Subgroups — #31797593

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	#22578452	C₂	1	2	yes	maximal, complemented
2	#22578431	C₂	1	2	yes	maximal, complemented
3	#22578430	C₂	1	4	yes	maximal, complemented
4	#22578421	C₂	1	2	yes	maximal, complemented
5	#22578419	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14555518	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577871	C₂	1	1	yes	maximal, characteristic
9	C₂×#14555518	C₂²	1	4	yes	—
10	C₂²×#11764047	C₂²	1	2	yes	—
11	#17984155	C₂²	1	4	yes	—
12	C₄⋊#11764047	C₂²	1	2	yes	—
13	#17984145	C₂²	1	4	yes	—
14	C₂²⋊#11764047	C₂²	1	2	yes	—
15	C₄×#11764047	C₂²	1	2	yes	—
16	C₄×#11764047	C₂²	1	4	yes	—
17	C₄×#11764047	C₂²	1	1	yes	characteristic
18	C₄×#11764047	C₂²	1	1	yes	characteristic
19	C₄×#11764047	C₂²	1	2	yes	—
20	C₂×#14555510	C₄	1	1	yes	characteristic
21	#17984053	C₄	1	1	yes	characteristic
22	#17984053	C₂²	1	1	yes	characteristic
23	#17984050	C₄	1	1	yes	characteristic
24	#17984050	C₂²	1	1	yes	characteristic
25	#17984048	C₄	1	2	yes	—
26	#17984048	C₂²	1	2	yes	—
27	#17984045	C₄	1	2	yes	—
28	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
29	C₄²×D₁₃₇	C₂²	1	2	yes	—
30	#117×C₁₃₇	C₄	1	1	yes	characteristic
31	#14555518	—	2	8	no	—
32	C₂×#11764047	—	2	4	no	—
33	C₂×#11764047	C₂³	1	4	yes	—
34	C₂×#11764047	C₂³	1	2	yes	—
35	C₂×#11764047	C₂³	1	2	yes	—
36	#14555510	C₈	1	8	yes	complemented
37	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
38	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
45	#14555489	C₂×C₄	1	2	yes	—
46	#14555489	C₂×C₄	1	2	yes	—
47	#14555487	C₂×C₄	1	2	yes	—
48	#14555487	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
51	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
52	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
53	C₄×Dic₁₃₇	C₂³	1	2	yes	—
54	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
59	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
60	#11764047	—	2	2	no	—
61	#11764047	—	2	4	no	—
62	#11764047	#41	1	4	yes	complemented
63	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
64	Dic₂₇₄	C₂×C₈	1	4	yes	complemented
65	D₅₄₈	C₂×C₈	1	4	yes	complemented
66	C₂²⋊D₁₃₇	C₂×C₈	1	8	yes	complemented
67	C₂×Dic₁₃₇	—	2	2	no	—
68	C₂×Dic₁₃₇	—	2	2	no	—
69	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₁₃₇	C₂⁴	1	1	yes	characteristic
74	C₄×D₁₃₇	—	2	4	no	—
75	C₄×D₁₃₇	—	2	4	no	—
76	C₄×D₁₃₇	C₂×C₈	1	8	yes	complemented
77	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	complemented
79	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
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	2	no	—
91	C₄×C₁₃₇	—	2	2	no	—
92	C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
93	C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
94	Dic₁₃₇	—	2	1	no	—
95	Dic₁₃₇	—	2	1	no	—
96	Dic₁₃₇	C₂²×C₈	1	4	yes	—
97	Dic₁₃₇	#131	1	2	yes	—
98	Dic₁₃₇	C₂×#41	1	2	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	1	no	—
113	C₂×C₄×C₈	—	137	2	no	—
114	C₂×#41	—	137	1	no	—
115	C₂²×C₈	—	137	2	no	—
116	C₂²×C₈	—	137	4	no	—
117	C₄×Q₈	—	137	1	no	—
118	C₄×Q₈	—	137	1	no	—
119	C₄×D₄	—	137	1	no	—
120	C₄×D₄	—	137	1	no	—
121	C₄×D₄	—	137	2	no	—
122	C₄×D₄	—	137	2	no	—
123	#117	—	137	2	no	—
