Subgroups — #30612016

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462734	C₂	1	2	yes	maximal
2	#21461370	C₂	1	6	yes	maximal, complemented
3	#21461366	C₂	1	3	yes	maximal, complemented
4	C₂×#16840625	C₂	1	3	yes	maximal, complemented
5	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×#92420	—	2	2	no	—
7	D₄×#92420	—	2	6	no	—
8	#16840778	C₂²	1	6	yes	—
9	C₂×C₄⋊#92420	C₂²	1	6	yes	—
10	#16840625	C₂²	1	3	yes	—
11	#16840625	C₂²	1	6	yes	—
12	#16840625	C₂²	1	1	yes	characteristic
13	#16840623	C₂²	1	6	yes	—
14	C₂×#14519784	C₄	1	1	yes	characteristic
15	#41×Dic₆₁	C₂²	1	1	yes	characteristic
16	#16839720	C₄	1	3	yes	—
17	#16839717	C₄	1	3	yes	—
18	M₁₆×Dic₆₁	C₂²	1	3	yes	—
19	C₂×C₈×Dic₆₁	C₂²	1	3	yes	—
20	C₂×#131×C₆₁	C₄	1	1	yes	characteristic
21	C₂²×#92420	—	2	6	no	—
22	C₄⋊#92420	—	2	6	no	—
23	C₄⋊#92420	C₂³	1	6	yes	—
24	C₄⋊#92420	C₂³	1	2	yes	—
25	C₂²⋊#92420	—	2	6	no	—
26	C₄×#92420	—	2	6	no	—
27	#14519784	—	2	4	no	—
28	C₂×#11758194	C₂×C₄	1	3	yes	—
29	C₂²×#92418	C₂×C₄	1	3	yes	—
30	Q₈×Dic₆₁	D₄	1	1	yes	characteristic
31	Q₈×Dic₆₁	Q₈	1	1	yes	characteristic
32	D₄×Dic₆₁	D₄	1	3	yes	—
33	D₄×Dic₆₁	Q₈	1	3	yes	—
34	#14519736	C₂×C₄	1	3	yes	—
35	C₂×C₄×Dic₆₁	C₂³	1	3	yes	—
36	#14519687	C₂×C₄	1	6	yes	—
37	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
38	C₈×Dic₆₁	C₂×C₄	1	3	yes	—
39	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₈×Dic₆₁	C₂³	1	3	yes	—
41	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
42	#131×C₆₁	C₈	1	8	yes	complemented
43	C₂×M₁₆×C₆₁	C₂×C₄	1	3	yes	—
44	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
45	#41×C₆₁	C₂×C₈	1	4	yes	—
46	#41×C₆₁	M₁₆	1	4	yes	—
47	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₆₁	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
50	C₂²×Dic₆₁	C₂×D₄	1	3	yes	—
51	C₂²×Dic₆₁	C₂×Q₈	1	3	yes	—
52	C₂×#92420	—	2	2	no	—
53	C₂×#92420	—	2	6	no	—
54	C₂×#92420	—	8	6	no	—
55	M₁₆×C₆₁	C₂×C₈	1	12	yes	complemented
56	C₂×C₈×C₆₁	C₂×C₈	1	12	yes	complemented
57	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
59	C₂²⋊Dic₆₁	C₄⋊C₄	1	6	yes	—
60	C₄⋊Dic₆₁	C₄⋊C₄	1	3	yes	—
61	C₄⋊Dic₆₁	C₄⋊C₄	1	3	yes	—
62	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
63	C₄×Dic₆₁	C₂×D₄	1	3	yes	—
64	C₄×Dic₆₁	C₂×Q₈	1	3	yes	—
65	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
66	#11758194	—	2	6	no	—
67	C₂×#92418	—	2	6	no	—
68	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
69	C₂×#92418	C₂²×C₄	1	3	yes	—
70	C₂³×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
71	Q₈×C₆₁	C₄⋊C₈	1	4	yes	—
72	D₄×C₆₁	C₄⋊C₈	1	12	yes	—
73	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
75	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
76	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₆₁	—	2	3	no	—
78	C₂×Dic₆₁	—	2	3	no	—
79	C₂×Dic₆₁	—	4	3	no	—
80	C₂×Dic₆₁	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×Dic₆₁	C₂×C₄⋊C₄	1	3	yes	—
82	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
84	#92420	—	4	6	no	—
85	#92420	—	4	2	no	—
86	C₈×C₆₁	C₂²×C₈	1	2	yes	—
87	C₈×C₆₁	C₂²×C₈	1	6	yes	—
88	#92418	—	2	1	no	—
89	#92418	—	2	3	no	—
90	#17789	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	—	2	3	no	—
92	C₂²×C₆₁	—	2	3	no	—
93	C₂²×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
94	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
96	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
98	Dic₆₁	—	2	3	no	—
99	Dic₆₁	—	2	1	no	—
100	Dic₆₁	—	2	3	no	—
101	Dic₆₁	#576	1	1	yes	characteristic
102	Dic₆₁	#578	1	1	yes	characteristic
103	#2988	—	61	2	no	—
104	C₄×#131	—	61	1	no	—
105	#1565	—	61	6	no	—
106	#1564	—	61	3	no	—
107	C₂×#335	—	61	3	no	—
