Subgroups — #29835522

Order 15616.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456358	C₂	1	2	yes	maximal
2	#21448105	C₂	1	4	yes	maximal, complemented
3	#21448098	C₂	1	2	yes	maximal, complemented
4	#21448094	C₂	1	2	yes	maximal, complemented
5	#21447607	C₂	1	2	yes	maximal
6	#21447603	C₂	1	1	yes	maximal, characteristic
7	C₂×#16839588	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic
9	#16840390	—	2	2	no	—
10	#16840171	—	2	4	no	—
11	#16840167	C₂²	1	4	yes	—
12	#16840164	C₂²	1	4	yes	—
13	#16839804	—	2	2	no	—
14	#16839800	C₂²	1	2	yes	—
15	C₂×#14519711	C₂²	1	2	yes	—
16	#16839590	C₂²	1	4	yes	—
17	#16839590	C₂²	1	2	yes	—
18	#16839588	C₂²	1	1	yes	characteristic
19	#16839588	C₂²	1	2	yes	—
20	#16839588	C₂²	1	4	yes	—
21	#16839588	C₂²	1	1	yes	characteristic
22	#16839588	C₂²	1	2	yes	—
23	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#405	C₂²	1	1	yes	characteristic
26	C₆₁×#346	C₂²	1	2	yes	—
27	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
28	#14519777	—	2	4	no	—
29	#14519776	—	4	2	no	—
30	#14519769	—	2	4	no	—
31	#14519768	—	4	2	no	—
32	#14519754	—	4	4	no	—
33	#14519736	—	2	2	no	—
34	C₂×C₄⋊Dic₆₁	—	2	2	no	—
35	#14519711	—	2	2	no	—
36	#14519711	—	2	4	no	—
37	#14519711	C₂³	1	4	yes	—
38	#14519711	C₂³	1	2	yes	—
39	#14519711	C₂³	1	2	yes	—
40	#14519709	—	2	2	no	—
41	#14519709	—	2	4	no	—
42	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	#117×C₆₁	D₄	1	1	yes	characteristic
45	M₃₂×C₆₁	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
49	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
50	#97×C₆₁	D₄	1	2	yes	—
51	C₄×C₈×C₆₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₆₁	—	4	2	no	—
54	C₂×Dic₁₂₂	—	4	4	no	—
55	M₁₆×C₆₁	Q₁₆	1	4	yes	—
56	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
62	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
64	C₄²×C₆₁	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₆₁	—	4	2	no	—
66	C₄⋊Dic₆₁	—	2	2	no	—
67	C₄⋊Dic₆₁	—	2	4	no	—
68	C₄⋊Dic₆₁	—	2	2	no	—
69	C₄⋊Dic₆₁	—	4	2	no	—
70	C₄⋊Dic₆₁	—	4	4	no	—
71	#11758196	—	4	4	no	—
72	#11758196	—	4	4	no	—
73	C₄×Dic₆₁	—	4	2	no	—
74	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
75	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
76	C₁₆×C₆₁	C₂×Q₈	1	4	yes	—
77	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₆₁	—	4	2	no	—
85	C₂×Dic₆₁	—	4	2	no	—
86	C₂×Dic₆₁	—	4	4	no	—
87	C₂×Dic₆₁	—	16	2	no	—
88	Dic₁₂₂	—	16	4	no	—
89	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
90	C₈×C₆₁	C₂×Q₁₆	1	4	yes	—
91	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
93	#33729	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	—	2	1	no	—
95	C₂²×C₆₁	—	2	1	no	—
96	C₂²×C₆₁	#500	1	2	yes	—
97	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₆₁	—	2	2	no	—
99	C₄×C₆₁	—	2	2	no	—
100	C₄×C₆₁	#500	1	2	yes	—
101	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₆₁	—	8	2	no	—
104	Dic₆₁	—	8	4	no	—
105	Dic₆₁	—	8	2	no	—
106	#3164	—	61	2	no	—
107	#2255	—	61	4	no	—
108	#2253	—	61	2	no	—
109	#2252	—	61	2	no	—
110	#2159	—	61	2	no	—
111	#2158	—	61	1	no	—
112	C₂×#366	—	61	1	no	—
113	#2108	D₆₁	1	1	yes	characteristic
114	C₂×C₆₁	—	2	2	no	—
115	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
116	C₂×C₆₁	#3415	1	1	yes	characteristic
117	C₂×C₆₁	#3417	1	1	yes	characteristic
118	#544	—	122	2	no	—
119	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
120	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
121	#501	—	122	4	no	—
122	#500	—	61	4	no	—
123	#499	—	61	4	no	—
124	#428	—	122	2	no	—
125	#427	—	61	2	no	—
