Subgroups — C₂×#22578404 #31797422

Order 17536.

640 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555518	C₂	1	2	yes	maximal
2	C₂×#17984155	C₂	1	2	yes	maximal, complemented
3	#22578404	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11764047	C₂	1	2	yes	maximal, complemented
5	C₂×C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555518	—	2	8	no	—
7	C₂×#14555518	C₂²	1	8	yes	—
8	C₂²×#11764047	C₂²	1	4	yes	—
9	#17984155	C₂²	1	8	yes	complemented
10	C₄⋊#11764047	C₂²	1	8	yes	complemented
11	C₂²×C₄×D₁₃₇	C₄	1	1	yes	characteristic
12	C₂²×C₄×D₁₃₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
17	C₄²×D₁₃₇	C₄	1	4	yes	—
18	C₄²×D₁₃₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
20	#14555518	—	2	16	no	—
21	C₂×#11764047	—	2	8	no	—
22	C₂×#11764047	C₂³	1	8	yes	—
23	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₁₃₇	—	2	4	no	—
28	C₂×C₄×D₁₃₇	C₈	1	8	yes	complemented
29	C₂×C₄×D₁₃₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₁₃₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×D₁₃₇	D₄	1	4	yes	—
33	C₂×C₄×D₁₃₇	Q₈	1	4	yes	—
34	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
35	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₃₇	C₂×C₄	1	4	yes	—
38	C₄×Dic₁₃₇	C₂³	1	2	yes	—
39	C₄×Dic₁₃₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
44	#11764047	—	2	8	no	—
45	C₂²×D₁₃₇	C₄⋊C₄	1	4	yes	—
46	C₂²×D₁₃₇	C₂×C₈	1	4	yes	—
47	C₂²×D₁₃₇	M₁₆	1	4	yes	—
48	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
49	C₂×Dic₁₃₇	—	2	2	no	—
50	C₂×Dic₁₃₇	C₂×C₈	1	4	yes	complemented
51	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
52	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
54	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
55	C₂×Dic₁₃₇	C₂×Q₈	1	2	yes	—
56	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
57	C₄×D₁₃₇	—	2	8	no	—
58	C₄×D₁₃₇	C₄⋊C₄	1	8	yes	—
59	C₄×D₁₃₇	C₂×C₈	1	16	yes	complemented
60	C₄×D₁₃₇	C₂×D₄	1	4	yes	—
61	C₄×D₁₃₇	C₂×Q₈	1	4	yes	—
62	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₇	—	2	2	no	—
64	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	complemented
66	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₂²×C₁₃₇	C₂²×C₈	1	2	yes	—
71	C₂²×C₁₃₇	C₂×M₁₆	1	2	yes	—
72	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×D₁₃₇	C₄⋊C₈	1	16	yes	complemented
74	C₂×D₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
75	C₂×D₁₃₇	C₂²×C₈	1	4	yes	—
76	C₂×D₁₃₇	C₂×M₁₆	1	4	yes	—
77	C₄×C₁₃₇	—	2	2	no	—
78	C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
79	C₄×C₁₃₇	C₂²×C₈	1	4	yes	complemented
80	Dic₁₃₇	—	2	2	no	—
81	Dic₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
82	Dic₁₃₇	C₂²×C₈	1	4	yes	complemented
83	Dic₁₃₇	C₂²×D₄	1	1	yes	characteristic
84	Dic₁₃₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₁₃₇	C₂×C₄⋊C₈	1	4	yes	complemented
86	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
88	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
89	D₁₃₇	C₂×C₄⋊C₈	1	8	yes	complemented
90	C₁₃₇	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₄⋊C₈	—	137	1	no	maximal, Sylow
92	C₂³×C₈	—	137	2	no	—
93	C₂²×C₄²	—	137	1	no	—
94	C₂×C₄⋊C₈	—	137	2	no	—
95	C₂×C₄⋊C₈	—	137	8	no	—
96	C₂×C₄⋊C₈	—	137	2	no	—
97	C₂³×C₄	—	137	1	no	—
98	C₂³×C₄	—	137	1	no	—
99	C₂³×C₄	—	137	1	no	—
100	C₂²×C₈	—	274	8	no	—
101	C₂²×C₈	—	137	4	no	—
