Subgroups — #30675823

Order 16768.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553940	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553619	C₂	1	2	yes	maximal, complemented
3	#22553612	C₂	1	2	yes	maximal, complemented
4	#22553602	C₂	1	2	yes	maximal, complemented
5	#22553597	C₂	1	2	yes	maximal, complemented
6	#22553352	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16862177	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553339	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553313	C₂	1	1	yes	maximal, characteristic
10	#22553308	C₂	1	1	yes	maximal, characteristic
11	#520×C₁₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#14523497	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	#16862246	C₂²	1	2	yes	—
15	#16862245	—	2	1	no	—
16	#16862238	C₂²	1	2	yes	—
17	#16862237	—	2	1	no	—
18	#16862223	—	2	2	no	—
19	#16862205	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
21	#16862179	C₂²	1	2	yes	—
22	#16862177	C₂²	1	2	yes	—
23	#16862177	C₂²	1	2	yes	—
24	#16862177	C₂²	1	2	yes	—
25	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
27	#16862170	C₂²	1	1	yes	characteristic
28	#16862170	C₂²	1	2	yes	—
29	#16862170	C₂²	1	1	yes	characteristic
30	#16862167	C₂²	1	1	yes	characteristic
31	#16862167	C₂²	1	2	yes	—
32	#16862167	C₂²	1	1	yes	characteristic
33	C₂×#41×C₁₃₁	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
35	C₄×D₄×C₁₃₁	—	2	2	no	—
36	C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
37	#117×C₁₃₁	—	2	1	no	—
38	#117×C₁₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₆₂	—	2	2	no	—
42	C₂²×Dic₁₃₁	—	2	1	no	—
43	#14523497	—	2	2	no	—
44	C₂×C₁₃₁⋊C₈	—	2	2	no	—
45	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₁₃₁	—	2	2	no	—
49	C₄⋊Dic₁₃₁	—	2	1	no	—
50	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
51	C₄⋊Dic₁₃₁	C₂³	1	2	yes	—
52	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
53	Dic₁₃₁⋊C₄	—	2	2	no	—
54	Dic₁₃₁⋊C₄	—	2	2	no	—
55	C₂²⋊Dic₁₃₁	—	2	1	no	—
56	C₄×Dic₁₃₁	—	2	1	no	—
57	#41×C₁₃₁	—	2	2	no	—
58	#41×C₁₃₁	Q₈	1	4	yes	complemented
59	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₃₁	—	2	1	no	—
61	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₁	—	2	1	no	—
63	C₂²×C₄×C₁₃₁	—	2	1	no	—
64	C₂²×C₄×C₁₃₁	—	2	2	no	—
65	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₁	—	2	1	no	—
67	C₄⋊C₄×C₁₃₁	—	2	2	no	—
68	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
71	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
72	C₂²⋊C₄×C₁₃₁	D₄	1	2	yes	—
73	C₄²×C₁₃₁	—	2	2	no	—
74	C₄²×C₁₃₁	C₂³	1	2	yes	—
75	Dic₂₆₂	—	8	2	no	—
76	C₂×Dic₁₃₁	—	2	1	no	—
77	C₂×Dic₁₃₁	—	2	1	no	—
78	C₂×Dic₁₃₁	—	2	2	no	—
79	C₂×Dic₁₃₁	—	8	1	no	—
80	C₁₃₁⋊C₈	—	2	1	no	—
81	C₁₃₁⋊C₈	—	2	1	no	—
82	C₁₃₁⋊C₈	—	2	2	no	—
83	C₂³×C₁₃₁	—	2	1	no	—
84	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₁₃₁	—	2	1	no	—
86	Q₈×C₁₃₁	C₂×Q₈	1	2	yes	—
87	D₄×C₁₃₁	—	2	4	no	—
88	D₄×C₁₃₁	—	2	1	no	—
89	D₄×C₁₃₁	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₁₃₁	—	2	1	no	—
91	C₂×C₄×C₁₃₁	—	2	4	no	—
92	C₂×C₄×C₁₃₁	—	2	2	no	—
93	C₂×C₄×C₁₃₁	—	8	2	no	—
94	C₂×C₄×C₁₃₁	—	8	1	no	—
95	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₁₃₁	#41	1	2	yes	—
100	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
101	C₂²×C₁₃₁	—	2	1	no	—
102	C₂²×C₁₃₁	—	2	1	no	—
103	C₂²×C₁₃₁	—	2	2	no	—
104	C₂²×C₁₃₁	—	4	1	no	—
105	C₂²×C₁₃₁	—	4	1	no	—
106	C₂²×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₁₃₁	—	2	2	no	—
109	C₄×C₁₃₁	—	2	2	no	—
110	C₄×C₁₃₁	—	2	2	no	—
111	C₄×C₁₃₁	—	4	2	no	—
112	C₄×C₁₃₁	C₂²⋊Q₈	1	2	yes	—
113	C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
114	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
115	Dic₁₃₁	—	4	1	no	—
116	Dic₁₃₁	—	4	2	no	—
117	Dic₁₃₁	—	4	1	no	—
118	C₂×C₁₃₁	—	2	2	no	—
119	C₂×C₁₃₁	—	4	2	no	—
120	C₂×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₁₃₁	#577	1	1	yes	characteristic
