Subgroups — #31789447

Order 17536.

402 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577464	C₂	1	4	yes	maximal, complemented
2	#22577460	C₂	1	2	yes	maximal, complemented
3	#22577456	C₂	1	2	yes	maximal, complemented
4	#22577430	C₂	1	2	yes	maximal, complemented
5	#22577367	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊#11764040	C₂	1	1	yes	maximal, characteristic
7	#22577359	C₂	1	2	yes	maximal, complemented
8	C₂×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14555494	C₂²	1	2	yes	—
10	C₂²×#11764040	C₂²	1	2	yes	—
11	C₄⋊#11764040	C₂²	1	4	yes	complemented
12	C₄⋊#11764040	C₂²	1	4	yes	complemented
13	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
14	C₂²⋊#11764040	C₂²	1	2	yes	—
15	C₂²⋊#11764040	C₂²	1	1	yes	characteristic
16	C₂²⋊#11764040	C₂²	1	2	yes	—
17	C₂²⋊#11764040	C₂²	1	2	yes	—
18	#17983999	C₂²	1	4	yes	complemented
19	C₄×#11764040	C₂²	1	4	yes	complemented
20	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
21	#117×C₁₃₇	C₄	1	1	yes	characteristic
22	#117×C₁₃₇	C₄	1	1	yes	characteristic
23	#117×C₁₃₇	C₄	1	2	yes	—
24	#117×C₁₃₇	C₂²	1	1	yes	characteristic
25	#117×C₁₃₇	C₂²	1	2	yes	—
26	#117×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
30	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
31	#14555494	—	2	4	no	—
32	C₂×#11764040	—	2	4	no	—
33	C₂×#11764040	C₂³	1	2	yes	—
34	C₂×#11764040	C₂³	1	2	yes	—
35	C₂×#11764040	C₂³	1	4	yes	—
36	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₇	—	2	4	no	—
38	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
43	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
56	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
57	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
58	C₄²×C₁₃₇	C₂³	1	2	yes	—
59	C₄²×C₁₃₇	C₂³	1	2	yes	—
60	#11764040	—	2	2	no	—
61	#11764040	—	2	2	no	—
62	#11764040	—	2	4	no	—
63	C₂³×C₁₃₇	—	2	2	no	—
64	C₂³×C₁₃₇	—	2	2	no	—
65	C₂³×C₁₃₇	—	4	1	no	—
66	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₄×C₁₃₇	—	2	4	no	—
71	C₂×C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₄×C₁₃₇	—	4	2	no	—
76	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
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	C₂×C₄×C₁₃₇	#41	1	4	yes	—
84	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
85	C₂²×C₁₃₇	—	2	1	no	—
86	C₂²×C₁₃₇	—	2	1	no	—
87	C₂²×C₁₃₇	—	2	1	no	—
88	C₂²×C₁₃₇	—	2	1	no	—
89	C₂²×C₁₃₇	—	2	1	no	—
90	C₂²×C₁₃₇	—	2	1	no	—
91	C₂²×C₁₃₇	—	2	1	no	—
92	C₂²×C₁₃₇	—	4	4	no	—
93	C₂²×C₁₃₇	#117	1	2	yes	—
94	C₂²×C₁₃₇	#131	1	2	yes	—
95	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
96	C₄×C₁₃₇	—	2	1	no	—
97	C₄×C₁₃₇	—	2	1	no	—
98	C₄×C₁₃₇	—	2	2	no	—
99	C₄×C₁₃₇	—	2	2	no	—
100	C₄×C₁₃₇	—	2	2	no	—
101	C₄×C₁₃₇	—	2	2	no	—
102	C₄×C₁₃₇	#117	1	2	yes	—
103	C₄×C₁₃₇	C₂×#41	1	2	yes	—
104	C₂×C₁₃₇	—	2	2	no	—
105	C₂×C₁₃₇	—	2	2	no	—
106	C₂×C₁₃₇	—	4	1	no	—
107	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
108	C₂×C₁₃₇	C₂×#131	1	1	yes	characteristic
109	C₂×C₁₃₇	#568	1	1	yes	characteristic
110	C₁₃₇	#2924	1	1	yes	characteristic, complemented, Sylow
111	#2924	—	137	1	no	maximal, Sylow
112	C₂×#117	D₁₃₇	1	1	yes	characteristic
113	#433	—	137	4	no	—
114	#432	—	137	2	no	—
115	#431	—	137	2	no	—
116	#424	—	137	2	no	—
117	#407	—	137	1	no	—
118	C₂×C₂²⋊C₈	—	137	1	no	—
119	#405	—	137	2	no	—
120	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
