Subgroups — #30612188

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462465	C₂	1	2	yes	maximal
2	#21461389	C₂	1	6	yes	maximal, complemented
3	#21461377	C₂	1	3	yes	maximal, complemented
4	D₆₁⋊#335	C₂	1	3	yes	maximal, complemented
5	C₈×#11758226	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519865	—	2	2	no	—
7	#16840769	—	2	6	no	—
8	#16840766	C₂²	1	6	yes	—
9	D₆₁⋊(C₄⋊C₈)	C₂²	1	6	yes	—
10	#16840626	C₂²	1	6	yes	—
11	#16840625	C₂²	1	1	yes	characteristic
12	#16840625	C₂²	1	3	yes	—
13	#16840625	C₂²	1	6	yes	—
14	C₄×#11758226	C₂²	1	1	yes	characteristic
15	Q₈×#92418	C₄	1	1	yes	characteristic
16	C₈×D₂₄₄	C₄	1	3	yes	—
17	#16839827	C₄	1	3	yes	—
18	#16839814	C₂²	1	3	yes	—
19	C₄×C₈×D₆₁	C₂²	1	3	yes	—
20	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
21	#14519865	—	4	8	no	—
22	C₂×#11758213	—	2	6	no	—
23	C₂×D₆₁⋊C₈	—	2	6	no	—
24	#14519844	—	2	6	no	—
25	C₄⋊#92420	—	2	6	no	—
26	C₄⋊#92420	C₂³	1	2	yes	—
27	C₄⋊#92420	C₂³	1	6	yes	—
28	C₂×#11758226	D₄	1	1	yes	characteristic
29	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
30	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
31	Q₈×Dic₆₁	Q₈	1	1	yes	characteristic
32	C₄×D₂₄₄	C₂×C₄	1	3	yes	—
33	#14519739	Q₈	1	3	yes	—
34	#14519735	D₄	1	3	yes	—
35	C₄²×D₆₁	C₂³	1	3	yes	—
36	C₄⋊#92418	C₂×C₄	1	3	yes	—
37	#14519689	C₂×C₄	1	6	yes	—
38	C₄×#92418	C₂×C₄	1	3	yes	—
39	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₈×Dic₆₁	C₂³	1	3	yes	—
41	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
44	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	#11758226	D₈	1	4	yes	—
46	#11758226	Q₁₆	1	4	yes	—
47	C₂×D₂₄₄	C₄⋊C₄	1	3	yes	—
48	C₂×C₄×D₆₁	C₂²×C₄	1	3	yes	—
49	C₂×C₄×D₆₁	C₂×D₄	1	3	yes	—
50	C₂×C₄×D₆₁	C₂×Q₈	1	3	yes	—
51	C₂×#92420	—	2	6	no	—
52	C₂×#92420	—	2	2	no	—
53	#11758213	—	4	12	no	—
54	D₆₁⋊C₈	—	4	12	no	—
55	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
58	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
59	#11758198	C₄⋊C₄	1	6	yes	—
60	C₄⋊Dic₆₁	C₄⋊C₄	1	3	yes	—
61	C₄×Dic₆₁	C₂×D₄	1	3	yes	—
62	C₄×Dic₆₁	C₂×Q₈	1	3	yes	—
63	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
64	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
65	C₂×#92418	C₂²×C₄	1	3	yes	—
66	C₈×D₆₁	—	2	6	no	—
67	C₈×D₆₁	—	4	3	no	—
68	C₂²×D₆₁	C₂×C₄⋊C₄	1	3	yes	—
69	Q₈×C₆₁	#107	1	4	yes	—
70	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₆₁	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
76	D₂₄₄	#107	1	12	yes	—
77	C₄×D₆₁	—	2	3	no	—
78	C₄×D₆₁	—	2	3	no	—
79	C₄×D₆₁	—	4	3	no	—
80	C₄×D₆₁	C₂×D₈	1	6	yes	—
81	C₄×D₆₁	C₂×Q₁₆	1	6	yes	—
82	#92420	—	4	2	no	—
83	#92420	—	4	6	no	—
84	C₈×C₆₁	—	2	3	no	—
85	C₈×C₆₁	#131	1	2	yes	complemented
86	#92418	—	2	3	no	—
87	#92418	—	2	1	no	—
88	#17828	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂×D₆₁	—	2	3	no	—
91	C₂×D₆₁	—	2	3	no	—
92	C₂×D₆₁	C₂×#107	1	6	yes	—
93	C₄×C₆₁	—	2	3	no	—