122	C₂×C₁₃₁	#577	1	1	yes	characteristic
123	C₁₃₁	#3084	1	1	yes	characteristic, complemented, Sylow
124	#3084	—	131	1	no	maximal, Sylow
125	#544	—	131	1	no	—
126	#520	D₁₃₁	1	1	yes	characteristic
127	#479	—	131	2	no	—
128	#478	—	131	2	no	—
129	#476	—	131	2	no	—
130	#475	—	131	2	no	—
131	#428	—	131	1	no	—
132	C₂×#106	—	131	1	no	—
133	#424	—	131	1	no	—
134	#417	—	131	1	no	—
135	#416	—	131	1	no	—
136	C₂×#41	C₂×D₁₃₁	1	1	yes	characteristic
137	C₂×M₁₆	—	131	1	no	—
138	C₂²×C₈	—	131	1	no	—
139	C₄⋊Q₈	—	262	1	no	—
140	C₄⋊Q₈	—	131	2	no	—
141	#125	—	262	1	no	—
142	#125	—	131	2	no	—
143	C₂²⋊Q₈	—	262	2	no	—
144	C₄×Q₈	C₂×D₁₃₁	1	2	yes	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	C₂×D₁₃₁	1	2	yes	—
147	#117	—	2	1	no	—
148	#117	—	131	1	no	—
149	#117	C₂×D₁₃₁	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	131	1	no	—
152	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
153	#107	—	131	2	no	—
154	#106	—	131	2	no	—
155	#106	—	131	2	no	—
156	#106	—	131	2	no	—
157	C₄⋊C₈	—	131	2	no	—
158	C₄⋊C₈	—	131	2	no	—
159	#103	—	131	1	no	—
160	#103	—	131	1	no	—
161	#103	—	131	2	no	—
162	#102	—	131	1	no	—
163	#102	—	131	1	no	—
164	#102	—	131	2	no	—
165	#41	—	2	2	no	—
166	#41	Dic₂₆₂	1	4	yes	complemented
167	C₂×Q₈	—	262	2	no	—
168	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	262	1	no	—
175	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
176	M₁₆	—	262	2	no	—
177	C₂×C₈	—	262	2	no	—
178	C₂×C₈	—	131	1	no	—
179	C₂×C₈	—	131	1	no	—
180	C₂×C₈	—	131	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	262	2	no	—
184	C₄⋊C₄	—	262	2	no	—
185	C₄⋊C₄	—	262	1	no	—
186	C₄⋊C₄	—	262	2	no	—
187	C₄⋊C₄	—	131	2	no	—
188	C₄⋊C₄	—	131	1	no	—
189	C₄⋊C₄	—	131	1	no	—
190	C₄⋊C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
191	C₄⋊C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×D₁₃₁	1	2	yes	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	262	1	no	—
195	C₂²⋊C₄	C₂²⋊D₁₃₁	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	262	1	no	—
198	C₄²	C₂²×D₁₃₁	1	2	yes	—
199	C₂³	—	2	1	no	—
200	C₂³	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	1048	2	no	—
203	Q₈	C₂×Dic₂₆₂	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₂₆₂	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	262	1	no	—
211	C₂×C₄	—	262	1	no	—
212	C₂×C₄	—	262	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	1048	1	no	—
216	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	2	yes	—
217	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
220	C₂×C₄	#14523513	1	2	yes	—
221	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
222	C₈	—	262	1	no	—
223	C₈	—	262	1	no	—
224	C₈	—	262	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#16862225	1	2	yes	—
231	C₂²	C₂²×C₂²⋊D₁₃₁	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	524	2	no	—
237	C₄	—	524	1	no	—
238	C₄	—	524	1	no	—
239	C₄	#16862225	1	2	yes	—
240	C₄	C₂²×Dic₂₆₂	1	1	yes	characteristic
241	C₄	C₂×#14523513	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#16862225	1	1	yes	characteristic
245	C₂	#22554120	1	1	yes	characteristic
246	C₂	#22554123	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#520×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×#41×C₁₃₁	C₂²
hypercenter	#520	D₁₃₁
hypocenter	C₁₃₁	#3084
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. #22553940	C₂	semidirect
2. #22553619	C₂	semidirect
3. #22553612	C₂	semidirect
4. #22553602	C₂	semidirect
5. #22553597	C₂	semidirect
6. #22553352	C₂	semidirect
7. C₂×#16862177	C₂	semidirect
8. #22553339	C₂	semidirect
58. #41×C₁₃₁	Q₈	semidirect
123. C₁₃₁	#3084	semidirect
166. #41	Dic₂₆₂	semidirect

Maximal subgroups

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

subgroup	index
#22553940	2
#22553619	2
#22553612	2
#22553602	2
#22553597	2
#22553352	2
C₂×#16862177	2
#22553339	2
#22553313	2
#22553308	2
#520×C₁₃₁	2
#3084	131