Subgroups — #29591288

Order 15616.

310 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456333	C₂	1	1	yes	maximal, characteristic
2	#21447575	C₂	1	1	yes	maximal, characteristic
3	#21443872	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic
5	#16840319	—	2	1	no	—
6	C₂×#14519717	—	2	1	no	—
7	#16840167	C₄	1	1	yes	characteristic
8	#16840167	C₄	1	1	yes	characteristic
9	#16840164	—	2	1	no	—
10	#16840164	C₂²	1	2	yes	—
11	#16840160	—	2	1	no	—
12	#16839800	—	2	1	no	—
13	#16839800	C₄	1	2	yes	—
14	C₄⋊#11758185	—	2	1	no	—
15	C₄⋊#11758185	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#334	—	2	1	no	—
19	C₆₁×#334	C₂²	1	2	yes	—
20	#14519777	—	2	1	no	—
21	#14519777	—	2	1	no	—
22	#14519769	—	2	2	no	—
23	#14519756	—	4	1	no	—
24	#14519756	—	4	1	no	—
25	C₄×Dic₁₂₂	—	4	1	no	—
26	C₄×Dic₁₂₂	—	4	1	no	—
27	#14519736	—	2	2	no	—
28	#14519717	—	4	2	no	—
29	C₂×#11758185	—	2	2	no	—
30	#14519711	—	2	1	no	—
31	#14519711	—	2	1	no	—
32	#14519711	C₂×C₄	1	2	yes	—
33	#14519711	C₂×C₄	1	2	yes	—
34	#14519709	—	2	2	no	—
35	#14519709	—	2	2	no	—
36	C₂²×C₈×C₆₁	—	4	1	no	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	—	2	2	no	—
41	C₄⋊C₈×C₆₁	—	4	1	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	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
49	C₂×Dic₁₂₂	—	4	1	no	—
50	C₂×Dic₁₂₂	—	4	1	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	4	2	no	—
53	C₂×C₈×C₆₁	—	8	2	no	—
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
57	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
58	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₆₁	D₈	1	1	yes	characteristic
62	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
63	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂²⋊Dic₆₁	—	4	2	no	—
65	C₄⋊Dic₆₁	—	2	2	no	—
66	C₄⋊Dic₆₁	—	2	2	no	—
67	C₄⋊Dic₆₁	—	4	1	no	—
68	C₄⋊Dic₆₁	—	4	1	no	—
69	#11758196	—	4	2	no	—
70	#11758196	—	4	2	no	—
71	C₄×Dic₆₁	—	4	2	no	—
72	#11758185	—	4	2	no	—
73	C₂³×C₆₁	#102	1	1	yes	characteristic
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	4	1	no	—
76	C₂×C₄×C₆₁	#101	1	2	yes	—
77	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#103	1	2	yes	—
81	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×Dic₆₁	—	4	2	no	—
85	C₂×Dic₆₁	—	4	2	no	—
86	Dic₁₂₂	—	8	2	no	—
87	Dic₁₂₂	—	8	2	no	—
88	C₈×C₆₁	—	2	1	no	—
89	C₈×C₆₁	—	2	1	no	—
90	C₈×C₆₁	—	8	2	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	#11936	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	—	2	1	no	—
96	C₂²×C₆₁	—	2	1	no	—
97	C₂²×C₆₁	#332	1	2	yes	—
98	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	#332	1	2	yes	—
103	C₄×C₆₁	#360	1	2	yes	—
104	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
105	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
106	Dic₆₁	—	8	2	no	—
107	Dic₆₁	—	8	2	no	—
108	#3160	—	61	1	no	—
109	#2150	—	61	1	no	—
110	C₂×#334	D₆₁	1	1	yes	characteristic
111	#1363	—	61	4	no	—
112	C₂×C₆₁	—	2	2	no	—
113	C₂×C₆₁	C₂×#332	1	1	yes	characteristic
114	C₂×C₆₁	#2143	1	1	yes	characteristic
115	C₂×C₆₁	C₂×#360	1	1	yes	characteristic
116	#533	—	122	1	no	—
117	C₂×M₃₂	—	122	1	no	—
118	#500	—	61	1	no	—
119	#500	—	61	1	no	—
120	#499	—	61	2	no	—
121	#499	—	122	1	no	—
122	#498	—	122	1	no	—
123	#427	—	61	2	no	—
124	#427	—	122	1	no	—
125	C₂×C₄⋊C₈	—	2	1	no	—
