Subgroups — C₈×#14555510 #31789746

Order 17536.

456 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14555510	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₂₇₄	C₂	1	2	yes	maximal, complemented
3	C₈×D₅₄₈	C₂	1	2	yes	maximal, complemented
4	C₈×C₂²⋊D₁₃₇	C₂	1	4	yes	maximal, complemented
5	#22577456	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577454	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×D₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14555510	C₄	1	1	yes	characteristic
10	C₂×C₈×D₁₃₇	C₂²	1	4	yes	—
11	C₄×Dic₂₇₄	C₄	1	1	yes	characteristic
12	C₄×Dic₂₇₄	C₂²	1	1	yes	characteristic
13	C₄×D₅₄₈	C₄	1	1	yes	characteristic
14	C₄×D₅₄₈	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₃₇	C₄	1	2	yes	—
16	C₄×C₂²⋊D₁₃₇	C₂²	1	2	yes	—
17	#17984047	C₄	1	1	yes	characteristic
18	#17984045	C₄	1	2	yes	—
19	C₄²×D₁₃₇	C₂²	1	2	yes	—
20	C₄⋊#11764040	C₂²	1	2	yes	—
21	#17984017	C₂²	1	4	yes	—
22	C₂²⋊#11764040	C₂²	1	2	yes	—
23	#17984000	C₂²	1	4	yes	—
24	C₄×#11764040	C₂²	1	2	yes	—
25	C₈×Dic₁₃₇	C₂²	1	4	yes	—
26	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
30	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
31	#14555510	C₈	1	8	yes	direct product factor
32	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
33	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
38	C₂×#11764040	C₂³	1	4	yes	—
39	C₈×D₁₃₇	—	2	8	no	—
40	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
42	#14555489	C₂×C₄	1	2	yes	—
43	#14555489	C₂×C₄	1	2	yes	—
44	C₂²⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
45	#14555487	C₂×C₄	1	2	yes	—
46	#14555487	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₃₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃₇	—	2	4	no	—
54	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
55	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
56	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
60	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
61	Dic₂₇₄	C₂×C₈	1	4	yes	complemented
62	D₅₄₈	C₂×C₈	1	4	yes	complemented
63	C₂²⋊D₁₃₇	C₂×C₈	1	8	yes	complemented
64	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
65	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
66	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
67	C₄×D₁₃₇	—	2	4	no	—
68	C₄×D₁₃₇	—	2	4	no	—
69	C₄×D₁₃₇	C₂×C₈	1	8	yes	complemented
70	#11764040	—	2	4	no	—
71	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	complemented
75	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
81	C₈×C₁₃₇	—	2	2	no	—
82	C₈×C₁₃₇	#41	1	4	yes	complemented
83	C₂²×C₁₃₇	—	2	1	no	—
84	C₂²×C₁₃₇	—	2	1	no	—
85	C₂²×C₁₃₇	C₂²×C₈	1	2	yes	—
86	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₁₃₇	—	2	2	no	—
88	C₂×D₁₃₇	—	2	2	no	—
89	C₂×D₁₃₇	C₂²×C₈	1	4	yes	—
90	C₄×C₁₃₇	—	2	1	no	—
91	C₄×C₁₃₇	—	2	1	no	—
92	C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
93	C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
94	C₄×C₁₃₇	#131	1	2	yes	—
95	C₄×C₁₃₇	C₂×#41	1	2	yes	—
96	Dic₁₃₇	—	2	2	no	—
97	Dic₁₃₇	—	2	2	no	—
98	Dic₁₃₇	C₂²×C₈	1	4	yes	—
99	C₂×C₁₃₇	—	2	2	no	—
100	C₂×C₁₃₇	C₄×#41	1	1	yes	characteristic
101	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
102	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
103	D₁₃₇	—	2	4	no	—
104	C₁₃₇	C₈×#41	1	1	yes	characteristic, complemented, Sylow
105	C₈×#41	—	137	1	no	maximal, Sylow
106	C₄×#41	—	137	1	no	—
107	C₈×Q₈	—	137	2	no	—
108	C₈×D₄	—	137	4	no	—
109	C₈×D₄	—	137	2	no	—
110	#431	—	137	1	no	—
111	#431	—	137	2	no	—
112	C₂×C₄×C₈	—	137	2	no	—
113	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic, complemented
