Subgroups — #30675086

Order 16768.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553443	C₂	1	2	yes	maximal, complemented
2	#22553384	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
4	#22553369	C₂	1	2	yes	maximal, complemented
5	#22553365	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553336	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523497	C₂²	1	2	yes	—
10	C₂²×C₁₃₁⋊C₈	C₂²	1	4	yes	—
11	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
14	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
15	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	—
16	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
17	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	—
18	#16862157	C₂²	1	4	yes	—
19	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
20	C₂×#41×C₁₃₁	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₃₁	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₃₁	C₄	1	2	yes	—
24	C₄×D₄×C₁₃₁	C₄	1	1	yes	characteristic
25	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
27	#117×C₁₃₁	C₄	1	2	yes	—
28	#117×C₁₃₁	C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₁	C₂²	1	2	yes	—
31	#14523497	—	4	2	no	—
32	C₂×C₁₃₁⋊C₈	—	4	4	no	—
33	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
34	C₂×C₁₃₁⋊C₈	C₂³	1	4	yes	—
35	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
36	#41×C₁₃₁	—	2	4	no	—
37	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
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	1	yes	characteristic
48	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
53	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
55	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
56	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
58	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
59	C₄²×C₁₃₁	C₂³	1	2	yes	—
60	C₁₃₁⋊C₈	—	2	2	no	—
61	C₁₃₁⋊C₈	—	2	4	no	—
62	C₁₃₁⋊C₈	—	2	2	no	—
63	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
65	Q₈×C₁₃₁	—	2	2	no	—
66	D₄×C₁₃₁	—	2	4	no	—
67	D₄×C₁₃₁	M₁₆	1	4	yes	complemented
68	C₂×C₄×C₁₃₁	—	2	4	no	—
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	1	no	—
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	2	1	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	1	yes	characteristic
79	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
85	C₂²×C₁₃₁	—	2	2	no	—
86	C₂²×C₁₃₁	—	2	2	no	—
87	C₂²×C₁₃₁	—	2	1	no	—
88	C₂²×C₁₃₁	—	2	1	no	—
89	C₂²×C₁₃₁	—	2	1	no	—
90	C₂²×C₁₃₁	C₂×M₁₆	1	4	yes	—
91	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₁₃₁	—	2	2	no	—
93	C₄×C₁₃₁	—	2	1	no	—
94	C₄×C₁₃₁	—	2	1	no	—
95	C₄×C₁₃₁	—	2	2	no	—
96	C₄×C₁₃₁	—	2	2	no	—
97	C₄×C₁₃₁	—	2	1	no	—
98	C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
99	C₄×C₁₃₁	#131	1	2	yes	—
100	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
101	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×C₁₃₁	—	2	4	no	—
103	C₂×C₁₃₁	—	4	1	no	—
104	C₂×C₁₃₁	#520	1	1	yes	characteristic
105	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
106	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
107	C₁₃₁	#2990	1	1	yes	characteristic, complemented, Sylow
108	#2990	—	131	1	no	maximal, Sylow
109	C₄×#41	D₁₃₁	1	1	yes	characteristic
110	#446	—	131	2	no	—
111	#435	—	131	4	no	—
112	C₈×D₄	—	131	2	no	—
113	#432	—	131	2	no	—
114	#431	—	131	1	no	—
115	#423	—	131	1	no	—
116	C₂×C₄⋊C₈	—	131	2	no	—
117	C₂×#41	Dic₁₃₁	1	1	yes	characteristic
118	C₂×M₁₆	—	131	2	no	—
119	C₂²×C₈	—	131	4	no	—
120	C₄×Q₈	Dic₁₃₁	1	1	yes	characteristic
121	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
122	C₄×D₄	Dic₁₃₁	1	2	yes	—
123	C₄×D₄	Dic₁₃₁	1	1	yes	characteristic
124	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
125	C₄×D₄	C₂×D₁₃₁	1	2	yes	—
126	#117	Dic₁₃₁	1	1	yes	characteristic
127	#117	Dic₁₃₁	1	2	yes	—
128	C₂×C₄²	C₂×D₁₃₁	1	2	yes	—
129	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
130	C₄⋊C₈	—	131	1	no	—
131	C₄⋊C₈	—	131	2	no	—
132	C₄⋊C₈	—	131	4	no	—
133	C₄⋊C₈	—	131	2	no	—
134	C₄⋊C₈	—	131	1	no	—
135	C₂²⋊C₈	—	131	2	no	—
136	C₂²⋊C₈	—	131	4	no	—
137	#97	—	131	4	no	—
138	C₄×C₈	—	131	2	no	—
139	#41	—	2	4	no	—
140	C₂×Q₈	C₂×Dic₁₃₁	1	1	yes	characteristic
141	C₂×D₄	C₂×Dic₁₃₁	1	2	yes	—
142	C₂×D₄	C₂×Dic₁₃₁	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
144	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
146	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
147	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
148	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
149	M₁₆	—	524	2	no	—
150	C₂×C₈	—	524	4	no	—
151	C₂×C₈	—	131	2	no	—
152	C₂×C₈	—	131	2	no	—
153	C₂×C₈	—	131	4	no	—
154	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
155	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
156	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
158	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
159	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
160	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
161	C₄²	C₂×Dic₁₃₁	1	2	yes	—
162	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
163	C₄²	C₂×Dic₁₃₁	1	2	yes	—
164	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
165	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
166	C₄²	C₂²×D₁₃₁	1	2	yes	—
167	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
168	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
169	C₂³	C₂²×Dic₁₃₁	1	2	yes	—
170	Q₈	—	2	2	no	—
171	D₄	—	2	4	no	—
172	D₄	#14523497	1	4	yes	complemented
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
183	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
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	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
190	C₈	—	262	2	no	—
191	C₈	—	262	4	no	—
192	C₈	—	262	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	C₂×#14523497	1	4	yes	—
199	C₂²	C₂³×Dic₁₃₁	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×#14523497	1	2	yes	—
207	C₄	#16862253	1	2	yes	—
208	C₄	#16862292	1	1	yes	characteristic
209	C₄	#16862293	1	1	yes	characteristic
210	C₂	—	2	4	no	—
211	C₂	—	4	1	no	—
212	C₂	#22553795	1	1	yes	characteristic
213	C₂	C₂²×#14523497	1	1	yes	characteristic
214	C₂	C₂×#16862253	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	C₄×#41×C₁₃₁	C₂
prime radical	C₂×#41×C₁₃₁	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	C₄×#41	D₁₃₁
hypocenter	C₁₃₁	#2990
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22553443	C₂	semidirect
2. #22553384	C₂	semidirect
3. D₄×C₁₃₁⋊C₈	C₂	semidirect
4. #22553369	C₂	semidirect
5. #22553365	C₂	semidirect
6. #22553336	C₂	semidirect
7. C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	semidirect
67. D₄×C₁₃₁	M₁₆	semidirect
107. C₁₃₁	#2990	semidirect
172. D₄	#14523497	semidirect

Maximal subgroups

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

subgroup	index
#22553443	2
#22553384	2
D₄×C₁₃₁⋊C₈	2
#22553369	2
#22553365	2
#22553336	2
C₂×C₄⋊(C₁₃₁⋊C₈)	2
C₄×#41×C₁₃₁	2
#2990	131