126	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	61	2	no	—
128	C₄⋊C₁₆	—	122	1	no	—
129	#334	—	2	1	no	—
130	#334	C₂×D₆₁	1	2	yes	—
131	C₆₁	#11936	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	4	1	no	—
133	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
134	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
135	C₄⋊Q₈	—	122	1	no	—
136	C₄⋊Q₈	—	122	1	no	—
137	#125	—	122	2	no	—
138	C₂²⋊Q₈	—	244	1	no	—
139	C₂²⋊Q₈	—	244	1	no	—
140	C₄×Q₈	—	244	1	no	—
141	C₄×Q₈	—	244	1	no	—
142	#117	—	122	2	no	—
143	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
144	M₃₂	—	244	2	no	—
145	C₂×C₁₆	—	122	2	no	—
146	#107	—	61	2	no	—
147	#107	—	61	2	no	—
148	#107	—	122	1	no	—
149	#107	—	122	1	no	—
150	#106	—	122	2	no	—
151	#106	—	122	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
155	C₄×C₈	D₂₄₄	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
158	C₂×Q₈	—	244	1	no	—
159	C₂×Q₈	—	244	1	no	—
160	C₂²×C₄	#11758198	1	1	yes	characteristic
161	C₂²×C₄	#11758199	1	1	yes	characteristic
162	C₂²×C₄	#11758201	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	8	2	no	—
166	C₂×C₈	#11758190	1	2	yes	—
167	C₂×C₈	Dic₂₄₄	1	2	yes	—
168	C₂×C₈	#11758198	1	1	yes	characteristic
169	C₂×C₈	#11758198	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
171	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
172	C₄⋊C₄	—	122	2	no	—
173	C₄⋊C₄	—	122	2	no	—
174	C₄⋊C₄	—	244	1	no	—
175	C₄⋊C₄	—	244	1	no	—
176	C₄⋊C₄	—	244	2	no	—
177	C₄⋊C₄	—	244	2	no	—
178	C₂²⋊C₄	—	244	2	no	—
179	C₄²	—	244	2	no	—
180	C₄²	#11758198	1	1	yes	characteristic
181	C₄²	#11758199	1	1	yes	characteristic
182	C₄²	#11758201	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
184	C₁₆	—	244	2	no	—
185	C₂³	#14519697	1	1	yes	characteristic
186	Q₈	—	488	2	no	—
187	Q₈	—	488	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	244	2	no	—
191	C₂×C₄	—	244	2	no	—
192	C₂×C₄	#14519695	1	2	yes	—
193	C₂×C₄	#14519697	1	1	yes	characteristic
194	C₂×C₄	#14519697	1	1	yes	characteristic
195	C₂×C₄	#14519697	1	1	yes	characteristic
196	C₂×C₄	#14519702	1	2	yes	—
197	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
198	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
199	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	8	2	no	—
203	C₈	#14519702	1	2	yes	—
204	C₈	C₂×#11758190	1	1	yes	characteristic
205	C₈	C₂×Dic₂₄₄	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16839509	1	2	yes	—
209	C₂²	C₂×#14519697	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	488	2	no	—
214	C₄	—	488	2	no	—
215	C₄	#16839509	1	2	yes	—
216	C₄	#16839571	1	2	yes	—
217	C₄	C₂×#14519695	1	1	yes	characteristic
218	C₄	C₂×#14519702	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	C₂×#16839509	1	1	yes	characteristic
221	C₂	#21447546	1	1	yes	characteristic
222	C₂	C₂×#16839571	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14519697
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle	C₂²×C₆₁	C₂×#102
prime radical	C₂³×C₆₁	#102
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	#11936
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #21443872	C₂	semidirect
131. C₆₁	#11936	semidirect

Maximal subgroups

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

subgroup	index
#21456333	2
#21447575	2
#21443872	2
C₂×C₆₁×#334	2
#11936	61