114	C₂×#41	—	137	1	no	—
115	C₂²×C₈	—	137	4	no	—
116	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
117	C₄×Q₈	—	137	1	no	—
118	C₄×Q₈	—	137	1	no	—
119	C₄×D₄	—	137	2	no	—
120	C₄×D₄	—	137	2	no	—
121	C₄×D₄	—	137	1	no	—
122	C₄×D₄	—	137	1	no	—
123	#117	—	137	1	no	—
124	#117	—	137	2	no	—
125	C₂×C₄²	—	137	2	no	—
126	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
127	C₄⋊C₈	—	137	4	no	—
128	C₄⋊C₈	—	137	2	no	—
129	C₂²⋊C₈	—	137	2	no	—
130	C₂²⋊C₈	—	137	4	no	—
131	C₄×C₈	—	137	4	no	—
132	C₄×C₈	—	137	2	no	—
133	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
134	C₄×C₈	C₂×D₁₃₇	1	2	yes	—
135	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
136	#41	—	137	8	no	—
137	C₂×Q₈	—	137	1	no	—
138	C₂×D₄	—	137	2	no	—
139	C₂×D₄	—	137	1	no	—
140	C₂²×C₄	—	137	2	no	—
141	C₂²×C₄	—	137	2	no	—
142	C₂²×C₄	—	137	2	no	—
143	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	274	8	no	—
148	C₂×C₈	—	137	4	no	—
149	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
150	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
151	C₄⋊C₄	—	137	2	no	—
152	C₄⋊C₄	—	137	2	no	—
153	C₄⋊C₄	—	137	1	no	—
154	C₄⋊C₄	—	137	1	no	—
155	C₂²⋊C₄	—	137	2	no	—
156	C₂²⋊C₄	—	137	2	no	—
157	C₂²⋊C₄	—	137	2	no	—
158	C₄²	—	137	2	no	—
159	C₄²	—	137	2	no	—
160	C₄²	—	137	2	no	—
161	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
162	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
163	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
164	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
165	C₂³	—	137	2	no	—
166	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
167	Q₈	—	137	4	no	—
168	D₄	—	137	8	no	—
169	D₄	—	137	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	274	4	no	—
173	C₂×C₄	—	274	4	no	—
174	C₂×C₄	—	137	2	no	—
175	C₂×C₄	—	137	2	no	—
176	C₂×C₄	—	137	2	no	—
177	C₂×C₄	—	137	8	no	—
178	C₂×C₄	C₈×D₁₃₇	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	274	4	no	—
187	C₈	#14555510	1	4	yes	direct product factor
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	274	2	no	—
191	C₂²	—	274	2	no	—
192	C₂²	—	137	4	no	—
193	C₂²	C₂×C₈×D₁₃₇	1	2	yes	—
194	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	274	2	no	—
198	C₄	—	274	2	no	—
199	C₄	—	137	4	no	—
200	C₄	C₂×C₈×D₁₃₇	1	2	yes	—
201	C₄	C₂×C₈×D₁₃₇	1	2	yes	—
202	C₄	#17984094	1	2	yes	—
203	C₄	C₂×#14555510	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	274	4	no	—
206	C₂	C₄×#14555510	1	1	yes	characteristic
207	C₂	C₂²×C₈×D₁₃₇	1	1	yes	characteristic
208	C₂	C₂×#17984094	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₁₃₇
commutator	C₂×C₁₃₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂³×C₄
prime radical	C₂×#14555510	C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	C₈×#41
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₂₇₄	C₂	semidirect
3. C₈×D₅₄₈	C₂	semidirect
4. C₈×C₂²⋊D₁₃₇	C₂	semidirect
5. #22577456	C₂	semidirect
6. #22577454	C₂	semidirect
7. C₄×C₈×D₁₃₇	C₂	semidirect
8. C₂×C₄×C₈×C₁₃₇	C₂	semidirect
31. #14555510	C₈	direct (N × H)
61. Dic₂₇₄	C₂×C₈	semidirect
62. D₅₄₈	C₂×C₈	semidirect
63. C₂²⋊D₁₃₇	C₂×C₈	semidirect
69. C₄×D₁₃₇	C₂×C₈	semidirect
74. C₂×C₄×C₁₃₇	C₂×C₈	semidirect
82. C₈×C₁₃₇	#41	semidirect
104. C₁₃₇	C₈×#41	semidirect
113. C₂×C₄×C₈	D₁₃₇	semidirect
178. C₂×C₄	C₈×D₁₃₇	semidirect
187. C₈	#14555510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#14555510	2
C₈×Dic₂₇₄	2
C₈×D₅₄₈	2
C₈×C₂²⋊D₁₃₇	2
#22577456	2
#22577454	2
C₄×C₈×D₁₃₇	2
C₂×C₄×C₈×C₁₃₇	2
C₈×#41	137

Subgroups — C8×#14555510 #31789746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14555510 #31789746