Subgroups — #30675047

Order 16768.

364 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#22553443	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553441	C₂	1	2	yes	maximal, complemented
4	#22553440	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553436	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553368	C₂	1	1	yes	maximal, characteristic
8	#22553367	C₂	1	2	yes	maximal
9	#22553364	C₂	1	2	yes	maximal
10	#22553363	C₂	1	1	yes	maximal, characteristic
11	#446×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₃₁	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₃₁	C₄	1	1	yes	characteristic
14	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₆₂	C₂²	1	2	yes	—
16	#16862211	C₄	1	2	yes	—
17	#16862211	C₄	1	1	yes	characteristic
18	C₄⋊C₄×D₁₃₁	C₄	1	2	yes	—
19	C₄⋊C₄×D₁₃₁	C₄	1	1	yes	characteristic
20	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
21	C₄²×D₁₃₁	C₂²	1	2	yes	—
22	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
23	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
24	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
27	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
31	#16862157	C₂²	1	2	yes	—
32	#16862156	C₂²	1	2	yes	—
33	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₃₁	C₂²	1	2	yes	—
35	C₈×Dic₁₃₁	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
38	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
39	#97×C₁₃₁	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₆₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₆₂	C₂×C₄	1	1	yes	characteristic
43	Q₈×D₁₃₁	—	2	4	no	—
44	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
51	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
54	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₁₃₁	C₂³	1	2	yes	—
62	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
67	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
68	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₄²×C₁₃₁	C₂³	1	2	yes	—
71	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
72	Dic₂₆₂	—	2	4	no	—
73	Dic₂₆₂	M₁₆	1	4	yes	—
74	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
79	C₄×D₁₃₁	—	2	2	no	—
80	C₄×D₁₃₁	—	2	4	no	—
81	C₄×D₁₃₁	—	2	2	no	—
82	C₄×D₁₃₁	—	2	1	no	—
83	C₄×D₁₃₁	—	2	1	no	—
84	C₄×D₁₃₁	—	2	4	no	—
85	C₁₃₁⋊C₈	—	2	1	no	—
86	C₁₃₁⋊C₈	—	2	1	no	—
87	C₁₃₁⋊C₈	—	2	2	no	—
88	Q₈×C₁₃₁	—	2	2	no	—
89	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₃₁	—	2	1	no	—
95	C₈×C₁₃₁	—	2	2	no	—
96	C₈×C₁₃₁	—	2	1	no	—
97	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₃₁	—	2	1	no	—
99	C₂×D₁₃₁	—	2	1	no	—
100	C₂×D₁₃₁	#131	1	2	yes	—
101	C₄×C₁₃₁	—	2	2	no	—
102	C₄×C₁₃₁	—	2	1	no	—
103	C₄×C₁₃₁	—	2	2	no	—
104	C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
105	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
106	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
107	Dic₁₃₁	—	2	1	no	—
108	Dic₁₃₁	—	2	1	no	—
109	Dic₁₃₁	—	2	2	no	—
110	Dic₁₃₁	—	2	1	no	—
111	Dic₁₃₁	C₂×M₁₆	1	4	yes	—
112	C₂×C₁₃₁	#519	1	1	yes	characteristic
113	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
114	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
115	D₁₃₁	—	2	2	no	—
116	C₁₃₁	#2984	1	1	yes	characteristic, complemented, Sylow
117	#2984	—	131	1	no	maximal, Sylow
118	C₂×C₄×Q₈	—	131	1	no	—
119	#446	—	131	1	no	—
120	#446	—	131	2	no	—
121	#446	—	131	2	no	—
