Subgroups — #31789494

Order 17536.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984094	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14555493	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577915	C₂	1	1	yes	maximal, characteristic
4	#22577482	C₂	1	4	yes	maximal, complemented
5	#22577476	C₂	1	4	yes	maximal, complemented
6	#22577430	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577371	C₂	1	1	yes	maximal, characteristic, complemented
8	#22577366	C₂	1	1	yes	maximal, characteristic, complemented
9	#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14555510	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₃₇	C₂²	1	1	yes	characteristic
12	#17984094	—	2	4	no	—
13	C₂×#14555494	C₂²	1	1	yes	characteristic
14	C₂×#14555493	—	2	4	no	—
15	C₂×#14555493	C₂²	1	2	yes	—
16	C₂×#14555493	C₂²	1	2	yes	—
17	C₂×#14555493	C₂²	1	2	yes	complemented
18	C₂²×#11764040	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₃₇	C₂²	1	2	yes	complemented
20	#17984072	C₄	1	2	yes	—
21	#17984068	C₄	1	2	yes	—
22	#17984063	C₄	1	2	yes	—
23	#17984058	C₄	1	2	yes	—
24	C₄×D₅₄₈	C₂²	1	4	yes	—
25	C₄⋊#11764040	C₂²	1	4	yes	complemented
26	#17984000	C₂²	1	4	yes	—
27	#17984000	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#117×C₁₃₇	C₂²	1	1	yes	characteristic
31	#97×C₁₃₇	C₂²	1	2	yes	complemented
32	C₄×C₈×C₁₃₇	C₂²	1	2	yes	complemented
33	C₂³×D₁₃₇	C₂×C₄	1	1	yes	characteristic
34	#14555510	—	2	2	no	—
35	#14555510	—	2	2	no	—
36	C₂×Dic₂₇₄	C₂×C₄	1	1	yes	characteristic
37	C₂×D₅₄₈	C₂×C₄	1	1	yes	characteristic
38	C₂×D₅₄₈	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₃₇	—	2	2	no	—
42	C₂×C₄×D₁₃₇	—	2	2	no	—
43	C₂×C₄×D₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
45	C₂×C₄×D₁₃₇	C₂³	1	2	yes	—
46	#14555494	—	2	2	no	—
47	#14555493	—	2	4	no	—
48	#14555493	—	2	4	no	—
49	#14555493	—	2	4	no	—
50	C₂×#11764040	—	2	2	no	—
51	C₂×#11764040	C₂³	1	2	yes	—
52	C₂×#11764040	C₂³	1	2	yes	—
53	C₈×D₁₃₇	—	2	4	no	—
54	C₄⋊Dic₁₃₇	C₂×C₄	1	4	yes	—
55	#14555487	C₂×C₄	1	4	yes	—
56	#14555487	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₃₇	—	2	2	no	—
59	C₂×C₈×C₁₃₇	D₄	1	4	yes	complemented
60	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
61	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
65	C₄²×C₁₃₇	C₂³	1	2	yes	—
66	C₂²×D₁₃₇	—	2	2	no	—
67	C₂²×D₁₃₇	—	4	2	no	—
68	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
69	C₂²×D₁₃₇	C₂²×C₄	1	2	yes	—
70	Dic₂₇₄	—	2	2	no	—
71	D₅₄₈	—	2	2	no	—
72	D₅₄₈	—	4	4	no	—
73	C₂²⋊D₁₃₇	—	2	4	no	—
74	C₂×Dic₁₃₇	—	2	2	no	—
75	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃₇	C₂²×C₄	1	2	yes	—
