Subgroups — #30674627

Order 16768.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 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	#22553336	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553279	C₂	1	2	yes	maximal
7	#22553268	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	2	yes	—
11	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
13	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
15	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
16	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	—
17	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	—
18	#16862157	C₂²	1	4	yes	—
19	C₄×C₁₃₁⋊C₈	C₂²	1	4	yes	—
20	C₂×#41×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₃₁	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₁₃₁	C₄	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	C₄	1	2	yes	—
26	C₄×D₄×C₁₃₁	C₄	1	1	yes	characteristic
27	#117×C₁₃₁	C₂²	1	2	yes	—
28	#117×C₁₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₁	C₂²	1	2	yes	—
30	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
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	#41×C₁₃₁	—	2	2	no	—
37	#41×C₁₃₁	—	2	2	no	—
38	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
40	C₂×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
54	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
55	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
57	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
59	C₄²×C₁₃₁	C₂³	1	2	yes	—
60	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
61	C₁₃₁⋊C₈	—	2	2	no	—
62	C₁₃₁⋊C₈	—	2	2	no	—
63	C₁₃₁⋊C₈	—	2	4	no	—
64	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
65	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
66	Q₈×C₁₃₁	—	2	2	no	—
67	D₄×C₁₃₁	—	2	4	no	—
68	D₄×C₁₃₁	—	2	2	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	—	4	1	no	—
74	C₂×C₄×C₁₃₁	—	4	2	no	—
75	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
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	2	yes	—
80	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
84	C₂×C₄×C₁₃₁	#41	1	2	yes	—
85	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
86	C₂²×C₁₃₁	—	2	1	no	—
87	C₂²×C₁₃₁	—	2	1	no	—
88	C₂²×C₁₃₁	—	2	2	no	—
89	C₂²×C₁₃₁	—	2	2	no	—
90	C₂²×C₁₃₁	—	2	2	no	—
91	C₂²×C₁₃₁	#117	1	2	yes	—
92	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₁₃₁	—	2	2	no	—
94	C₄×C₁₃₁	—	2	1	no	—
95	C₄×C₁₃₁	—	2	2	no	—
96	C₄×C₁₃₁	—	2	1	no	—
97	C₄×C₁₃₁	—	2	1	no	—
98	C₄×C₁₃₁	—	2	2	no	—
99	C₄×C₁₃₁	#117	1	2	yes	—
100	C₄×C₁₃₁	#131	1	2	yes	—
101	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
102	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
103	C₂×C₁₃₁	—	2	2	no	—
104	C₂×C₁₃₁	—	4	2	no	—
105	C₂×C₁₃₁	C₂×#117	1	1	yes	characteristic
106	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
107	C₂×C₁₃₁	#568	1	1	yes	characteristic
108	C₁₃₁	#2925	1	1	yes	characteristic, complemented, Sylow
109	#2925	—	131	1	no	maximal, Sylow
110	C₄×#41	D₁₃₁	1	1	yes	characteristic
111	#433	—	131	4	no	—
112	#432	—	131	2	no	—
113	#431	—	131	2	no	—
114	#423	—	131	1	no	—
115	C₂×C₄⋊C₈	—	131	1	no	—
116	#408	—	131	2	no	—
117	#405	—	131	2	no	—
118	C₂×#41	C₂×D₁₃₁	1	1	yes	characteristic
119	C₂×M₁₆	—	131	2	no	—
120	C₂²×C₈	—	131	2	no	—
121	C₄×Q₈	Dic₁₃₁	1	1	yes	characteristic
122	C₄×Q₈	Dic₁₃₁	1	1	yes	characteristic
123	C₄×D₄	Dic₁₃₁	1	2	yes	—
124	C₄×D₄	Dic₁₃₁	1	1	yes	characteristic
125	C₄×D₄	Dic₁₃₁	1	2	yes	—
126	C₄×D₄	Dic₁₃₁	1	1	yes	characteristic
127	#117	C₂×D₁₃₁	1	1	yes	characteristic
128	#117	C₂×D₁₃₁	1	2	yes	—
129	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
130	C₂×C₄²	C₂×D₁₃₁	1	2	yes	—
131	C₄⋊C₈	—	131	1	no	—
132	C₄⋊C₈	—	131	1	no	—
133	C₄⋊C₈	—	131	2	no	—
134	C₄⋊C₈	—	131	2	no	—
135	C₄⋊C₈	—	131	2	no	—
136	C₂²⋊C₈	—	131	4	no	—
137	C₂²⋊C₈	—	131	4	no	—
138	#97	—	131	4	no	—
139	C₄×C₈	—	131	4	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×Dic₁₃₁	1	1	yes	characteristic
143	C₂×D₄	C₂×Dic₁₃₁	1	2	yes	—
144	C₂×D₄	C₂×Dic₁₃₁	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
146	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
148	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
150	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
151	M₁₆	—	262	4	no	—
152	C₂×C₈	—	262	4	no	—
153	C₂×C₈	—	131	2	no	—
154	C₂×C₈	—	131	4	no	—
155	C₂×C₈	—	131	2	no	—
156	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
159	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
160	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
161	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
162	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
163	C₄²	C₂×Dic₁₃₁	1	2	yes	—
164	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
165	C₄²	C₂×Dic₁₃₁	1	2	yes	—
166	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
167	C₄²	C₂²×D₁₃₁	1	2	yes	—
168	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
169	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
170	C₂³	C₂²×Dic₁₃₁	1	2	yes	—
171	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
183	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
184	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
186	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
189	C₂×C₄	#14523513	1	2	yes	—
190	C₂×C₄	#14523513	1	2	yes	—
191	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
192	C₈	—	262	2	no	—
193	C₈	—	262	2	no	—
194	C₈	—	262	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	#16862205	1	2	yes	—
201	C₂²	C₂³×Dic₁₃₁	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	#16862205	1	2	yes	—
209	C₄	#16862253	1	2	yes	—
210	C₄	C₂×#14523513	1	1	yes	characteristic
211	C₄	C₂×#14523513	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×#16862205	1	1	yes	characteristic
215	C₂	C₂×#16862253	1	1	yes	characteristic
216	C₂	#22554079	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	C₂²
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	C₄×#41	D₁₃₁
hypocenter	C₁₃₁	#2925
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. #22553373	C₂	semidirect
2. #22553369	C₂	semidirect
3. #22553365	C₂	semidirect
4. #22553336	C₂	semidirect
5. C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	semidirect
7. #22553268	C₂	semidirect
108. C₁₃₁	#2925	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
#22553336	2
C₂×C₄⋊(C₁₃₁⋊C₈)	2
#22553279	2
#22553268	2
C₄×#41×C₁₃₁	2
#2925	131