121	C₂×M₁₆	—	137	2	no	—
122	C₂²×C₈	—	137	2	no	—
123	#117	Dic₁₃₇	1	1	yes	characteristic
124	#117	Dic₁₃₇	1	1	yes	characteristic
125	#117	Dic₁₃₇	1	2	yes	—
126	#117	C₂×D₁₃₇	1	1	yes	characteristic
127	#117	C₂×D₁₃₇	1	1	yes	characteristic
128	#117	C₂×D₁₃₇	1	2	yes	—
129	C₂×C₄⋊C₄	Dic₁₃₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	Dic₁₃₇	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	Dic₁₃₇	1	2	yes	—
132	C₂×C₄²	C₂×D₁₃₇	1	2	yes	—
133	C₄⋊C₈	—	137	4	no	—
134	C₄⋊C₈	—	137	4	no	—
135	C₂²⋊C₈	—	137	1	no	—
136	C₂²⋊C₈	—	137	1	no	—
137	C₂²⋊C₈	—	137	2	no	—
138	C₂²⋊C₈	—	137	2	no	—
139	C₂²⋊C₈	—	137	2	no	—
140	#97	—	137	4	no	—
141	C₄×C₈	—	137	4	no	—
142	C₂⁴	C₂×Dic₁₃₇	1	1	yes	characteristic
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
149	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
152	M₁₆	—	274	4	no	—
153	C₂×C₈	—	274	4	no	—
154	C₂×C₈	—	137	2	no	—
155	C₂×C₈	—	137	4	no	—
156	C₂×C₈	—	137	2	no	—
157	C₄⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
159	C₄⋊C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
161	C₄⋊C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
162	C₄⋊C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
163	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
164	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
165	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
166	C₂²⋊C₄	C₂×Dic₁₃₇	1	2	yes	—
167	C₄²	C₂×Dic₁₃₇	1	2	yes	—
168	C₄²	C₂×Dic₁₃₇	1	2	yes	—
169	C₄²	C₂²×D₁₃₇	1	2	yes	—
170	C₄²	C₂²×D₁₃₇	1	2	yes	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	4	1	no	—
174	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
175	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
176	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
186	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
187	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
188	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
189	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
191	C₂×C₄	#14555510	1	4	yes	—
192	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
193	C₈	—	274	2	no	—
194	C₈	—	274	2	no	—
195	C₈	—	274	4	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	4	no	—
204	C₂²	#17984047	1	2	yes	—
205	C₂²	#17984095	1	2	yes	—
206	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	#17984047	1	2	yes	—
214	C₄	C₂×#14555510	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	2	no	—
217	C₂	—	4	1	no	—
218	C₂	C₂×#17984047	1	1	yes	characteristic
219	C₂	C₂×#17984095	1	1	yes	characteristic
220	C₂	#22578170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	C₂×#117×C₁₃₇	C₂
prime radical	C₂⁴×C₁₃₇	C₂×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂×#117	D₁₃₇
hypocenter	C₁₃₇	#2924
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22577464	C₂	semidirect
2. #22577460	C₂	semidirect
3. #22577456	C₂	semidirect
4. #22577430	C₂	semidirect
7. #22577359	C₂	semidirect
11. C₄⋊#11764040	C₂²	semidirect
12. C₄⋊#11764040	C₂²	semidirect
18. #17983999	C₂²	semidirect
19. C₄×#11764040	C₂²	semidirect
110. C₁₃₇	#2924	semidirect

Maximal subgroups

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

subgroup	index
#22577464	2
#22577460	2
#22577456	2
#22577430	2
#22577367	2
C₂×C₂²⋊#11764040	2
#22577359	2
C₂×#117×C₁₃₇	2
#2924	137