Subgroups — #30675089

Order 16768.

394 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523512	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553436	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553390	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553388	C₂	1	2	yes	maximal, complemented
6	#22553387	C₂	1	2	yes	maximal, complemented
7	#22553363	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553362	C₂	1	2	yes	maximal, complemented
9	#22553335	C₂	1	2	yes	maximal, complemented
10	#22553334	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523512	C₄	1	1	yes	characteristic
13	C₂×#14523497	C₂²	1	2	yes	—
14	C₂×#14523496	C₂²	1	2	yes	—
15	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
16	#16862211	C₄	1	1	yes	characteristic
17	D₄×Dic₁₃₁	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₃₁	C₂²	1	2	yes	—
19	(C₂²⋊D₁₃₁)⋊C₄	C₄	1	2	yes	—
20	#16862204	C₄	1	1	yes	characteristic
21	#16862202	C₄	1	2	yes	—
22	C₂×C₄×Dic₁₃₁	C₂²	1	2	yes	—
23	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₃₁⋊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	1	yes	characteristic
28	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
29	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
30	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
31	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
33	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₃₁	C₂²	1	2	yes	—
35	C₈×Dic₁₃₁	C₂²	1	2	yes	—
36	C₂×M₁₆×C₁₃₁	C₂²	1	2	yes	—
37	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
41	#14523512	—	2	4	no	—
42	C₂×Dic₂₆₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₃₁	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
49	#14523497	—	4	2	no	—
50	#14523496	—	4	2	no	—
51	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
52	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
54	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
55	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
58	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃₁	C₂³	1	2	yes	—
64	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
67	M₁₆×C₁₃₁	—	4	2	no	—
68	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
69	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
73	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
74	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
75	Dic₂₆₂	M₁₆	1	4	yes	complemented
76	C₂²⋊D₁₃₁	—	2	4	no	—
77	C₂×Dic₁₃₁	—	2	4	no	—
78	C₂×Dic₁₃₁	—	2	4	no	—
79	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
83	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₁₃₁	—	2	2	no	—
85	C₄×D₁₃₁	—	2	1	no	—
86	C₄×D₁₃₁	—	2	1	no	—
87	C₄×D₁₃₁	—	2	2	no	—
88	C₁₃₁⋊C₈	—	2	2	no	—
89	C₁₃₁⋊C₈	—	2	1	no	—
90	C₁₃₁⋊C₈	—	2	1	no	—
91	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
92	D₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	—	2	2	no	—
95	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₃₁	—	2	1	no	—
100	C₈×C₁₃₁	—	2	2	no	—
101	C₈×C₁₃₁	—	2	1	no	—
102	C₂²×C₁₃₁	—	2	2	no	—
103	C₂²×C₁₃₁	—	2	2	no	—
104	C₂²×C₁₃₁	—	2	2	no	—
105	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×D₁₃₁	—	2	1	no	—
107	C₂×D₁₃₁	—	2	1	no	—
108	C₂×D₁₃₁	—	2	1	no	—
109	C₄×C₁₃₁	—	2	2	no	—
110	C₄×C₁₃₁	—	2	1	no	—
111	C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
112	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
114	Dic₁₃₁	—	2	1	no	—
115	Dic₁₃₁	—	2	1	no	—
116	Dic₁₃₁	—	2	2	no	—
117	Dic₁₃₁	C₂×M₁₆	1	4	yes	—
118	Dic₁₃₁	#131	1	2	yes	—
119	C₂×C₁₃₁	—	4	2	no	—
120	C₂×C₁₃₁	#520	1	1	yes	characteristic
121	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
122	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
123	D₁₃₁	—	4	1	no	—
124	C₁₃₁	#2991	1	1	yes	characteristic, complemented, Sylow
125	#2991	—	131	1	no	maximal, Sylow