122	#446	D₁₃₁	1	1	yes	characteristic, complemented
123	C₈×Q₈	—	131	1	no	—
124	C₈×Q₈	—	131	1	no	—
125	#432	—	131	1	no	—
126	#432	—	131	2	no	—
127	#431	—	131	2	no	—
128	#431	—	131	1	no	—
129	C₂²×Q₈	—	131	1	no	—
130	C₄×Q₈	—	131	1	no	—
131	C₄×Q₈	—	131	2	no	—
132	C₄×Q₈	—	131	1	no	—
133	C₄×Q₈	—	131	1	no	—
134	C₄×Q₈	—	131	2	no	—
135	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	131	1	no	—
137	C₂×C₄⋊C₄	—	131	2	no	—
138	C₂×C₄²	—	131	2	no	—
139	C₂×C₄²	—	131	1	no	—
140	C₄⋊C₈	—	131	1	no	—
141	C₄⋊C₈	—	131	1	no	—
142	C₄⋊C₈	—	131	2	no	—
143	C₄⋊C₈	—	131	2	no	—
144	C₄⋊C₈	—	131	2	no	—
145	C₄⋊C₈	—	131	1	no	—
146	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
147	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
148	C₂²⋊C₈	—	131	1	no	—
149	C₂²⋊C₈	—	131	2	no	—
150	C₂²⋊C₈	—	131	1	no	—
151	#97	—	131	2	no	—
152	#97	—	131	2	no	—
153	#97	C₂×D₁₃₁	1	2	yes	—
154	C₄×C₈	—	131	2	no	—
155	C₄×C₈	—	131	2	no	—
156	C₄×C₈	—	131	1	no	—
157	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
158	C₂×Q₈	—	262	4	no	—
159	C₂×Q₈	—	131	2	no	—
160	C₂×Q₈	—	131	1	no	—
161	C₂×Q₈	C₄×D₁₃₁	1	1	yes	characteristic
162	C₂²×C₄	—	131	2	no	—
163	C₂²×C₄	—	131	1	no	—
164	C₂²×C₄	—	131	2	no	—
165	C₂²×C₄	—	131	1	no	—
166	C₂²×C₄	—	131	1	no	—
167	C₂×C₈	—	131	1	no	—
168	C₂×C₈	—	131	1	no	—
169	C₂×C₈	—	131	2	no	—
170	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
172	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
173	C₄⋊C₄	—	131	2	no	—
174	C₄⋊C₄	—	131	2	no	—
175	C₄⋊C₄	—	131	2	no	—
176	C₄⋊C₄	—	131	2	no	—
177	C₄⋊C₄	—	131	1	no	—
178	C₄⋊C₄	C₄×D₁₃₁	1	2	yes	—
179	C₄⋊C₄	C₄×D₁₃₁	1	1	yes	characteristic
180	C₄²	—	131	2	no	—
181	C₄²	—	131	1	no	—
182	C₄²	—	131	1	no	—
183	C₄²	—	131	2	no	—
184	C₄²	—	131	2	no	—
185	C₄²	—	131	1	no	—
186	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
187	C₄²	C₂²×D₁₃₁	1	2	yes	—
188	C₂³	—	131	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	262	4	no	—
191	Q₈	—	131	4	no	—
192	C₂×C₄	—	262	2	no	—
193	C₂×C₄	—	262	4	no	—
194	C₂×C₄	—	262	4	no	—
195	C₂×C₄	—	262	2	no	—
196	C₂×C₄	—	262	1	no	—
197	C₂×C₄	—	262	1	no	—
198	C₂×C₄	—	131	1	no	—
199	C₂×C₄	—	131	2	no	—
200	C₂×C₄	—	131	1	no	—
201	C₂×C₄	—	131	1	no	—
202	C₂×C₄	—	131	2	no	—
203	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	1	no	—
211	C₈	—	262	1	no	—
212	C₈	—	262	1	no	—
213	C₈	—	262	2	no	—
214	C₂²	—	262	1	no	—
215	C₂²	—	262	1	no	—
216	C₂²	—	131	2	no	—
217	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	262	1	no	—
222	C₄	—	262	1	no	—
223	C₄	—	262	2	no	—
224	C₄	—	262	1	no	—
225	C₄	—	131	4	no	—
226	C₄	M₁₆×D₁₃₁	1	2	yes	—
227	C₄	#16862293	1	1	yes	characteristic
228	C₄	#16862294	1	1	yes	characteristic
229	C₂	—	262	2	no	—
230	C₂	#22553788	1	1	yes	characteristic
231	C₂	C₂×M₁₆×D₁₃₁	1	1	yes	characteristic
232	C₂	#131×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#446×C₁₃₁	C₂
prime radical	C₂²×D₁₃₁	C₂²×C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#446	D₁₃₁
hypocenter	C₁₃₁	#2984
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #22553443	C₂	semidirect
3. #22553441	C₂	semidirect
4. #22553440	C₂	semidirect
5. C₈×Dic₂₆₂	C₂	semidirect
6. #22553436	C₂	semidirect
11. #446×C₁₃₁	C₂	semidirect
116. C₁₃₁	#2984	semidirect
122. #446	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₁₃₁	2
#22553443	2
#22553441	2
#22553440	2
C₈×Dic₂₆₂	2
#22553436	2
#22553368	2
#22553367	2
#22553364	2
#22553363	2
#446×C₁₃₁	2
#2984	131