77	C₄×D₁₃₇	—	2	4	no	—
78	C₄×D₁₃₇	—	2	2	no	—
79	C₄×D₁₃₇	—	2	2	no	—
80	C₄×D₁₃₇	—	2	2	no	—
81	C₄×D₁₃₇	—	2	2	no	—
82	#11764040	—	2	2	no	—
83	#11764040	—	2	2	no	—
84	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
86	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₃₇	#41	1	2	yes	—
89	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₃₇	—	2	2	no	—
91	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₈×C₁₃₇	C₂×D₄	1	2	yes	—
93	C₂²×C₁₃₇	—	2	2	no	—
94	C₂²×C₁₃₇	C₄×D₄	1	2	yes	—
95	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₃₇	—	2	2	no	—
97	C₂×D₁₃₇	—	2	2	no	—
98	C₂×D₁₃₇	—	2	2	no	—
99	C₂×D₁₃₇	—	2	4	no	—
100	C₂×D₁₃₇	—	2	2	no	—
101	C₂×D₁₃₇	—	4	4	no	—
102	C₄×C₁₃₇	—	2	4	no	—
103	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
104	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
106	Dic₁₃₇	—	2	2	no	—
107	Dic₁₃₇	—	2	2	no	—
108	C₂×C₁₃₇	—	2	2	no	—
109	C₂×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂×C₁₃₇	#568	1	2	yes	—
111	D₁₃₇	—	4	2	no	—
112	D₁₃₇	—	4	2	no	—
113	C₁₃₇	#2930	1	1	yes	characteristic, complemented, Sylow
114	#2930	—	137	1	no	maximal, Sylow
115	C₂×#131	—	137	1	no	—
116	C₂²×M₁₆	—	137	1	no	—
117	#525	—	137	1	no	—
118	#436	—	137	4	no	—
119	#435	—	137	4	no	—
120	#424	—	137	1	no	—
121	#408	—	137	1	no	—
122	#407	—	137	1	no	—
123	#405	D₁₃₇	1	1	yes	characteristic, complemented
124	C₂×#41	—	137	1	no	—
125	C₂³×C₄	—	137	1	no	—
126	#131	—	274	4	no	—
127	C₂×M₁₆	—	274	4	no	—
128	C₂×M₁₆	—	137	1	no	—
129	C₂×M₁₆	—	137	2	no	—
130	C₂×M₁₆	—	137	2	no	—
131	C₂×M₁₆	—	137	2	no	—
132	C₂×M₁₆	C₂×D₁₃₇	1	1	yes	characteristic
133	C₂²×C₈	—	137	1	no	—
134	C₂²×C₈	—	137	2	no	—
135	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
136	#123	—	137	2	no	—
137	C₂²⋊Q₈	—	137	2	no	—
138	#121	—	137	2	no	—
139	#120	—	137	2	no	—
140	C₄×D₄	—	137	4	no	—
141	#117	C₂×D₁₃₇	1	1	yes	characteristic
142	C₄⋊C₈	—	137	4	no	—
143	C₂²⋊C₈	—	137	4	no	—
144	C₂²⋊C₈	—	137	4	no	—
145	#97	C₂×D₁₃₇	1	2	yes	complemented
146	C₄×C₈	C₂×D₁₃₇	1	2	yes	complemented
147	C₂⁴	—	137	1	no	—
148	#41	—	274	2	no	—
149	#41	—	274	2	no	—
150	C₂×Q₈	—	137	1	no	—
151	C₂×D₄	—	137	2	no	—
152	C₂×D₄	—	137	1	no	—
153	C₂×D₄	—	137	4	no	—
154	C₂²×C₄	—	274	2	no	—
155	C₂²×C₄	—	274	2	no	—
156	C₂²×C₄	—	137	1	no	—
157	C₂²×C₄	—	137	2	no	—
158	C₂²×C₄	—	137	2	no	—
159	C₂²×C₄	—	137	2	no	—
160	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	274	2	no	—
163	M₁₆	—	274	4	no	—
164	M₁₆	—	274	4	no	—
165	M₁₆	—	274	4	no	—
166	C₂×C₈	—	274	2	no	—
167	C₂×C₈	—	274	4	no	—
168	C₂×C₈	—	137	2	no	—
169	C₂×C₈	—	137	2	no	—
