Subgroups — #29589976

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	#21447622	C₂	1	1	yes	maximal, characteristic
2	#21447618	C₂	1	1	yes	maximal, characteristic
3	#21443801	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#446	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11758185	—	2	1	no	—
6	C₄⋊#11758185	C₂²	1	2	yes	—
7	C₂²⋊#11758185	—	2	1	no	—
8	C₂²⋊#11758185	—	2	1	no	—
9	#16839538	C₂²	1	2	yes	—
10	C₄×#11758185	—	2	1	no	—
11	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
12	C₆₁×#446	—	2	1	no	—
13	C₆₁×#446	—	2	1	no	—
14	C₆₁×#446	C₄	1	2	yes	—
15	C₆₁×#446	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
17	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
18	C₂×#97×C₆₁	—	2	1	no	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758185	—	2	2	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₂²×Q₈×C₆₁	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	—	2	1	no	—
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₆₁	—	2	1	no	—
27	C₄×Q₈×C₆₁	—	2	1	no	—
28	C₄×Q₈×C₆₁	C₈	1	2	yes	—
29	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₁	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	—	2	1	no	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₆₁	—	2	2	no	—
35	C₄⋊C₈×C₆₁	—	2	2	no	—
36	C₄⋊C₈×C₆₁	—	2	1	no	—
37	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
38	#97×C₆₁	—	2	2	no	—
39	#97×C₆₁	—	2	2	no	—
40	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₆₁	—	2	1	no	—
45	C₂×Q₈×C₆₁	—	4	2	no	—
46	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₆₁	—	2	1	no	—
48	C₂²×C₄×C₆₁	—	2	1	no	—
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	—	2	2	no	—
54	C₂×C₈×C₆₁	—	4	1	no	—
55	C₂×C₈×C₆₁	—	4	2	no	—
56	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₆₁	—	2	2	no	—
61	C₄⋊C₄×C₆₁	—	2	1	no	—
62	C₄⋊C₄×C₆₁	—	2	2	no	—
63	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
64	C₄²×C₆₁	—	2	2	no	—
65	C₄²×C₆₁	—	2	2	no	—
66	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
69	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	#11758185	—	4	2	no	—
71	#11758185	—	4	2	no	—
72	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
73	Q₈×C₆₁	—	4	2	no	—
74	Q₈×C₆₁	—	4	2	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	—	2	1	no	—
78	C₂×C₄×C₆₁	—	2	1	no	—
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	—	2	2	no	—
81	C₂×C₄×C₆₁	—	4	2	no	—
82	C₂×C₄×C₆₁	—	4	2	no	—
83	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
89	C₈×C₆₁	—	2	1	no	—
90	C₈×C₆₁	—	2	1	no	—
91	C₈×C₆₁	—	2	1	no	—
92	C₈×C₆₁	—	4	2	no	—
93	C₈×C₆₁	#137	1	2	yes	—
94	#11642	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	—	2	1	no	—
96	C₂²×C₆₁	—	2	1	no	—
97	C₂²×C₆₁	#350	1	2	yes	—
98	C₂²×C₆₁	C₂×C₂²⋊C₈	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	—	2	1	no	—
103	C₄×C₆₁	—	4	2	no	—
104	C₄×C₆₁	—	4	2	no	—
105	C₄×C₆₁	#349	1	2	yes	—
106	C₄×C₆₁	#419	1	1	yes	characteristic
107	C₄×C₆₁	#421	1	1	yes	characteristic
108	C₂×#446	D₆₁	1	1	yes	characteristic
109	#2163	—	61	1	no	—
110	#2162	—	61	1	no	—
111	#1338	—	61	4	no	—
112	C₂×C₆₁	—	2	2	no	—
113	C₂×C₆₁	#1479	1	1	yes	characteristic
114	C₂×C₆₁	C₂×#349	1	1	yes	characteristic
115	C₂×C₆₁	C₂×#350	1	1	yes	characteristic
116	C₂×C₄×Q₈	Dic₆₁	1	1	yes	characteristic
117	#446	—	2	1	no	—
118	#446	—	2	1	no	—
119	#446	Dic₆₁	1	2	yes	—
120	#446	C₂×D₆₁	1	2	yes	—
121	C₂×C₄⋊C₈	—	2	1	no	—
122	C₂×C₄⋊C₈	Dic₆₁	1	1	yes	characteristic
123	C₂×#97	—	2	1	no	—
124	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
125	C₄⋊C₁₆	—	61	2	no	—
126	C₄⋊C₁₆	—	122	1	no	—
127	C₂²⋊C₁₆	—	122	1	no	—
128	C₂²⋊C₁₆	—	122	1	no	—
129	#346	—	61	2	no	—
130	C₄×C₁₆	—	122	1	no	—
131	C₆₁	#11642	1	1	yes	characteristic, complemented, Sylow
132	C₂²×Q₈	#92418	1	1	yes	characteristic
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	#92418	1	2	yes	—
139	C₄×Q₈	C₂×Dic₆₁	1	2	yes	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	#92418	1	1	yes	characteristic
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
144	C₂×C₁₆	—	122	2	no	—
145	C₂×C₁₆	—	122	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	1	no	—
149	C₄⋊C₈	C₂×Dic₆₁	1	2	yes	—
150	#97	—	2	2	no	—
151	#97	—	2	2	no	—
152	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	C₂×#92418	1	2	yes	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	C₂×#92418	1	1	yes	characteristic
162	C₂²×C₄	#11758194	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
169	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	C₂×#92418	1	2	yes	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₄²	C₂×#92418	1	1	yes	characteristic
179	C₄²	#11758194	1	1	yes	characteristic
180	C₄²	C₂²⋊Dic₆₁	1	1	yes	characteristic
181	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
182	C₁₆	—	244	2	no	—
183	C₁₆	—	244	2	no	—
184	C₂³	C₂²⋊#92418	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
199	C₂×C₄	C₂²×#92418	1	1	yes	characteristic
200	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	#14519808	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16839546	1	2	yes	—
209	C₂²	C₂×C₂²⋊#92418	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	#16839544	1	2	yes	—
217	C₄	#16839771	1	1	yes	characteristic
218	C₄	#16839779	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#21444214	1	1	yes	characteristic
221	C₂	C₂×#16839544	1	1	yes	characteristic
222	C₂	C₂×#16839546	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#446	C₂
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂²⋊C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×#446	D₆₁
hypocenter	C₆₁	#11642
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #21443801	C₂	semidirect
131. C₆₁	#11642	semidirect

Maximal subgroups

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

subgroup	index
#21447622	2
#21447618	2
#21443801	2
C₂×C₆₁×#446	2
#11642	61