	2	16	no	—
26	C₂×Dic₂₇₄	C₂³	1	16	yes	—
27	C₂²×Dic₁₃₇	C₂³	1	8	yes	—
28	C₄⋊Dic₁₃₇	C₂³	1	16	yes	—
29	C₄⋊Dic₁₃₇	C₂³	1	16	yes	—
30	#14555489	C₂³	1	32	yes	—
31	C₂²⋊Dic₁₃₇	C₂³	1	16	yes	—
32	C₄×Dic₁₃₇	C₂³	1	16	yes	—
33	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₃₇	—	2	8	no	—
35	C₂²×C₄×C₁₃₇	Q₈	1	8	yes	complemented
36	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃₇	C₂³	1	8	yes	—
38	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
40	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₄²×C₁₃₇	C₂³	1	8	yes	—
43	C₄²×C₁₃₇	C₂³	1	4	yes	—
44	C₄²×C₁₃₇	C₂³	1	4	yes	—
45	Dic₂₇₄	—	2	32	no	—
46	C₂×Dic₁₃₇	—	2	16	no	—
47	C₂×Dic₁₃₇	C₂⁴	1	16	yes	—
48	C₂³×C₁₃₇	—	2	4	no	—
49	C₂³×C₁₃₇	C₂×Q₈	1	4	yes	—
50	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
51	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	—	2	4	no	—
53	C₂×C₄×C₁₃₇	—	2	16	no	—
54	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
55	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
56	C₂×C₄×C₁₃₇	C₂×Q₈	1	16	yes	complemented
57	C₂×C₄×C₁₃₇	#41	1	8	yes	—
58	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
59	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
60	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
61	C₂×C₄×C₁₃₇	C₂⁴	1	4	yes	—
62	C₂²×C₁₃₇	—	2	8	no	—
63	C₂²×C₁₃₇	—	2	4	no	—
64	C₂²×C₁₃₇	C₂²×Q₈	1	4	yes	—
65	C₂²×C₁₃₇	C₂²×Q₈	1	2	yes	—
66	C₂²×C₁₃₇	C₂×#41	1	4	yes	—
67	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
68	C₄×C₁₃₇	—	2	4	no	—
69	C₄×C₁₃₇	C₂²×Q₈	1	4	yes	—
70	C₄×C₁₃₇	C₂²×Q₈	1	4	yes	—
71	C₄×C₁₃₇	C₂×#41	1	8	yes	—
72	Dic₁₃₇	—	2	16	no	—
73	C₂×C₁₃₇	—	2	4	no	—
74	C₂×C₁₃₇	#533	1	4	yes	complemented
75	C₂×C₁₃₇	C₂³×Q₈	1	1	yes	characteristic
76	C₂×C₁₃₇	C₂²×#41	1	2	yes	—
77	C₁₃₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
78	C₂×#533	—	137	1	no	maximal, Sylow
79	#533	—	137	16	no	—
80	C₂×C₄⋊Q₈	—	137	2	no	—
81	C₂×#125	—	137	2	no	—
82	C₂×C₂²⋊Q₈	—	137	4	no	—
83	C₂×C₄×Q₈	—	137	4	no	—
84	C₂×#117	—	137	2	no	—
85	C₂²×C₄²	D₁₃₇	1	1	yes	characteristic
86	C₂²×Q₈	—	137	4	no	—
87	C₂³×C₄	C₂×D₁₃₇	1	2	yes	—
88	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
89	C₄⋊Q₈	—	137	16	no	—
90	#125	—	137	16	no	—
91	C₂²⋊Q₈	—	137	32	no	—
92	C₄×Q₈	—	137	32	no	—
93	#117	—	137	16	no	—
94	C₂×C₄⋊C₄	—	137	4	no	—
95	C₂×C₄⋊C₄	—	137	8	no	—
96	C₂×C₄⋊C₄	—	137	4	no	—
97	C₂×C₂²⋊C₄	—	137	4	no	—
98	C₂×C₄²	—	137	4	no	—
99	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₁₃₇	1	8	yes	—
102	C₂×C₄²	C₂×D₁₃₇	1	2	yes	—
103	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
104	C₂×Q₈	—	274	16	no	—
105	C₂×Q₈	—	137	16	no	—
106	C₂²×C₄	—	2	8	no	—