126	C₄×#41	—	131	1	no	—
127	C₈×Q₈	—	131	1	no	—
128	C₈×Q₈	—	131	1	no	—
129	#436	—	131	2	no	—
130	#436	—	131	1	no	—
131	#436	—	131	2	no	—
132	#436	D₁₃₁	1	1	yes	characteristic, complemented
133	#431	—	131	2	no	—
134	#431	—	131	1	no	—
135	#423	—	131	2	no	—
136	#423	—	131	1	no	—
137	C₂×#41	—	131	1	no	—
138	C₂×M₁₆	—	131	2	no	—
139	C₂×M₁₆	—	131	2	no	—
140	C₂×M₁₆	C₂×D₁₃₁	1	2	yes	—
141	C₄×Q₈	—	131	1	no	—
142	C₄×Q₈	—	131	1	no	—
143	C₄×D₄	—	131	1	no	—
144	C₄×D₄	—	131	2	no	—
145	C₄×D₄	—	131	2	no	—
146	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
147	#117	—	131	1	no	—
148	#117	—	131	2	no	—
149	C₂×C₄²	—	131	2	no	—
150	C₂×C₄²	—	131	1	no	—
151	C₄⋊C₈	—	131	2	no	—
152	C₄⋊C₈	—	131	1	no	—
153	C₄⋊C₈	—	131	2	no	—
154	C₄⋊C₈	—	131	2	no	—
155	C₄⋊C₈	—	131	1	no	—
156	C₄⋊C₈	—	131	1	no	—
157	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
158	C₂²⋊C₈	—	131	2	no	—
159	C₂²⋊C₈	—	131	1	no	—
160	C₂²⋊C₈	—	131	1	no	—
161	C₂²⋊C₈	C₂×D₁₃₁	1	2	yes	—
162	C₄×C₈	—	131	2	no	—
163	C₄×C₈	—	131	2	no	—
164	C₄×C₈	—	131	1	no	—
165	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
166	#41	—	262	4	no	—
167	C₂×Q₈	—	131	1	no	—
168	C₂×D₄	—	131	2	no	—
169	C₂×D₄	C₄×D₁₃₁	1	1	yes	characteristic
170	C₂²×C₄	—	131	2	no	—
171	C₂²×C₄	—	131	2	no	—
172	C₂²×C₄	—	131	1	no	—
173	C₂²×C₄	—	131	1	no	—
174	C₂²×C₄	—	131	1	no	—
175	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
176	M₁₆	—	4	2	no	—
177	M₁₆	—	524	2	no	—
178	M₁₆	—	524	2	no	—
179	C₂×C₈	—	131	1	no	—
180	C₂×C₈	—	131	2	no	—
181	C₂×C₈	—	131	1	no	—
182	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
184	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
185	C₄⋊C₄	—	131	2	no	—
186	C₄⋊C₄	—	131	2	no	—
187	C₄⋊C₄	—	131	1	no	—
188	C₄⋊C₄	C₄×D₁₃₁	1	1	yes	characteristic
189	C₂²⋊C₄	—	131	2	no	—
190	C₂²⋊C₄	—	131	2	no	—
191	C₂²⋊C₄	C₄×D₁₃₁	1	2	yes	—
192	C₄²	—	131	2	no	—
193	C₄²	—	131	1	no	—
194	C₄²	—	131	1	no	—
195	C₄²	—	131	2	no	—
196	C₄²	—	131	2	no	—
197	C₄²	—	131	1	no	—
198	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
199	C₂³	—	131	1	no	—
200	C₂³	C₂×C₄×D₁₃₁	1	2	yes	—
201	Q₈	—	131	4	no	—
202	D₄	—	2	2	no	—
203	D₄	—	262	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	262	2	no	—
207	C₂×C₄	—	262	4	no	—
208	C₂×C₄	—	262	4	no	—
209	C₂×C₄	—	262	2	no	—
210	C₂×C₄	—	262	1	no	—
211	C₂×C₄	—	262	1	no	—
212	C₂×C₄	—	131	1	no	—
213	C₂×C₄	—	131	1	no	—
214	C₂×C₄	—	131	2	no	—
215	C₂×C₄	—	131	1	no	—
216	C₂×C₄	—	131	2	no	—
217	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	262	2	no	—
225	C₈	—	262	1	no	—
226	C₈	—	262	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	262	1	no	—
231	C₂²	—	262	1	no	—
232	C₂²	—	262	1	no	—
233	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	262	1	no	—
237	C₄	—	262	1	no	—
238	C₄	—	262	2	no	—
239	C₄	—	131	4	no	—
240	C₄	—	131	2	no	—
241	C₄	M₁₆×D₁₃₁	1	2	yes	—
242	C₄	#16862291	1	1	yes	characteristic
243	C₄	#16862293	1	1	yes	characteristic
244	C₂	—	4	2	no	—
245	C₂	—	524	1	no	—
246	C₂	#22553791	1	1	yes	characteristic
247	C₂	C₂×M₁₆×D₁₃₁	1	1	yes	characteristic
248	C₂	#131×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#436×C₁₃₁	C₂
prime radical	C₂×#14523512	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#436	D₁₃₁
hypocenter	C₁₃₁	#2991
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₂₆₂	C₂	semidirect
3. #22553436	C₂	semidirect
4. #22553390	C₂	semidirect
5. #22553388	C₂	semidirect
6. #22553387	C₂	semidirect
7. #22553363	C₂	semidirect
8. #22553362	C₂	semidirect
9. #22553335	C₂	semidirect
10. #22553334	C₂	semidirect
11. #436×C₁₃₁	C₂	semidirect
75. Dic₂₆₂	M₁₆	semidirect
124. C₁₃₁	#2991	semidirect
132. #436	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523512	2
C₈×Dic₂₆₂	2
#22553436	2
#22553390	2
#22553388	2
#22553387	2
#22553363	2
#22553362	2
#22553335	2
#22553334	2
#436×C₁₃₁	2
#2991	131