124	#117	#10587739	1	1	yes	characteristic
125	C₂×C₄²	—	137	1	no	—
126	C₂×C₄²	—	137	2	no	—
127	C₄⋊C₈	—	137	2	no	—
128	C₄⋊C₈	—	137	4	no	—
129	C₂²⋊C₈	—	137	2	no	—
130	C₂²⋊C₈	—	137	4	no	—
131	C₄×C₈	—	137	1	no	—
132	C₄×C₈	—	137	1	no	—
133	C₄×C₈	—	137	2	no	—
134	C₄×C₈	—	137	2	no	—
135	C₄×C₈	—	137	4	no	—
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	1	no	—
141	C₂²×C₄	—	137	1	no	—
142	C₂²×C₄	—	137	2	no	—
143	C₂²×C₄	—	137	2	no	—
144	C₂²×C₄	—	137	2	no	—
145	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
146	C₂×C₈	—	274	4	no	—
147	C₂×C₈	—	274	8	no	—
148	C₂×C₈	—	137	4	no	—
149	C₂×C₈	—	137	2	no	—
150	C₂×C₈	—	137	2	no	—
151	C₄⋊C₄	—	137	2	no	—
152	C₄⋊C₄	—	137	2	no	—
153	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
154	C₄⋊C₄	C₂×#10587739	1	1	yes	characteristic
155	C₂²⋊C₄	—	137	2	no	—
156	C₂²⋊C₄	—	137	2	no	—
157	C₂²⋊C₄	C₂×#10587739	1	2	yes	—
158	C₄²	—	137	1	no	—
159	C₄²	—	137	2	no	—
160	C₄²	—	137	1	no	—
161	C₄²	—	137	2	no	—
162	C₄²	—	137	1	no	—
163	C₄²	—	137	1	no	—
164	C₄²	C₂×#10587739	1	2	yes	—
165	C₂³	—	137	2	no	—
166	C₂³	C₂²×#10587739	1	1	yes	characteristic
167	Q₈	—	137	4	no	—
168	D₄	—	137	8	no	—
169	D₄	—	137	4	no	—
170	C₂×C₄	—	274	4	no	—
171	C₂×C₄	—	274	4	no	—
172	C₂×C₄	—	274	2	no	—
173	C₂×C₄	—	274	2	no	—
174	C₂×C₄	—	137	1	no	—
175	C₂×C₄	—	137	1	no	—
176	C₂×C₄	—	137	1	no	—
177	C₂×C₄	—	137	1	no	—
178	C₂×C₄	—	137	2	no	—
179	C₂×C₄	—	137	8	no	—
180	C₂×C₄	#14555518	1	4	yes	complemented
181	C₂×C₄	C₂²×#10587739	1	2	yes	—
182	C₂×C₄	C₂²×#10587739	1	2	yes	—
183	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
184	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
185	C₈	—	274	2	no	—
186	C₈	—	274	4	no	—
187	C₈	—	137	4	no	—
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×#14555518	1	2	yes	—
194	C₂²	C₂³×#10587739	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	274	1	no	—
198	C₄	—	274	1	no	—
199	C₄	—	137	2	no	—
200	C₄	—	137	4	no	—
201	C₄	—	137	2	no	—
202	C₄	C₂×#14555518	1	2	yes	—
203	C₄	C₂×#14555518	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	274	4	no	—
206	C₂	#41×#10587739	1	1	yes	characteristic
207	C₂	C₂²×#14555518	1	1	yes	characteristic
208	C₂	D₁₃₇⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10587739
commutator	C₂×C₁₃₇	C₂³×C₈
Frattini	C₂²	C₂³×#10587739
Fitting	#117×C₁₃₇	C₄
socle	C₂²×C₁₃₇	C₂³×C₄
prime radical	C₂×#14555510	C₄
cosocle	C₂×Dic₁₃₇	C₂⁴
hypercenter	#117	#10587739
hypocenter	C₁₃₇	C₈×#41
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. #22578452	C₂	semidirect
2. #22578431	C₂	semidirect
3. #22578430	C₂	semidirect
4. #22578421	C₂	semidirect
5. #22578419	C₂	semidirect
6. C₄×#14555518	C₂	semidirect
7. C₂×C₄×#11764047	C₂	semidirect
36. #14555510	C₈	semidirect
62. #11764047	#41	semidirect
64. Dic₂₇₄	C₂×C₈	semidirect
65. D₅₄₈	C₂×C₈	semidirect
66. C₂²⋊D₁₃₇	C₂×C₈	semidirect
76. C₄×D₁₃₇	C₂×C₈	semidirect
78. C₂×C₄×C₁₃₇	C₂×C₈	semidirect
104. C₁₃₇	C₈×#41	semidirect
180. C₂×C₄	#14555518	semidirect

Maximal subgroups

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

subgroup	index
#22578452	2
#22578431	2
#22578430	2
#22578421	2
#22578419	2
C₄×#14555518	2
C₂×C₄×#11764047	2
#22577871	2
C₈×#41	137