Subgroups — C₂×#22577425 #31789342

Order 17536.

576 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555493	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#22577425	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11764040	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17984017	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₃₇	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₁₃₇	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#14555493	—	2	8	no	—
11	C₂×#14555493	C₂²	1	8	yes	—
12	C₂²×#11764040	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
16	C₄²×D₁₃₇	C₄	1	4	yes	—
17	C₄²×D₁₃₇	C₂²	1	4	yes	—
18	C₄⋊#11764040	C₂²	1	4	yes	complemented
19	#17984017	C₂²	1	8	yes	complemented
20	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	complemented
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	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₁₃₇	—	2	4	no	—
28	C₂×C₄×D₁₃₇	—	2	4	no	—
29	C₂×C₄×D₁₃₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₁₃₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×D₁₃₇	C₂×C₄	1	2	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	#14555493	—	2	16	no	—
36	C₂×#11764040	—	2	4	no	—
37	C₂×#11764040	C₂³	1	4	yes	—
38	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
39	C₄×Dic₁₃₇	C₂×C₄	1	4	yes	—
40	C₄×Dic₁₃₇	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₃₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₃₇	—	2	4	no	—
46	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
47	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
48	C₄²×C₁₃₇	C₂³	1	2	yes	—
49	C₂²×D₁₃₇	—	2	2	no	—
50	C₂²×D₁₃₇	—	2	2	no	—
51	C₂²×D₁₃₇	C₄⋊C₄	1	4	yes	—
52	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
53	C₂×Dic₁₃₇	—	2	2	no	—
54	C₂×Dic₁₃₇	C₄⋊C₄	1	4	yes	—
55	C₂×Dic₁₃₇	M₁₆	1	4	yes	complemented
56	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
57	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
58	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
59	C₄×D₁₃₇	—	2	8	no	—
60	C₄×D₁₃₇	—	2	8	no	—
61	C₄×D₁₃₇	C₄⋊C₄	1	8	yes	—
62	C₄×D₁₃₇	C₂×D₄	1	4	yes	—
63	C₄×D₁₃₇	C₂×Q₈	1	4	yes	—
64	#11764040	—	2	4	no	—
65	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₇	—	2	2	no	—
67	C₂×C₄×C₁₃₇	M₁₆	1	4	yes	complemented
68	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
74	C₈×C₁₃₇	—	2	4	no	—
75	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₁₃₇	C₂×M₁₆	1	2	yes	—
77	C₂²×C₁₃₇	C₂×M₁₆	1	2	yes	—
78	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
79	C₂×D₁₃₇	—	2	2	no	—
80	C₂×D₁₃₇	—	2	2	no	—
81	C₂×D₁₃₇	—	2	8	no	—
82	C₂×D₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₄×C₁₃₇	—	2	2	no	—
84	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₄×C₁₃₇	C₂×M₁₆	1	4	yes	complemented
86	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
88	Dic₁₃₇	—	2	2	no	—
89	Dic₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
90	Dic₁₃₇	C₂×M₁₆	1	4	yes	complemented
91	C₂×C₁₃₇	#423	1	4	yes	complemented
92	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
94	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
95	D₁₃₇	—	2	4	no	—
96	C₁₃₇	C₂×#423	1	1	yes	characteristic, complemented, Sylow
97	C₂×#423	—	137	1	no	maximal, Sylow
98	C₂²×M₁₆	—	137	2	no	—
99	C₂²×C₄²	—	137	1	no	—
100	#423	—	137	8	no	—
101	C₂×C₄⋊C₈	—	137	2	no	—
102	C₂×C₄⋊C₈	—	137	1	no	—
103	C₂×C₄⋊C₈	D₁₃₇	1	1	yes	characteristic, complemented
104	C₂³×C₄	—	137	1	no	—
105	C₂³×C₄	—	137	1	no	—
106	C₂³×C₄	—	137	1	no	—
107	C₂×M₁₆	—	274	8	no	—
