Subgroups — C₂×#22577956 #31794076

Order 17536.

1168 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577956	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17984087	C₂	1	2	yes	maximal, complemented
3	C₂×#17984079	C₂	1	2	yes	maximal, complemented
4	C₂×#17984065	C₂	1	4	yes	maximal
5	C₂×C₄×Dic₂₇₄	C₂	1	4	yes	maximal, complemented
6	C₂×#17984047	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₇₄	C₂²	1	4	yes	—
10	C₂³×Dic₁₃₇	C₂²	1	2	yes	—
11	#17984087	C₂²	1	16	yes	complemented
12	#17984079	C₂²	1	16	yes	complemented
13	#17984065	C₂²	1	32	yes	—
14	C₄×Dic₂₇₄	C₂²	1	32	yes	complemented
15	#17984047	C₂²	1	8	yes	—
16	C₂×C₄⋊Dic₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₃₇	C₂²	1	4	yes	—
18	C₂×#14555489	C₂²	1	4	yes	—
19	C₂×#14555489	C₂²	1	4	yes	—
20	C₂×C₂²⋊Dic₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃₇	C₂²	1	8	yes	—
24	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
26	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	#117×C₁₃₇	C₂²	1	8	yes	—
28	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
31	C₂×Dic₂₇₄	—	2	16	no	—
32	C₂×Dic₂₇₄	C₂³	1	16	yes	—
33	C₂²×Dic₁₃₇	—	2	8	no	—
34	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
35	C₂²×Dic₁₃₇	C₂³	1	8	yes	—
36	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
37	C₂²×Dic₁₃₇	C₂³	1	2	yes	—
38	C₄⋊Dic₁₃₇	C₂³	1	8	yes	—
39	C₄⋊Dic₁₃₇	C₂³	1	16	yes	—
40	#14555489	C₂³	1	16	yes	—
41	#14555489	C₂³	1	16	yes	—
42	C₂²⋊Dic₁₃₇	C₂³	1	8	yes	—
43	C₄×Dic₁₃₇	C₂³	1	8	yes	—
44	C₄×Dic₁₃₇	C₂³	1	4	yes	—
45	C₄×Dic₁₃₇	C₂³	1	4	yes	—
46	C₄×Dic₁₃₇	C₂³	1	8	yes	—
47	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₇	Q₈	1	8	yes	complemented
49	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
50	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₇	C₂³	1	4	yes	—
52	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
54	C₂²⋊C₄×C₁₃₇	C₂³	1	8	yes	—
55	C₄²×C₁₃₇	C₂³	1	8	yes	—
56	Dic₂₇₄	—	2	32	no	—
57	C₂×Dic₁₃₇	—	2	4	no	—
58	C₂×Dic₁₃₇	—	2	8	no	—
59	C₂×Dic₁₃₇	—	2	16	no	—
60	C₂×Dic₁₃₇	#41	1	8	yes	—
61	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
62	C₂×Dic₁₃₇	C₂⁴	1	8	yes	—
63	C₂×Dic₁₃₇	C₂⁴	1	4	yes	—
64	C₂³×C₁₃₇	—	2	4	no	—
65	C₂³×C₁₃₇	C₂×Q₈	1	4	yes	—
66	C₂³×C₁₃₇	C₂⁴	1	2	yes	—
67	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
68	C₂×C₄×C₁₃₇	—	2	8	no	—
69	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
70	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
71	C₂×C₄×C₁₃₇	C₂×Q₈	1	16	yes	complemented
72	C₂×C₄×C₁₃₇	C₂⁴	1	8	yes	—
73	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
74	C₂×C₄×C₁₃₇	C₂⁴	1	2	yes	—
75	C₂²×C₁₃₇	—	2	8	no	—
76	C₂²×C₁₃₇	—	2	4	no	—
77	C₂²×C₁₃₇	C₂²×Q₈	1	4	yes	—
78	C₂²×C₁₃₇	C₂²×Q₈	1	2	yes	—
79	C₂²×C₁₃₇	C₂×#41	1	4	yes	—
80	C₂²×C₁₃₇	C₂⁵	1	1	yes	characteristic
81	C₄×C₁₃₇	—	2	8	no	—
82	C₄×C₁₃₇	C₂²×Q₈	1	4	yes	—
83	C₄×C₁₃₇	C₂²×Q₈	1	4	yes	—
84	Dic₁₃₇	—	2	4	no	—
85	Dic₁₃₇	—	2	8	no	—
86	Dic₁₃₇	C₂×#41	1	8	yes	—
87	C₂×C₁₃₇	—	2	4	no	—
88	C₂×C₁₃₇	#533	1	4	yes	complemented
89	C₂×C₁₃₇	C₂³×Q₈	1	1	yes	characteristic
90	C₂×C₁₃₇	C₂²×#41	1	2	yes	—
91	C₁₃₇	C₂×#533	1	1	yes	characteristic, complemented, Sylow
92	C₂×#533	—	137	1	no	maximal, Sylow
93	#533	—	137	16	no	—
94	C₂×C₄⋊Q₈	—	137	2	no	—
95	C₂×#125	—	137	2	no	—
96	C₂×C₂²⋊Q₈	—	137	4	no	—
97	C₂×C₄×Q₈	—	137	4	no	—
98	C₂×#117	—	137	1	no	—
99	C₂×#117	D₁₃₇	1	1	yes	characteristic
100	C₂²×C₄²	—	137	1	no	—
101	C₂²×Q₈	—	137	4	no	—
102	C₂³×C₄	—	137	2	no	—