108	C₂×C₆₁	—	2	6	no	—
109	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₆₁	#2913	1	1	yes	characteristic
111	C₂×C₆₁	#2916	1	1	yes	characteristic
112	C₂×#131	—	61	1	no	—
113	C₂×#131	#6923	1	1	yes	characteristic
114	C₄×#41	—	61	1	no	—
115	C₈×Q₈	—	122	2	no	—
116	C₈×D₄	—	122	6	no	—
117	#431	—	61	6	no	—
118	C₂×C₄⋊C₈	—	61	6	no	—
119	#405	—	61	3	no	—
120	#405	—	61	3	no	—
121	C₄×M₁₆	—	61	3	no	—
122	C₂×C₄×C₈	—	61	3	no	—
123	#335	—	61	3	no	—
124	#335	—	61	6	no	—
125	#335	—	61	1	no	—
126	#334	—	61	6	no	—
127	C₆₁	#17789	1	1	yes	characteristic, complemented, Sylow
128	C₂×#41	C₂×#6923	1	1	yes	characteristic
129	#131	—	122	4	no	—
130	#131	#92420	1	8	yes	complemented
131	C₂×M₁₆	—	61	3	no	—
132	C₂×M₁₆	C₂×#6923	1	3	yes	—
133	C₂²×C₈	—	61	3	no	—
134	C₂²×C₈	—	122	6	no	—
135	C₂²×C₈	C₂×#6923	1	3	yes	—
136	C₄×Q₈	—	61	1	no	—
137	C₄×Q₈	—	61	1	no	—
138	C₄×D₄	—	61	3	no	—
139	C₄×D₄	—	61	3	no	—
140	#117	—	61	3	no	—
141	C₂×C₄²	—	61	3	no	—
142	C₄⋊C₈	—	61	2	no	—
143	C₄⋊C₈	—	61	6	no	—
144	C₄⋊C₈	—	122	6	no	—
145	C₂²⋊C₈	—	122	6	no	—
146	#97	—	61	6	no	—
147	C₄×C₈	—	61	3	no	—
148	C₄×C₈	—	61	1	no	—
149	C₄×C₈	—	61	3	no	—
150	C₄×C₈	—	61	3	no	—
151	C₄×C₈	—	122	6	no	—
152	#41	#11758213	1	4	yes	—
153	#41	C₂×#92420	1	4	yes	—
154	C₂×Q₈	C₄⋊#6923	1	1	yes	characteristic
155	C₂×D₄	C₄⋊#6923	1	3	yes	—
156	C₂²×C₄	—	61	3	no	—
157	C₂²×C₄	—	61	3	no	—
158	C₂²×C₄	C₂²×#6923	1	3	yes	—
159	M₁₆	—	122	6	no	—
160	M₁₆	C₂×#92420	1	12	yes	complemented
161	C₂×C₈	—	61	1	no	—
162	C₂×C₈	—	61	3	no	—
163	C₂×C₈	—	122	6	no	—
164	C₂×C₈	—	122	2	no	—
165	C₂×C₈	—	122	6	no	—
166	C₂×C₈	—	488	6	no	—
167	C₂×C₈	C₂×#92420	1	12	yes	complemented
168	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
169	C₂×C₈	C₂²×#6923	1	3	yes	—
170	C₄⋊C₄	—	61	3	no	—
171	C₄⋊C₄	—	61	3	no	—
172	C₂²⋊C₄	—	61	6	no	—
173	C₄²	—	61	3	no	—
174	C₄²	—	61	3	no	—
175	C₄²	—	61	1	no	—
176	C₄²	—	61	3	no	—
177	C₂³	C₂×C₄⋊#6923	1	3	yes	—
178	Q₈	C₄⋊#92420	1	4	yes	—
179	D₄	C₄⋊#92420	1	12	yes	—
180	C₂×C₄	—	61	1	no	—
181	C₂×C₄	—	61	3	no	—
182	C₂×C₄	—	61	3	no	—
183	C₂×C₄	—	61	1	no	—
184	C₂×C₄	—	122	3	no	—
185	C₂×C₄	—	122	3	no	—
186	C₂×C₄	—	244	3	no	—
187	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
188	C₂×C₄	C₂²×#92420	1	6	yes	—
189	C₂×C₄	C₂×#11758213	1	6	yes	—
190	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
191	C₈	—	122	1	no	—
192	C₈	—	122	3	no	—
193	C₈	—	244	2	no	—
194	C₈	—	244	6	no	—
195	C₈	C₂²×#92420	1	2	yes	—
196	C₈	C₂²×#92420	1	6	yes	—
197	C₂²	—	2	3	no	—
198	C₂²	—	2	3	no	—
199	C₂²	C₂×C₄⋊#92420	1	6	yes	—
200	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
201	C₄	—	61	1	no	—
202	C₄	—	61	1	no	—
203	C₄	—	122	3	no	—
204	C₄	—	122	3	no	—
205	C₄	—	122	1	no	—
206	C₄	C₂×C₄⋊#92420	1	6	yes	—
207	C₄	C₂³×#92420	1	1	yes	characteristic
208	C₄	C₂²×#11758213	1	1	yes	characteristic
209	C₂	—	2	6	no	—
210	C₂	C₂²×C₄⋊#92420	1	1	yes	characteristic
211	C₂	#21462474	1	1	yes	characteristic
212	C₂	#21462481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂×#131×C₆₁	C₄
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂×#131	#6923
hypocenter	C₆₁	#17789
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #21461370	C₂	semidirect
3. #21461366	C₂	semidirect
4. C₂×#16840625	C₂	semidirect
42. #131×C₆₁	C₈	semidirect
55. M₁₆×C₆₁	C₂×C₈	semidirect
56. C₂×C₈×C₆₁	C₂×C₈	semidirect
127. C₆₁	#17789	semidirect
130. #131	#92420	semidirect
160. M₁₆	C₂×#92420	semidirect
167. C₂×C₈	C₂×#92420	semidirect

Maximal subgroups

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

subgroup	index
#21462734	2
#21461370	2
#21461366	2
C₂×#16840625	2
#131×Dic₆₁	2
#17789	61