126	C₂×#107	—	61	2	no	—
127	#405	C₂×D₆₁	1	1	yes	characteristic
128	#367	—	61	2	no	—
129	#367	—	61	4	no	—
130	#366	—	61	2	no	—
131	#366	—	61	2	no	—
132	#366	—	61	1	no	—
133	#366	—	61	1	no	—
134	#366	—	61	4	no	—
135	#346	C₂×D₆₁	1	2	yes	—
136	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
137	C₆₁	#33729	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	D₂₄₄	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	122	4	no	—
141	C₄⋊Q₈	—	244	2	no	—
142	#125	—	122	4	no	—
143	#125	—	244	2	no	—
144	C₂²⋊Q₈	—	244	4	no	—
145	#117	—	122	2	no	—
146	#117	D₂₄₄	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	122	2	no	—
148	M₃₂	Dic₁₂₂	1	4	yes	complemented
149	C₂×C₁₆	Dic₁₂₂	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
152	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
153	#107	—	61	4	no	—
154	#107	—	61	2	no	—
155	#107	—	61	2	no	—
156	#107	—	122	2	no	—
157	#107	—	122	4	no	—
158	#106	—	122	2	no	—
159	#106	—	122	4	no	—
160	#97	D₂₄₄	1	2	yes	—
161	C₄×C₈	C₂²×D₆₁	1	2	yes	—
162	C₂×Q₈	—	244	4	no	—
163	C₂²×C₄	—	244	2	no	—
164	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
165	M₁₆	Dic₂₄₄	1	4	yes	—
166	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
167	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
168	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
170	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
171	C₄⋊C₄	—	122	2	no	—
172	C₄⋊C₄	—	122	4	no	—
173	C₄⋊C₄	—	122	2	no	—
174	C₄⋊C₄	—	244	2	no	—
175	C₄⋊C₄	—	244	4	no	—
176	C₄⋊C₄	—	244	4	no	—
177	C₄⋊C₄	—	244	4	no	—
178	C₄⋊C₄	D₄₈₈	1	1	yes	characteristic
179	C₄⋊C₄	D₄₈₈	1	1	yes	characteristic
180	C₂²⋊C₄	—	244	2	no	—
181	C₂²⋊C₄	D₄₈₈	1	2	yes	—
182	C₄²	—	244	2	no	—
183	C₄²	C₂×D₂₄₄	1	2	yes	—
184	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
185	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
186	C₁₆	C₂×Dic₁₂₂	1	4	yes	—
187	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
188	Q₈	—	976	4	no	—
189	C₂×C₄	—	976	2	no	—
190	C₂×C₄	—	244	2	no	—
191	C₂×C₄	—	244	2	no	—
192	C₂×C₄	—	244	4	no	—
193	C₂×C₄	#14519777	1	2	yes	—
194	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
195	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
196	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
197	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
198	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
199	C₈	#14519777	1	2	yes	—
200	C₈	C₂×Dic₂₄₄	1	4	yes	—
201	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#16840167	1	2	yes	—
206	C₂²	C₂²×D₄₈₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	488	4	no	—
210	C₄	—	488	2	no	—
211	C₄	—	488	2	no	—
212	C₄	#16840167	1	2	yes	—
213	C₄	C₂×#14519777	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₄₄	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#16840167	1	1	yes	characteristic
217	C₂	#21458960	1	1	yes	characteristic
218	C₂	#21458972	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₈₈
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2108	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#2108	D₆₁
hypocenter	C₆₁	#33729
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #21448105	C₂	semidirect
3. #21448098	C₂	semidirect
4. #21448094	C₂	semidirect
45. M₃₂×C₆₁	Q₈	semidirect
46. C₂×C₁₆×C₆₁	Q₈	semidirect
137. C₆₁	#33729	semidirect
148. M₃₂	Dic₁₂₂	semidirect
149. C₂×C₁₆	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21456358	2
#21448105	2
#21448098	2
#21448094	2
#21447607	2
#21447603	2
C₂×#16839588	2
C₆₁×#2108	2
#33729	61