Subgroups — C₂×#22578269 #31796449

Order 17536.

528 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11764047	C₂	1	2	yes	maximal
2	#22578269	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₄×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11764047	—	2	4	no	—
5	C₂²×#11764047	—	2	4	no	—
6	C₂²×#11764047	C₄	1	8	yes	complemented
7	C₂²×#11764047	C₂²	1	4	yes	—
8	C₂³×Dic₁₃₇	C₄	1	1	yes	characteristic
9	C₂³×Dic₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₇	—	2	4	no	—
11	C₂×C₄×Dic₁₃₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃₇	C₄	1	1	yes	characteristic
14	C₂×#11764047	—	2	16	no	—
15	C₂×#11764047	—	2	4	no	—
16	C₂×#11764047	—	2	4	no	—
17	C₂×#11764047	C₂×C₄	1	8	yes	complemented
18	C₂²×Dic₁₃₇	—	2	2	no	—
19	C₂²×Dic₁₃₇	—	2	2	no	—
20	C₂²×Dic₁₃₇	C₈	1	4	yes	complemented
21	C₂²×Dic₁₃₇	C₂×C₄	1	4	yes	—
22	C₂²×Dic₁₃₇	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₁₃₇	D₄	1	4	yes	—
25	C₂²×Dic₁₃₇	D₄	1	2	yes	—
26	C₂²×Dic₁₃₇	Q₈	1	2	yes	—
27	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
28	C₄×Dic₁₃₇	—	2	4	no	—
29	C₄×Dic₁₃₇	—	2	4	no	—
30	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₇	—	2	2	no	—
32	C₂²×C₄×C₁₃₇	—	2	2	no	—
33	C₂²×C₄×C₁₃₇	C₈	1	4	yes	complemented
34	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
35	#11764047	—	2	8	no	—
36	C₂×Dic₁₃₇	—	2	8	no	—
37	C₂×Dic₁₃₇	—	2	2	no	—
38	C₂×Dic₁₃₇	—	2	2	no	—
39	C₂×Dic₁₃₇	C₄²	1	2	yes	—
40	C₂×Dic₁₃₇	C₂²⋊C₄	1	2	yes	—
41	C₂×Dic₁₃₇	C₂²⋊C₄	1	8	yes	—
42	C₂×Dic₁₃₇	C₄⋊C₄	1	2	yes	—
43	C₂×Dic₁₃₇	C₄⋊C₄	1	8	yes	—
44	C₂×Dic₁₃₇	C₂×C₈	1	4	yes	complemented
45	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
46	C₂×Dic₁₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₁₃₇	C₂×D₄	1	2	yes	—
48	C₂×Dic₁₃₇	C₂×Q₈	1	1	yes	characteristic
49	C₂³×C₁₃₇	C₄²	1	2	yes	—
50	C₂³×C₁₃₇	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₁₃₇	C₄⋊C₄	1	2	yes	—
52	C₂³×C₁₃₇	C₂×C₈	1	2	yes	—
53	C₂³×C₁₃₇	C₂×C₈	1	2	yes	—
54	C₂³×C₁₃₇	M₁₆	1	2	yes	—
55	C₂³×C₁₃₇	M₁₆	1	2	yes	—
56	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	—	2	8	no	—
58	C₂×C₄×C₁₃₇	—	2	2	no	—
59	C₂×C₄×C₁₃₇	—	2	2	no	—
60	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	complemented
61	C₂²×C₁₃₇	#95	1	4	yes	—
62	C₂²×C₁₃₇	C₄×C₈	1	4	yes	—
63	C₂²×C₁₃₇	#97	1	4	yes	—
64	C₂²×C₁₃₇	C₂²⋊C₈	1	4	yes	—
65	C₂²×C₁₃₇	C₂²⋊C₈	1	4	yes	—
66	C₂²×C₁₃₇	C₄⋊C₈	1	4	yes	—
67	C₂²×C₁₃₇	C₄⋊C₈	1	4	yes	—
68	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
69	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
72	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
73	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
75	C₄×C₁₃₇	—	2	4	no	—
76	Dic₁₃₇	—	2	4	no	—
77	Dic₁₃₇	#95	1	4	yes	—
78	Dic₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
79	Dic₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₁₃₇	#336	1	8	yes	complemented
81	C₂×C₁₃₇	C₂×#95	1	1	yes	characteristic
82	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂×C₁₃₇	C₂×#97	1	1	yes	characteristic
84	C₂×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₁₃₇	C₂×#336	1	1	yes	characteristic, complemented, Sylow
89	C₂×#336	—	137	1	no	maximal, Sylow
90	C₂³×C₈	—	137	2	no	—
91	C₂²×C₄²	—	137	1	no	—
92	#336	—	137	4	no	—
93	C₂³×C₄	—	137	1	no	—
94	C₂³×C₄	—	137	1	no	—
95	C₂³×C₄	#10587739	1	1	yes	characteristic