107	C₂²×C₄	—	137	8	no	—
108	C₂²×C₄	Dic₂₇₄	1	8	yes	complemented
109	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
110	C₂²×C₄	C₂²×D₁₃₇	1	8	yes	—
111	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
113	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
114	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
115	C₄⋊C₄	—	137	32	no	—
116	C₄⋊C₄	—	137	16	no	—
117	C₄⋊C₄	—	137	16	no	—
118	C₂²⋊C₄	—	137	16	no	—
119	C₄²	—	137	16	no	—
120	C₄²	C₂²×D₁₃₇	1	8	yes	—
121	C₄²	C₂²×D₁₃₇	1	4	yes	—
122	C₄²	C₂²×D₁₃₇	1	4	yes	—
123	C₂³	—	2	4	no	—
124	C₂³	C₂×Dic₂₇₄	1	4	yes	—
125	C₂³	C₂³×D₁₃₇	1	2	yes	—
126	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
127	Q₈	—	274	32	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	16	no	—
130	C₂×C₄	—	274	16	no	—
131	C₂×C₄	—	137	16	no	—
132	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
133	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
134	C₂×C₄	C₂×Dic₂₇₄	1	16	yes	complemented
135	C₂×C₄	#14555510	1	8	yes	—
136	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
137	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
138	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
139	C₂×C₄	C₂³×D₁₃₇	1	4	yes	—
140	C₂²	—	2	8	no	—
141	C₂²	—	2	4	no	—
142	C₂²	C₂²×Dic₂₇₄	1	4	yes	—
143	C₂²	C₂²×Dic₂₇₄	1	2	yes	—
144	C₂²	C₂×#14555510	1	4	yes	—
145	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	274	16	no	—
148	C₄	C₂²×Dic₂₇₄	1	4	yes	—
149	C₄	C₂²×Dic₂₇₄	1	4	yes	—
150	C₄	C₂×#14555510	1	8	yes	—
151	C₂	—	2	4	no	—
152	C₂	#22577960	1	4	yes	direct product factor
153	C₂	C₂³×Dic₂₇₄	1	1	yes	characteristic
154	C₂	C₂²×#14555510	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂²×C₄²×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂⁴×C₁₃₇	C₂³
hypercenter	C₂²×C₄²	D₁₃₇
hypocenter	C₁₃₇	C₂×#533
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #22577960	C₂	direct (N × H)
2. C₂×#17984088	C₂	semidirect
3. C₂×#17984080	C₂	semidirect
5. C₂×C₄×Dic₂₇₄	C₂	semidirect
9. #17984088	C₂²	semidirect
10. #17984080	C₂²	semidirect
12. C₄×Dic₂₇₄	C₂²	semidirect
35. C₂²×C₄×C₁₃₇	Q₈	semidirect
56. C₂×C₄×C₁₃₇	C₂×Q₈	semidirect
74. C₂×C₁₃₇	#533	semidirect
77. C₁₃₇	C₂×#533	semidirect
108. C₂²×C₄	Dic₂₇₄	semidirect
134. C₂×C₄	C₂×Dic₂₇₄	semidirect
152. C₂	#22577960	direct (N × H)

Maximal subgroups

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

subgroup	index
#22577960	2
C₂×#17984088	2
C₂×#17984080	2
C₂×#17984067	2
C₂×C₄×Dic₂₇₄	2
C₂×#17984047	2
C₂²×C₄²×C₁₃₇	2
C₂×#533	137

Subgroups — C2×#22577960 #31794080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577960 #31794080