Subgroups — #30674619

Order 16768.

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	#22553373	C₂	1	4	yes	maximal, complemented
2	#22553369	C₂	1	2	yes	maximal, complemented
3	#22553365	C₂	1	2	yes	maximal, complemented
4	#22553339	C₂	1	2	yes	maximal, complemented
5	#22553276	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#22553268	C₂	1	2	yes	maximal, complemented
8	C₂×#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523497	C₂²	1	2	yes	—
10	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
11	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	complemented
12	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	complemented
13	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
15	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
17	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
18	#16862157	C₂²	1	4	yes	complemented
19	C₄×C₁₃₁⋊C₈	C₂²	1	4	yes	complemented
20	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
21	#117×C₁₃₁	C₄	1	1	yes	characteristic
22	#117×C₁₃₁	C₄	1	1	yes	characteristic
23	#117×C₁₃₁	C₄	1	2	yes	—
24	#117×C₁₃₁	C₂²	1	2	yes	—
25	#117×C₁₃₁	C₂²	1	1	yes	characteristic
26	#117×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₃₁	C₄	1	2	yes	—
30	C₂×C₄²×C₁₃₁	C₂²	1	2	yes	—
31	#14523497	—	2	4	no	—
32	C₂×C₁₃₁⋊C₈	—	2	4	no	—
33	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
35	C₂×C₁₃₁⋊C₈	C₂³	1	4	yes	—
36	C₂⁴×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₁	—	2	4	no	—
38	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
56	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
57	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
58	C₄²×C₁₃₁	C₂³	1	2	yes	—
59	C₄²×C₁₃₁	C₂³	1	2	yes	—
60	C₁₃₁⋊C₈	—	2	2	no	—
61	C₁₃₁⋊C₈	—	2	2	no	—
62	C₁₃₁⋊C₈	—	2	4	no	—
63	C₂³×C₁₃₁	—	2	2	no	—
64	C₂³×C₁₃₁	—	2	2	no	—
65	C₂³×C₁₃₁	—	4	1	no	—
66	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃₁	—	2	2	no	—
70	C₂×C₄×C₁₃₁	—	2	4	no	—
71	C₂×C₄×C₁₃₁	—	2	2	no	—
72	C₂×C₄×C₁₃₁	—	2	2	no	—
73	C₂×C₄×C₁₃₁	—	2	2	no	—
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	4	2	no	—
76	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₁	#41	1	4	yes	—
84	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
85	C₂²×C₁₃₁	—	2	1	no	—
86	C₂²×C₁₃₁	—	2	1	no	—
87	C₂²×C₁₃₁	—	2	1	no	—
88	C₂²×C₁₃₁	—	2	1	no	—
89	C₂²×C₁₃₁	—	2	1	no	—
90	C₂²×C₁₃₁	—	2	1	no	—
91	C₂²×C₁₃₁	—	2	1	no	—
92	C₂²×C₁₃₁	—	4	4	no	—
93	C₂²×C₁₃₁	#117	1	2	yes	—
94	C₂²×C₁₃₁	#131	1	2	yes	—
95	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
96	C₄×C₁₃₁	—	2	1	no	—
97	C₄×C₁₃₁	—	2	1	no	—
98	C₄×C₁₃₁	—	2	2	no	—
99	C₄×C₁₃₁	—	2	2	no	—
100	C₄×C₁₃₁	—	2	2	no	—
101	C₄×C₁₃₁	—	2	2	no	—
102	C₄×C₁₃₁	#117	1	2	yes	—
103	C₄×C₁₃₁	C₂×#41	1	2	yes	—
104	C₂×C₁₃₁	—	2	2	no	—
105	C₂×C₁₃₁	—	2	2	no	—
106	C₂×C₁₃₁	—	4	1	no	—
107	C₂×C₁₃₁	C₂×#117	1	1	yes	characteristic
108	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
109	C₂×C₁₃₁	#568	1	1	yes	characteristic
110	C₁₃₁	#2924	1	1	yes	characteristic, complemented, Sylow
111	#2924	—	131	1	no	maximal, Sylow
112	C₂×#117	D₁₃₁	1	1	yes	characteristic
113	#433	—	131	4	no	—
114	#432	—	131	2	no	—
115	#431	—	131	2	no	—
116	#424	—	131	2	no	—
117	#407	—	131	1	no	—
118	C₂×C₂²⋊C₈	—	131	1	no	—
119	#405	—	131	2	no	—
120	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
121	C₂×M₁₆	—	131	2	no	—
122	C₂²×C₈	—	131	2	no	—
123	#117	Dic₁₃₁	1	1	yes	characteristic
124	#117	Dic₁₃₁	1	1	yes	characteristic