103	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
104	C₄⋊Q₈	—	137	16	no	—
105	#125	—	137	16	no	—
106	C₂²⋊Q₈	—	137	32	no	—
107	C₄×Q₈	—	137	32	no	—
108	#117	—	137	8	no	—
109	#117	C₂×D₁₃₇	1	8	yes	—
110	C₂×C₄⋊C₄	—	137	2	no	—
111	C₂×C₄⋊C₄	—	137	4	no	—
112	C₂×C₄⋊C₄	—	137	4	no	—
113	C₂×C₄⋊C₄	—	137	4	no	—
114	C₂×C₄⋊C₄	C₂×D₁₃₇	1	2	yes	—
115	C₂×C₂²⋊C₄	—	137	2	no	—
116	C₂×C₂²⋊C₄	C₂×D₁₃₇	1	2	yes	—
117	C₂×C₄²	—	137	1	no	—
118	C₂×C₄²	—	137	1	no	—
119	C₂×C₄²	—	137	8	no	—
120	C₂×C₄²	—	137	2	no	—
121	C₂×C₄²	—	137	2	no	—
122	C₂×C₄²	C₂×D₁₃₇	1	2	yes	—
123	C₂⁴	C₂²×D₁₃₇	1	1	yes	characteristic
124	C₂×Q₈	—	274	16	no	—
125	C₂×Q₈	—	137	16	no	—
126	C₂²×C₄	—	274	8	no	—
127	C₂²×C₄	—	137	2	no	—
128	C₂²×C₄	—	137	8	no	—
129	C₂²×C₄	—	137	4	no	—
130	C₂²×C₄	—	137	2	no	—
131	C₂²×C₄	Dic₂₇₄	1	8	yes	complemented
132	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
133	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₁₃₇	1	4	yes	—
135	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
136	C₄⋊C₄	—	137	16	no	—
137	C₄⋊C₄	—	137	16	no	—
138	C₄⋊C₄	—	137	8	no	—
139	C₄⋊C₄	—	137	16	no	—
140	C₄⋊C₄	C₂²×D₁₃₇	1	8	yes	—
141	C₂²⋊C₄	—	137	8	no	—
142	C₂²⋊C₄	C₂²×D₁₃₇	1	8	yes	—
143	C₄²	—	137	8	no	—
144	C₄²	—	137	4	no	—
145	C₄²	—	137	4	no	—
146	C₄²	—	137	8	no	—
147	C₄²	C₂²×D₁₃₇	1	8	yes	—
148	C₂³	—	2	4	no	—
149	C₂³	C₂×Dic₂₇₄	1	4	yes	—
150	C₂³	C₂³×D₁₃₇	1	2	yes	—
151	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
152	Q₈	—	274	32	no	—
153	C₂×C₄	—	2	8	no	—
154	C₂×C₄	—	274	4	no	—
155	C₂×C₄	—	274	8	no	—
156	C₂×C₄	—	274	16	no	—
157	C₂×C₄	—	137	4	no	—
158	C₂×C₄	—	137	8	no	—
159	C₂×C₄	—	137	8	no	—
160	C₂×C₄	—	137	4	no	—
161	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
162	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
163	C₂×C₄	C₂×Dic₂₇₄	1	16	yes	complemented
164	C₂×C₄	C₂³×D₁₃₇	1	8	yes	—
165	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
166	C₂×C₄	C₂³×D₁₃₇	1	2	yes	—
167	C₂²	—	2	8	no	—
168	C₂²	—	2	4	no	—
169	C₂²	C₂²×Dic₂₇₄	1	4	yes	—
170	C₂²	C₂²×Dic₂₇₄	1	2	yes	—
171	C₂²	#41×D₁₃₇	1	4	yes	—
172	C₂²	C₂⁴×D₁₃₇	1	1	yes	characteristic
173	C₄	—	2	8	no	—
174	C₄	—	274	4	no	—
175	C₄	—	274	8	no	—
176	C₄	—	137	8	no	—
177	C₄	C₂²×Dic₂₇₄	1	4	yes	—
178	C₄	C₂²×Dic₂₇₄	1	4	yes	—
179	C₂	—	2	4	no	—
180	C₂	#22577956	1	4	yes	direct product factor
181	C₂	C₂³×Dic₂₇₄	1	1	yes	characteristic
182	C₂	C₂×#41×D₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₇
commutator, cosocle	C₂²×C₁₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₇
Fitting	C₂×#117×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂⁴
prime radical	C₂⁴×C₁₃₇	C₂³
hypercenter	C₂×#117	D₁₃₇
hypocenter	C₁₃₇	C₂×#533
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22577956	C₂	direct (N × H)
2. C₂×#17984087	C₂	semidirect
3. C₂×#17984079	C₂	semidirect
5. C₂×C₄×Dic₂₇₄	C₂	semidirect
11. #17984087	C₂²	semidirect
12. #17984079	C₂²	semidirect
14. C₄×Dic₂₇₄	C₂²	semidirect
48. C₂²×C₄×C₁₃₇	Q₈	semidirect
71. C₂×C₄×C₁₃₇	C₂×Q₈	semidirect
88. C₂×C₁₃₇	#533	semidirect
91. C₁₃₇	C₂×#533	semidirect
131. C₂²×C₄	Dic₂₇₄	semidirect
163. C₂×C₄	C₂×Dic₂₇₄	semidirect
180. C₂	#22577956	direct (N × H)

Maximal subgroups

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

subgroup	index
#22577956	2
C₂×#17984087	2
C₂×#17984079	2
C₂×#17984065	2
C₂×C₄×Dic₂₇₄	2
C₂×#17984047	2
C₂²×C₄×Dic₁₃₇	2
C₂×#117×C₁₃₇	2
C₂×#533	137

Subgroups — C2×#22577956 #31794076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577956 #31794076