Subgroups — #29589825

Order 15616.

334 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454753	C₂	1	1	yes	maximal, characteristic
2	#21447575	C₂	1	1	yes	maximal, characteristic
3	#21443799	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	#16840319	C₄	1	1	yes	characteristic
6	C₂×#14519717	—	2	1	no	—
7	#16839892	—	2	1	no	—
8	#16839892	C₄	1	2	yes	—
9	C₈×Dic₁₂₂	—	2	1	no	—
10	C₈×Dic₁₂₂	C₂²	1	2	yes	—
11	#16839788	—	2	1	no	—
12	#16839786	C₄	1	1	yes	characteristic
13	#16839717	—	2	1	no	—
14	C₄⋊#11758185	—	2	1	no	—
15	C₄⋊#11758185	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₆₁	—	2	1	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₆₁	—	2	1	no	—
19	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
20	C₂×#11758194	—	2	1	no	—
21	#14519777	C₈	1	1	yes	characteristic
22	#14519777	C₈	1	1	yes	characteristic
23	#14519769	—	2	1	no	—
24	#14519756	—	2	1	no	—
25	#14519756	C₈	1	2	yes	—
26	C₄×Dic₁₂₂	—	2	1	no	—
27	C₄×Dic₁₂₂	C₂×C₄	1	2	yes	—
28	#14519736	—	2	1	no	—
29	#14519717	—	4	2	no	—
30	C₂×#11758185	—	2	2	no	—
31	C₄⋊#92418	—	2	1	no	—
32	C₄⋊#92418	C₂×C₄	1	2	yes	—
33	#14519706	—	2	2	no	—
34	#14519706	—	2	2	no	—
35	#14519687	—	2	2	no	—
36	C₈×Dic₆₁	—	2	2	no	—
37	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₆₁	—	2	4	no	—
41	C₂×C₁₆×C₆₁	—	2	2	no	—
42	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
49	C₂×Dic₁₂₂	—	2	1	no	—
50	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
55	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
56	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
61	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²⋊Dic₆₁	—	2	2	no	—
63	C₄⋊Dic₆₁	—	2	1	no	—
64	C₄⋊Dic₆₁	—	2	1	no	—
65	C₄⋊Dic₆₁	—	2	1	no	—
66	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
67	#11758196	—	2	2	no	—
68	#11758196	—	2	2	no	—
69	C₄×Dic₆₁	—	2	2	no	—
70	#11758194	—	4	2	no	—
71	C₂×#92418	—	2	2	no	—
72	C₁₆×C₆₁	—	2	4	no	—
73	#11758185	—	4	2	no	—
74	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#103	1	2	yes	—
81	C₂×C₄×C₆₁	#104	1	2	yes	—
82	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₆₁	—	2	2	no	—
86	C₂×Dic₆₁	—	2	2	no	—
87	Dic₁₂₂	—	2	4	no	—
88	Dic₁₂₂	—	4	2	no	—
89	C₈×C₆₁	—	2	1	no	—
90	C₈×C₆₁	—	2	1	no	—
91	C₈×C₆₁	#103	1	2	yes	—
92	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
94	#92418	—	4	2	no	—
95	#11608	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	—	2	1	no	—
97	C₂²×C₆₁	—	2	1	no	—
98	C₂²×C₆₁	Q₈⋊C₈	1	2	yes	—
99	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	Q₈⋊C₈	1	2	yes	—
103	C₄×C₆₁	#350	1	2	yes	—
104	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
105	C₄×C₆₁	C₂×#104	1	1	yes	characteristic
106	C₄×C₆₁	#504	1	2	yes	—
107	Dic₆₁	—	4	2	no	—
108	Dic₆₁	—	4	2	no	—
109	#2960	—	61	1	no	—
110	#2150	—	61	1	no	—
111	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic
112	Q₈⋊C₁₆	—	61	4	no	—
113	C₂×C₆₁	—	2	2	no	—
114	C₂×C₆₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
115	C₂×C₆₁	#2113	1	1	yes	characteristic
116	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
117	#533	—	61	1	no	—
118	C₂×M₃₂	—	122	1	no	—
119	C₂²×C₁₆	—	2	1	no	—
120	#446	—	61	2	no	—
121	#446	—	122	1	no	—
122	C₈×Q₈	—	61	2	no	—
123	C₈×Q₈	—	122	1	no	—
124	#424	—	122	1	no	—
125	#423	—	61	1	no	—
126	#405	—	122	1	no	—