96	C₂²×C₈	—	274	4	no	—
97	C₂²×C₈	—	274	4	no	—
98	C₂²×C₈	—	137	4	no	—
99	C₂²×C₈	—	137	8	no	—
100	C₂×C₄²	—	274	4	no	—
101	C₂×C₄²	—	137	2	no	—
102	C₂×C₄²	—	137	2	no	—
103	C₂⁴	C₂×#10587739	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	274	2	no	—
107	C₂²×C₄	—	274	2	no	—
108	C₂²×C₄	—	137	4	no	—
109	C₂²×C₄	—	137	2	no	—
110	C₂²×C₄	—	137	4	no	—
111	C₂²×C₄	—	137	4	no	—
112	C₂²×C₄	—	137	2	no	—
113	C₂²×C₄	—	137	2	no	—
114	C₂²×C₄	—	137	1	no	—
115	C₂²×C₄	—	137	1	no	—
116	C₂²×C₄	#11764047	1	4	yes	complemented
117	C₂²×C₄	C₂×#10587739	1	2	yes	—
118	C₂×C₈	—	274	16	no	—
119	C₂×C₈	—	274	4	no	—
120	C₂×C₈	—	274	4	no	—
121	C₂×C₈	—	137	8	no	—
122	C₄²	—	274	4	no	—
123	C₄²	—	274	4	no	—
124	C₂³	C₄×#10587739	1	2	yes	—
125	C₂³	C₂²⋊#10587739	1	2	yes	—
126	C₂³	C₄⋊#10587739	1	2	yes	—
127	C₂³	C₂×#11764047	1	2	yes	—
128	C₂³	#14555518	1	2	yes	—
129	C₂³	#14555519	1	2	yes	—
130	C₂³	#14555520	1	2	yes	—
131	C₂³	C₂²×#10587739	1	1	yes	characteristic
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	274	8	no	—
136	C₂×C₄	—	274	2	no	—
137	C₂×C₄	—	274	2	no	—
138	C₂×C₄	—	137	4	no	—
139	C₂×C₄	—	137	8	no	—
140	C₂×C₄	—	137	8	no	—
141	C₂×C₄	—	137	2	no	—
142	C₂×C₄	—	137	2	no	—
143	C₂×C₄	—	137	2	no	—
144	C₂×C₄	—	137	2	no	—
145	C₂×C₄	—	137	1	no	—
146	C₂×C₄	—	137	2	no	—
147	C₂×C₄	—	137	1	no	—
148	C₂×C₄	C₂×#11764047	1	4	yes	complemented
149	C₈	—	274	8	no	—
150	C₂²	(C₂×#10587739)⋊C₄	1	4	yes	—
151	C₂²	C₄×#11764047	1	4	yes	—
152	C₂²	#17984143	1	4	yes	—
153	C₂²	C₂²⋊#11764047	1	4	yes	—
154	C₂²	#17984145	1	4	yes	—
155	C₂²	C₄⋊#11764047	1	4	yes	—
156	C₂²	#17984155	1	4	yes	—
157	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
158	C₂²	C₂×C₂²⋊#10587739	1	1	yes	characteristic
159	C₂²	C₂×C₄⋊#10587739	1	1	yes	characteristic
160	C₂²	C₂²×#11764047	1	1	yes	characteristic
161	C₂²	C₂×#14555518	1	1	yes	characteristic
162	C₂²	C₂×#14555519	1	1	yes	characteristic
163	C₂²	C₂×#14555520	1	1	yes	characteristic
164	C₄	—	2	4	no	—
165	C₄	—	274	4	no	—
166	C₄	—	137	2	no	—
167	C₄	—	137	4	no	—
168	C₄	—	137	2	no	—
169	C₂	#22578269	1	8	yes	direct product factor
170	C₂	C₂×(C₂×#10587739)⋊C₄	1	1	yes	characteristic
171	C₂	C₂×C₄×#11764047	1	1	yes	characteristic
172	C₂	C₂×#17984143	1	1	yes	characteristic
173	C₂	C₂×C₂²⋊#11764047	1	1	yes	characteristic
174	C₂	C₂×#17984145	1	1	yes	characteristic
175	C₂	C₂×C₄⋊#11764047	1	1	yes	characteristic
176	C₂	C₂×#17984155	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×#10587739
commutator	C₂×C₁₃₇	C₂×C₄×C₈
Frattini	C₂³	C₂²×#10587739
Fitting	C₂³×C₄×C₁₃₇	C₄
socle, prime radical	C₂⁴×C₁₃₇	C₂×C₄
cosocle	C₂²×Dic₁₃₇	C₂³
hypercenter	C₂³×C₄	#10587739
hypocenter	C₁₃₇	C₂×#336
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. #22578269	C₂	direct (N × H)
6. C₂²×#11764047	C₄	semidirect
17. C₂×#11764047	C₂×C₄	semidirect
20. C₂²×Dic₁₃₇	C₈	semidirect
33. C₂²×C₄×C₁₃₇	C₈	semidirect
44. C₂×Dic₁₃₇	C₂×C₈	semidirect
60. C₂×C₄×C₁₃₇	C₂×C₈	semidirect
80. C₂×C₁₃₇	#336	semidirect
88. C₁₃₇	C₂×#336	semidirect
116. C₂²×C₄	#11764047	semidirect
148. C₂×C₄	C₂×#11764047	semidirect
169. C₂	#22578269	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11764047	2
#22578269	2
C₂²×C₄×Dic₁₃₇	2
C₂×#336	137

Subgroups — C2×#22578269 #31796449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22578269 #31796449