108	C₂×M₁₆	—	137	8	no	—
109	C₂²×C₈	—	137	2	no	—
110	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
111	C₂×C₄²	—	137	1	no	—
112	C₂×C₄²	—	137	1	no	—
113	C₂×C₄²	—	137	1	no	—
114	C₂×C₄²	—	137	4	no	—
115	C₂×C₄²	—	137	4	no	—
116	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
117	C₄⋊C₈	—	137	8	no	—
118	C₄⋊C₈	—	137	4	no	—
119	C₄⋊C₈	C₂×D₁₃₇	1	4	yes	complemented
120	C₂⁴	—	137	1	no	—
121	C₂²×C₄	—	274	4	no	—
122	C₂²×C₄	—	274	4	no	—
123	C₂²×C₄	—	137	1	no	—
124	C₂²×C₄	—	137	4	no	—
125	C₂²×C₄	—	137	4	no	—
126	C₂²×C₄	—	137	1	no	—
127	C₂²×C₄	—	137	1	no	—
128	C₂²×C₄	—	137	2	no	—
129	C₂²×C₄	—	137	4	no	—
130	C₂²×C₄	—	137	4	no	—
131	C₂²×C₄	—	137	2	no	—
132	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
133	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
135	M₁₆	—	274	16	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	274	4	no	—
138	C₂×C₈	—	137	4	no	—
139	C₂×C₈	C₂²×D₁₃₇	1	4	yes	—
140	C₄²	—	137	4	no	—
141	C₄²	—	137	2	no	—
142	C₄²	—	137	4	no	—
143	C₄²	—	137	2	no	—
144	C₄²	C₄×D₁₃₇	1	2	yes	—
145	C₄²	C₂²×D₁₃₇	1	2	yes	—
146	C₂³	—	274	2	no	—
147	C₂³	—	274	2	no	—
148	C₂³	—	137	4	no	—
149	C₂³	—	137	2	no	—
150	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	274	8	no	—
153	C₂×C₄	—	274	2	no	—
154	C₂×C₄	—	274	8	no	—
155	C₂×C₄	—	137	2	no	—
156	C₂×C₄	—	137	4	no	—
157	C₂×C₄	—	137	4	no	—
158	C₂×C₄	—	137	2	no	—
159	C₂×C₄	—	137	2	no	—
160	C₂×C₄	—	137	8	no	—
161	C₂×C₄	—	137	4	no	—
162	C₂×C₄	—	137	4	no	—
163	C₂×C₄	#14555493	1	4	yes	complemented
164	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
165	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
167	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
168	C₂×C₄	Q₈×D₁₃₇	1	2	yes	—
169	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	274	4	no	—
172	C₂²	—	274	8	no	—
173	C₂²	—	274	2	no	—
174	C₂²	—	274	2	no	—
175	C₂²	—	137	4	no	—
176	C₂²	C₄⋊C₄×D₁₃₇	1	2	yes	—
177	C₂²	M₁₆×D₁₃₇	1	2	yes	—
178	C₂²	C₂×#14555493	1	2	yes	—
179	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	274	2	no	—
182	C₄	—	137	4	no	—
183	C₄	—	137	4	no	—
184	C₄	C₄⋊C₄×D₁₃₇	1	2	yes	—
185	C₄	C₂×#14555493	1	4	yes	complemented
186	C₄	C₂×D₄×D₁₃₇	1	1	yes	characteristic
187	C₄	C₂×Q₈×D₁₃₇	1	1	yes	characteristic
188	C₂	—	274	4	no	—
189	C₂	#22577425	1	4	yes	direct product factor
190	C₂	C₂×C₄⋊C₄×D₁₃₇	1	1	yes	characteristic
191	C₂	C₂×M₁₆×D₁₃₇	1	1	yes	characteristic
192	C₂	C₂²×#14555493	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⋊C₈×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂²×C₄
prime radical	C₂³×D₁₃₇	C₂×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₃₇
hypocenter	C₁₃₇	C₂×#423
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #22577425	C₂	direct (N × H)
4. C₂×C₄⋊#11764040	C₂	semidirect
5. C₂×#17984017	C₂	semidirect
6. C₂×C₄⋊C₈×C₁₃₇	C₂	semidirect
18. C₄⋊#11764040	C₂²	semidirect
19. #17984017	C₂²	semidirect
22. C₄⋊C₈×C₁₃₇	C₂²	semidirect
55. C₂×Dic₁₃₇	M₁₆	semidirect
67. C₂×C₄×C₁₃₇	M₁₆	semidirect
85. C₄×C₁₃₇	C₂×M₁₆	semidirect
90. Dic₁₃₇	C₂×M₁₆	semidirect
91. C₂×C₁₃₇	#423	semidirect
96. C₁₃₇	C₂×#423	semidirect
103. C₂×C₄⋊C₈	D₁₃₇	semidirect
119. C₄⋊C₈	C₂×D₁₃₇	semidirect
163. C₂×C₄	#14555493	semidirect
185. C₄	C₂×#14555493	semidirect
189. C₂	#22577425	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×#14555493	2
C₂×C₄²×D₁₃₇	2
#22577425	2
C₂×C₄⋊#11764040	2
C₂×#17984017	2
C₂×C₄⋊C₈×C₁₃₇	2
C₂×#423	137

Subgroups — C2×#22577425 #31789342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577425 #31789342