127	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
128	C₄⋊C₁₆	—	61	2	no	—
129	C₄⋊C₁₆	—	122	1	no	—
130	C₄×C₁₆	—	2	1	no	—
131	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
132	C₆₁	#11608	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	122	1	no	—
134	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	61	1	no	—
137	C₄⋊Q₈	—	61	1	no	—
138	#125	—	122	1	no	—
139	C₂²⋊Q₈	—	61	2	no	—
140	C₂²⋊Q₈	—	122	1	no	—
141	C₄×Q₈	—	61	2	no	—
142	C₄×Q₈	—	122	1	no	—
143	#117	—	122	1	no	—
144	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
145	M₃₂	—	244	2	no	—
146	C₂×C₁₆	—	2	4	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	122	2	no	—
149	C₄⋊C₈	—	61	2	no	—
150	C₄⋊C₈	—	122	2	no	—
151	C₄⋊C₈	—	122	2	no	—
152	C₄⋊C₈	—	122	1	no	—
153	#97	—	122	2	no	—
154	C₄×C₈	—	122	2	no	—
155	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
156	C₄×C₈	D₂₄₄	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
159	C₂×Q₈	—	61	2	no	—
160	C₂×Q₈	—	122	1	no	—
161	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
162	C₂²×C₄	#11758189	1	1	yes	characteristic
163	C₂²×C₄	#11758198	1	1	yes	characteristic
164	M₁₆	—	244	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	#11758190	1	2	yes	—
168	C₂×C₈	Dic₂₄₄	1	2	yes	—
169	C₂×C₈	#11758198	1	1	yes	characteristic
170	C₂×C₈	#11758198	1	1	yes	characteristic
171	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
172	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
173	C₄⋊C₄	—	61	2	no	—
174	C₄⋊C₄	—	122	2	no	—
175	C₄⋊C₄	—	122	1	no	—
176	C₄⋊C₄	—	122	1	no	—
177	C₄⋊C₄	—	122	1	no	—
178	C₄⋊C₄	—	122	2	no	—
179	C₂²⋊C₄	—	122	2	no	—
180	C₄²	—	122	2	no	—
181	C₄²	C₈×D₆₁	1	1	yes	characteristic
182	C₄²	#11758189	1	1	yes	characteristic
183	C₄²	#11758198	1	1	yes	characteristic
184	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
185	C₁₆	—	2	4	no	—
186	C₁₆	—	244	2	no	—
187	C₂³	#14519689	1	1	yes	characteristic
188	Q₈	—	122	4	no	—
189	Q₈	—	244	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	122	2	no	—
193	C₂×C₄	—	122	2	no	—
194	C₂×C₄	#14519689	1	1	yes	characteristic
195	C₂×C₄	#14519689	1	1	yes	characteristic
196	C₂×C₄	#14519689	1	1	yes	characteristic
197	C₂×C₄	#14519702	1	2	yes	—
198	C₂×C₄	#14519705	1	2	yes	—
199	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
201	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	244	2	no	—
205	C₈	#14519702	1	2	yes	—
206	C₈	C₂×#11758190	1	1	yes	characteristic
207	C₈	C₂×Dic₂₄₄	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#16839494	1	2	yes	—
211	C₂²	C₂×#14519689	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	244	2	no	—
215	C₄	—	244	2	no	—
216	C₄	#16839494	1	2	yes	—
217	C₄	#16839545	1	2	yes	—
218	C₄	C₂×#14519702	1	1	yes	characteristic
219	C₄	C₂×#14519705	1	1	yes	characteristic
220	C₄	#16840178	1	2	yes	—
221	C₂	—	2	2	no	—
222	C₂	C₂×#16839494	1	1	yes	characteristic
223	C₂	#21447432	1	1	yes	characteristic
224	C₂	C₂×#16839545	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂²⋊C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	#11608
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #21443799	C₂	semidirect
132. C₆₁	#11608	semidirect

Maximal subgroups

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

subgroup	index
#21454753	2
#21447575	2
#21443799	2
C₂×C₄×C₁₆×C₆₁	2
#11608	61