Subgroups — #29578548

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446535	C₂	1	2	yes	maximal
2	C₄×#14519709	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
4	#21443658	C₂	1	2	yes	maximal
5	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519709	C₄	1	1	yes	characteristic
7	C₂×#14519709	C₄	1	1	yes	characteristic
8	C₂×#14519706	—	2	2	no	—
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	#16839525	C₄	1	2	yes	—
13	#16839516	—	2	2	no	—
14	C₈×#92418	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#334	—	2	2	no	—
18	C₂²×#92418	—	2	1	no	—
19	C₂²×#92418	D₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₆₁	—	2	2	no	—
23	#14519709	—	2	2	no	—
24	#14519709	C₈	1	4	yes	complemented
25	#14519706	—	4	4	no	—
26	C₄×#92418	—	2	1	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	—	2	1	no	—
29	C₄×#92418	Q₈	1	2	yes	—
30	#14519684	—	2	2	no	—
31	C₂²×C₈×C₆₁	—	2	2	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	—	2	2	no	—
36	C₄⋊C₈×C₆₁	—	2	2	no	—
37	C₄×C₈×C₆₁	D₄	1	2	yes	—
38	C₄×C₈×C₆₁	Q₈	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
42	C₂²×Dic₆₁	—	2	2	no	—
43	C₂²×Dic₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	2	1	no	—
45	C₂×C₈×C₆₁	—	2	1	no	—
46	C₂×C₈×C₆₁	—	4	2	no	—
47	C₂×C₈×C₆₁	—	4	2	no	—
48	C₂×C₈×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
53	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	1	yes	characteristic
57	C₄⋊Dic₆₁	—	2	2	no	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄⋊Dic₆₁	—	2	2	no	—
60	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
61	C₄×Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	—	4	2	no	—
63	C₂×#92418	—	2	4	no	—
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	—	2	2	no	—
66	C₂×#92418	—	2	2	no	—
67	C₂×#92418	SD₁₆	1	4	yes	—
68	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	2	2	no	—
70	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₆₁	#135	1	2	yes	—
82	C₂×Dic₆₁	—	2	4	no	—
83	C₂×Dic₆₁	—	4	2	no	—
84	C₂×Dic₆₁	—	4	2	no	—
85	C₂×Dic₆₁	—	4	2	no	—
86	C₂×Dic₆₁	—	4	2	no	—
87	C₂×Dic₆₁	—	4	2	no	—
88	C₈×C₆₁	—	2	2	no	—
89	C₈×C₆₁	—	8	2	no	—
90	C₈×C₆₁	C₄⋊C₈	1	4	yes	complemented
91	#92418	—	2	4	no	—
92	#92418	—	2	4	no	—
93	#9593	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	#389	1	1	yes	characteristic
95	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
96	C₂²×C₆₁	#436	1	1	yes	characteristic
97	C₂²×C₆₁	C₄×SD₁₆	1	2	yes	—
98	C₂²×C₆₁	#443	1	2	yes	—
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₄×C₆₁	#424	1	1	yes	characteristic
103	C₄×C₆₁	C₈×Q₈	1	1	yes	characteristic
104	C₄×C₆₁	#446	1	1	yes	characteristic
105	C₄×C₆₁	#465	1	2	yes	—
106	C₄×C₆₁	#498	1	1	yes	characteristic
107	C₄×C₆₁	#499	1	1	yes	characteristic
108	Dic₆₁	—	4	4	no	—
109	Dic₆₁	—	8	2	no	—
110	#1939	—	61	2	no	—
111	C₄×#106	—	61	1	no	—
112	C₂×#334	D₆₁	1	1	yes	characteristic
113	C₂×C₈²	—	61	1	no	—
114	#1279	—	61	2	no	—
115	C₂×C₆₁	C₈×SD₁₆	1	2	yes	—
116	C₂×C₆₁	#1590	1	2	yes	—
117	C₂×C₆₁	#1939	1	1	yes	characteristic
118	C₂×C₆₁	#1941	1	1	yes	characteristic
119	C₂×C₆₁	#1945	1	1	yes	characteristic
120	C₂×#106	—	61	1	no	—
121	C₂×#106	—	61	1	no	—
122	C₂×C₄⋊C₈	—	122	2	no	—
123	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
124	C₂×C₄×C₈	—	61	1	no	—
125	C₂×C₄×C₈	—	61	1	no	—
126	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
127	C₄×C₄⋊C₄	—	61	2	no	—
128	#340	—	61	2	no	—