125	#117	Dic₁₃₁	1	2	yes	—
126	#117	C₂×D₁₃₁	1	1	yes	characteristic
127	#117	C₂×D₁₃₁	1	1	yes	characteristic
128	#117	C₂×D₁₃₁	1	2	yes	—
129	C₂×C₄⋊C₄	Dic₁₃₁	1	1	yes	characteristic
130	C₂×C₄⋊C₄	Dic₁₃₁	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	Dic₁₃₁	1	2	yes	—
132	C₂×C₄²	C₂×D₁₃₁	1	2	yes	—
133	C₄⋊C₈	—	131	4	no	—
134	C₄⋊C₈	—	131	4	no	—
135	C₂²⋊C₈	—	131	1	no	—
136	C₂²⋊C₈	—	131	1	no	—
137	C₂²⋊C₈	—	131	2	no	—
138	C₂²⋊C₈	—	131	2	no	—
139	C₂²⋊C₈	—	131	2	no	—
140	#97	—	131	4	no	—
141	C₄×C₈	—	131	4	no	—
142	C₂⁴	C₂×Dic₁₃₁	1	1	yes	characteristic
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
145	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
149	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
152	M₁₆	—	262	4	no	—
153	C₂×C₈	—	262	4	no	—
154	C₂×C₈	—	131	2	no	—
155	C₂×C₈	—	131	4	no	—
156	C₂×C₈	—	131	2	no	—
157	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
159	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
160	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
161	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
162	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
163	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
164	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
165	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
166	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
167	C₄²	C₂×Dic₁₃₁	1	2	yes	—
168	C₄²	C₂×Dic₁₃₁	1	2	yes	—
169	C₄²	C₂²×D₁₃₁	1	2	yes	—
170	C₄²	C₂²×D₁₃₁	1	2	yes	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	4	1	no	—
174	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
175	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
176	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	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×Dic₁₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
186	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
187	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
188	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
189	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
191	C₂×C₄	#14523513	1	4	yes	—
192	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
193	C₈	—	262	2	no	—
194	C₈	—	262	2	no	—
195	C₈	—	262	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	#16862205	1	2	yes	—
205	C₂²	#16862253	1	2	yes	—
206	C₂²	C₂³×Dic₁₃₁	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	#16862205	1	2	yes	—
214	C₄	C₂×#14523513	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	2	no	—
217	C₂	—	4	1	no	—
218	C₂	C₂×#16862205	1	1	yes	characteristic
219	C₂	C₂×#16862253	1	1	yes	characteristic
220	C₂	#22554079	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	C₂×#117×C₁₃₁	C₂
prime radical	C₂⁴×C₁₃₁	C₂×C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	C₂×#117	D₁₃₁
hypocenter	C₁₃₁	#2924
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22553373	C₂	semidirect
2. #22553369	C₂	semidirect
3. #22553365	C₂	semidirect
4. #22553339	C₂	semidirect
7. #22553268	C₂	semidirect
11. C₄⋊(C₁₃₁⋊C₈)	C₂²	semidirect
12. C₄⋊(C₁₃₁⋊C₈)	C₂²	semidirect
18. #16862157	C₂²	semidirect
19. C₄×C₁₃₁⋊C₈	C₂²	semidirect
110. C₁₃₁	#2924	semidirect

Maximal subgroups

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

subgroup	index
#22553373	2
#22553369	2
#22553365	2
#22553339	2
#22553276	2
C₂×C₂²⋊(C₁₃₁⋊C₈)	2
#22553268	2
C₂×#117×C₁₃₁	2
#2924	131