102	C₂²×C₈	—	137	8	no	—
103	C₂×C₄²	—	137	1	no	—
104	C₂×C₄²	—	137	1	no	—
105	C₂×C₄²	—	137	1	no	—
106	C₂×C₄²	—	137	4	no	—
107	C₂×C₄²	—	137	4	no	—
108	C₂×C₄²	#10587739	1	1	yes	characteristic
109	C₄⋊C₈	—	137	8	no	—
110	C₄⋊C₈	—	137	8	no	—
111	C₂⁴	—	137	1	no	—
112	C₂²×C₄	—	274	4	no	—
113	C₂²×C₄	—	137	1	no	—
114	C₂²×C₄	—	137	4	no	—
115	C₂²×C₄	—	137	1	no	—
116	C₂²×C₄	—	137	1	no	—
117	C₂²×C₄	—	137	4	no	—
118	C₂²×C₄	—	137	4	no	—
119	C₂²×C₄	—	137	8	no	—
120	C₂²×C₄	—	137	2	no	—
121	C₂²×C₄	—	137	4	no	—
122	C₂²×C₄	—	137	2	no	—
123	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
124	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
125	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
126	C₂×C₈	—	274	8	no	—
127	C₂×C₈	—	274	16	no	—
128	C₂×C₈	—	137	8	no	—
129	C₄²	—	137	4	no	—
130	C₄²	—	137	2	no	—
131	C₄²	—	137	2	no	—
132	C₄²	—	137	2	no	—
133	C₄²	—	137	2	no	—
134	C₄²	C₂×#10587739	1	4	yes	—
135	C₂³	—	137	4	no	—
136	C₂³	—	137	4	no	—
137	C₂³	—	137	4	no	—
138	C₂³	—	137	2	no	—
139	C₂³	C₂²×#10587739	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	274	8	no	—
142	C₂×C₄	—	274	2	no	—
143	C₂×C₄	—	137	2	no	—
144	C₂×C₄	—	137	2	no	—
145	C₂×C₄	—	137	2	no	—
146	C₂×C₄	—	137	4	no	—
147	C₂×C₄	—	137	1	no	—
148	C₂×C₄	—	137	1	no	—
149	C₂×C₄	—	137	2	no	—
150	C₂×C₄	—	137	4	no	—
151	C₂×C₄	—	137	8	no	—
152	C₂×C₄	—	137	4	no	—
153	C₂×C₄	—	137	16	no	—
154	C₂×C₄	C₄⋊#10587739	1	4	yes	—
155	C₂×C₄	#14555518	1	4	yes	complemented
156	C₂×C₄	C₂²×#10587739	1	2	yes	—
157	C₂×C₄	C₂²×#10587739	1	2	yes	—
158	C₂×C₄	C₂²×#10587739	1	2	yes	—
159	C₈	—	274	8	no	—
160	C₂²	—	137	4	no	—
161	C₂²	—	137	4	no	—
162	C₂²	—	137	4	no	—
163	C₂²	—	137	16	no	—
164	C₂²	C₂×C₄⋊#10587739	1	2	yes	—
165	C₂²	C₂×#14555518	1	2	yes	—
166	C₂²	D₁₃₇⋊M₁₆	1	2	yes	—
167	C₂²	C₂³×#10587739	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	274	2	no	—
170	C₄	—	137	1	no	—
171	C₄	—	137	2	no	—
172	C₄	—	137	1	no	—
173	C₄	—	137	4	no	—
174	C₄	C₂×C₄⋊#10587739	1	4	yes	—
175	C₄	C₂×#14555518	1	4	yes	complemented
176	C₂	—	137	8	no	—
177	C₂	#22578404	1	4	yes	direct product factor
178	C₂	C₂²×C₄⋊#10587739	1	1	yes	characteristic
179	C₂	C₂²×#14555518	1	1	yes	characteristic
180	C₂	C₂×D₁₃₇⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#10587739
commutator	C₂×C₁₃₇	C₂³×C₈
Frattini	C₂²	C₂³×#10587739
Fitting	C₂×C₄²×C₁₃₇	C₄
socle	C₂³×C₁₃₇	C₂²×C₄
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	C₂×C₄²	#10587739
hypocenter	C₁₃₇	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#17984155	C₂	semidirect
3. #22578404	C₂	direct (N × H)
4. C₂×C₄⋊#11764047	C₂	semidirect
9. #17984155	C₂²	semidirect
10. C₄⋊#11764047	C₂²	semidirect
28. C₂×C₄×D₁₃₇	C₈	semidirect
50. C₂×Dic₁₃₇	C₂×C₈	semidirect
59. C₄×D₁₃₇	C₂×C₈	semidirect
65. C₂×C₄×C₁₃₇	C₂×C₈	semidirect
73. C₂×D₁₃₇	C₄⋊C₈	semidirect
79. C₄×C₁₃₇	C₂²×C₈	semidirect
82. Dic₁₃₇	C₂²×C₈	semidirect
85. C₂×C₁₃₇	C₂×C₄⋊C₈	semidirect
89. D₁₃₇	C₂×C₄⋊C₈	semidirect
90. C₁₃₇	C₂²×C₄⋊C₈	semidirect
155. C₂×C₄	#14555518	semidirect
175. C₄	C₂×#14555518	semidirect
177. C₂	#22578404	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14555518	2
C₂×#17984155	2
#22578404	2
C₂×C₄⋊#11764047	2
C₂×C₄²×D₁₃₇	2
C₂²×C₄⋊C₈	137

Subgroups — C2×#22578404 #31797422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578404 #31797422