129	#336	—	122	2	no	—
130	#334	—	2	2	no	—
131	C₈²	—	122	2	no	—
132	C₆₁	#9593	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	61	2	no	—
135	C₂²×C₈	—	122	1	no	—
136	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
137	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	61	2	no	—
139	C₂×C₄⋊C₄	—	61	2	no	—
140	C₂×C₄²	—	122	2	no	—
141	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
142	#106	—	61	4	no	—
143	#106	—	122	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	244	4	no	—
147	C₄×C₈	—	61	2	no	—
148	C₄×C₈	—	122	1	no	—
149	C₄×C₈	—	122	1	no	—
150	C₄×C₈	—	122	1	no	—
151	C₄×C₈	Dic₁₂₂	1	2	yes	—
152	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
153	#95	—	122	2	no	—
154	C₂²×C₄	—	122	2	no	—
155	C₂²×C₄	—	122	2	no	—
156	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
158	C₂²×C₄	#11758226	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	61	4	no	—
165	C₂×C₈	—	122	4	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	122	2	no	—
168	C₂×C₈	—	122	2	no	—
169	C₂×C₈	C₈×D₆₁	1	2	yes	—
170	C₂×C₈	#11758189	1	2	yes	—
171	C₂×C₈	#11758196	1	2	yes	—
172	C₂×C₈	#11758196	1	2	yes	—
173	C₄⋊C₄	—	61	4	no	—
174	C₄⋊C₄	—	122	2	no	—
175	C₄⋊C₄	—	122	2	no	—
176	C₄⋊C₄	—	122	2	no	—
177	C₄²	—	244	2	no	—
178	C₄²	—	244	2	no	—
179	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
180	C₄²	#11758224	1	1	yes	characteristic
181	C₄²	Q₈×D₆₁	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
183	C₂³	#14519739	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	122	4	no	—
186	C₂×C₄	—	244	2	no	—
187	C₂×C₄	—	244	2	no	—
188	C₂×C₄	—	244	2	no	—
189	C₂×C₄	—	244	2	no	—
190	C₂×C₄	—	244	2	no	—
191	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
192	C₂×C₄	#14519742	1	1	yes	characteristic
193	C₂×C₄	#14519748	1	1	yes	characteristic
194	C₂×C₄	#14519753	1	1	yes	characteristic
195	C₂×C₄	#14519768	1	1	yes	characteristic
196	C₂×C₄	#14519776	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
198	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
199	C₂×C₄	#14519785	1	1	yes	characteristic
200	C₂×C₄	#14519785	1	1	yes	characteristic
201	C₂×C₄	SD₁₆×D₆₁	1	2	yes	—
202	C₂×C₄	#14519798	1	2	yes	—
203	C₈	—	2	2	no	—
204	C₈	—	8	2	no	—
205	C₈	—	122	4	no	—
206	C₈	—	122	4	no	—
207	C₈	#14519706	1	4	yes	complemented
208	C₂²	#16839658	1	1	yes	characteristic
209	C₂²	#16839827	1	1	yes	characteristic
210	C₂²	#16839839	1	1	yes	characteristic
211	C₂²	#16839849	1	2	yes	—
212	C₂²	#16839878	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	488	2	no	—
216	C₄	—	244	4	no	—
217	C₄	C₂×#14519706	1	1	yes	characteristic
218	C₄	#16839790	1	1	yes	characteristic
219	C₄	#16839886	1	1	yes	characteristic
220	C₄	#16839889	1	1	yes	characteristic
221	C₄	#16839988	1	2	yes	—
222	C₄	#16840159	1	1	yes	characteristic
223	C₄	#16840162	1	1	yes	characteristic
224	C₂	#21444618	1	2	yes	—
225	C₂	#21444708	1	2	yes	—
226	C₂	#21446535	1	1	yes	characteristic
227	C₂	#21446545	1	1	yes	characteristic
228	C₂	#21446565	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	#9593
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
24. #14519709	C₈	semidirect
90. C₈×C₆₁	C₄⋊C₈	semidirect
132. C₆₁	#9593	semidirect
207. C₈	#14519706	semidirect

Maximal subgroups

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

subgroup	index
#21446535	2
C₄×#14519709	2
C₂×C₈×#92418	2
#21443658	2
C₂×C₆₁×#334	2
#9593	61