170	C₂×C₈	D₅₄₈	1	4	yes	complemented
171	C₂×C₈	C₂²×D₁₃₇	1	2	yes	—
172	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
174	C₄⋊C₄	—	137	4	no	—
175	C₄⋊C₄	C₄×D₁₃₇	1	2	yes	—
176	C₂²⋊C₄	—	137	4	no	—
177	C₂²⋊C₄	—	137	4	no	—
178	C₂²⋊C₄	C₄×D₁₃₇	1	2	yes	—
179	C₄²	C₂²×D₁₃₇	1	2	yes	—
180	C₂³	—	274	2	no	—
181	C₂³	—	548	2	no	—
182	C₂³	—	137	2	no	—
183	C₂³	—	137	2	no	—
184	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
185	Q₈	—	274	2	no	—
186	D₄	—	274	4	no	—
187	D₄	—	274	2	no	—
188	D₄	—	548	4	no	—
189	C₂×C₄	—	274	2	no	—
190	C₂×C₄	—	274	4	no	—
191	C₂×C₄	—	274	2	no	—
192	C₂×C₄	—	274	2	no	—
193	C₂×C₄	—	274	2	no	—
194	C₂×C₄	—	274	2	no	—
195	C₂×C₄	—	137	2	no	—
196	C₂×C₄	—	137	2	no	—
197	C₂×C₄	C₂×C₄×D₁₃₇	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
199	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
200	C₂×C₄	#14555510	1	2	yes	—
201	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	274	2	no	—
204	C₈	—	274	2	no	—
205	C₈	C₂×D₅₄₈	1	2	yes	—
206	C₈	C₂×D₅₄₈	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	274	2	no	—
209	C₂²	—	274	2	no	—
210	C₂²	—	274	2	no	—
211	C₂²	—	274	4	no	—
212	C₂²	—	274	2	no	—
213	C₂²	—	548	4	no	—
214	C₂²	C₄×D₅₄₈	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₃₇	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	274	2	no	—
218	C₄	—	274	2	no	—
219	C₄	C₄×D₅₄₈	1	2	yes	—
220	C₄	C₂²×D₅₄₈	1	1	yes	characteristic
221	C₄	C₂×#14555510	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	548	2	no	—
224	C₂	—	548	2	no	—
225	C₂	C₂×C₄×D₅₄₈	1	1	yes	characteristic
226	C₂	#22578168	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₇
commutator, socle	C₂²×C₁₃₇	C₂³×C₄
Fitting	#405×C₁₃₇	C₂
prime radical	#22577915	C₂
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	#405	D₁₃₇
hypocenter	C₁₃₇	#2930
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#17984094	C₂	semidirect
2. C₂²×#14555493	C₂	semidirect
4. #22577482	C₂	semidirect
5. #22577476	C₂	semidirect
6. #22577430	C₂	semidirect
7. #22577371	C₂	semidirect
8. #22577366	C₂	semidirect
9. #405×C₁₃₇	C₂	semidirect
17. C₂×#14555493	C₂²	semidirect
19. C₂×C₈×D₁₃₇	C₂²	semidirect
25. C₄⋊#11764040	C₂²	semidirect
27. #17984000	C₂²	semidirect
31. #97×C₁₃₇	C₂²	semidirect
32. C₄×C₈×C₁₃₇	C₂²	semidirect
59. C₂×C₈×C₁₃₇	D₄	semidirect
113. C₁₃₇	#2930	semidirect
123. #405	D₁₃₇	semidirect
145. #97	C₂×D₁₃₇	semidirect
146. C₄×C₈	C₂×D₁₃₇	semidirect
170. C₂×C₈	D₅₄₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984094	2
C₂²×#14555493	2
#22577915	2
#22577482	2
#22577476	2
#22577430	2
#22577371	2
#22577366	2
#405×C₁₃₇	2
#2930	137