94	C₄×C₆₁	C₂×#107	1	6	yes	—
95	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
96	C₄×C₆₁	#568	1	1	yes	characteristic
97	Dic₆₁	—	2	3	no	—
98	Dic₆₁	—	2	1	no	—
99	Dic₆₁	C₂²×D₈	1	1	yes	characteristic
100	Dic₆₁	C₂²×Q₁₆	1	1	yes	characteristic
101	C₈×#41	—	61	1	no	—
102	#2906	—	61	2	no	—
103	#1571	—	61	6	no	—
104	#1567	—	61	3	no	—
105	C₂×#335	—	61	3	no	—
106	C₂×C₆₁	#2906	1	1	yes	characteristic
107	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
108	C₂×C₆₁	#2916	1	1	yes	characteristic
109	D₆₁	—	2	6	no	—
110	C₂×#131	—	122	2	no	—
111	C₄×#41	—	61	1	no	—
112	C₈×Q₈	—	61	1	no	—
113	C₈×Q₈	#6923	1	1	yes	characteristic
114	C₈×D₄	—	61	3	no	—
115	C₈×D₄	—	61	3	no	—
116	#431	—	61	3	no	—
117	#424	—	122	6	no	—
118	#423	—	61	6	no	—
119	C₂×C₄⋊C₈	—	61	6	no	—
120	C₂×C₄×C₈	—	61	3	no	—
121	#335	—	61	1	no	—
122	#335	—	61	3	no	—
123	#335	—	61	6	no	—
124	#335	—	61	6	no	—
125	C₆₁	#17828	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	—	61	1	no	—
127	#131	—	244	8	no	—
128	C₂×M₁₆	—	122	6	no	—
129	C₂²×C₈	—	61	3	no	—
130	C₂²×C₈	—	61	3	no	—
131	C₂²×C₈	—	122	6	no	—
132	C₄×Q₈	—	61	1	no	—
133	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
134	C₄×D₄	—	61	3	no	—
135	C₄×D₄	—	61	3	no	—
136	#117	—	61	3	no	—
137	C₂×C₄²	—	61	3	no	—
138	C₄⋊C₈	—	61	2	no	—
139	C₄⋊C₈	—	61	6	no	—
140	C₄⋊C₈	—	61	3	no	—
141	C₄⋊C₈	—	122	6	no	—
142	C₄⋊C₈	—	122	6	no	—
143	C₄⋊C₈	C₂×#6923	1	3	yes	—
144	C₂²⋊C₈	—	61	6	no	—
145	C₄×C₈	—	61	1	no	—
146	C₄×C₈	—	61	3	no	—
147	C₄×C₈	—	61	3	no	—
148	C₄×C₈	C₂×#6923	1	3	yes	—
149	#41	—	61	4	no	—
150	#41	—	61	4	no	—
151	C₂×Q₈	C₄⋊#6923	1	1	yes	characteristic
152	C₂×D₄	—	61	3	no	—
153	C₂²×C₄	—	61	3	no	—
154	C₂²×C₄	—	61	3	no	—
155	C₂²×C₄	—	61	3	no	—
156	M₁₆	—	244	12	no	—
157	C₂×C₈	—	61	1	no	—
158	C₂×C₈	—	61	3	no	—
159	C₂×C₈	—	122	6	no	—
160	C₂×C₈	—	122	2	no	—
161	C₂×C₈	—	122	6	no	—
162	C₂×C₈	—	244	3	no	—
163	C₂×C₈	—	244	12	no	—
164	C₂×C₈	C₂²×#6923	1	3	yes	—
165	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
166	C₄⋊C₄	—	61	3	no	—
167	C₄⋊C₄	C₄⋊#6923	1	3	yes	—
168	C₂²⋊C₄	—	61	6	no	—
169	C₄²	—	61	3	no	—
170	C₄²	—	61	3	no	—
171	C₄²	—	61	1	no	—
172	C₄²	C₂²×#6923	1	3	yes	—
173	C₂³	—	61	3	no	—
174	Q₈	#14519846	1	4	yes	—
175	D₄	—	61	12	no	—
176	C₂×C₄	—	61	1	no	—
177	C₂×C₄	—	61	6	no	—
178	C₂×C₄	—	61	6	no	—
179	C₂×C₄	—	61	3	no	—
180	C₂×C₄	—	61	1	no	—
181	C₂×C₄	—	122	3	no	—
182	C₂×C₄	—	122	3	no	—
183	C₂×C₄	—	244	3	no	—
184	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
185	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
186	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
187	C₈	—	2	3	no	—
188	C₈	—	122	3	no	—
189	C₈	—	122	1	no	—
190	C₈	—	244	2	no	—
191	C₈	—	244	6	no	—
192	C₈	#14519865	1	2	yes	complemented
193	C₂²	—	61	6	no	—
194	C₂²	—	122	3	no	—
195	C₂²	—	122	3	no	—
196	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	61	1	no	—
199	C₄	—	61	1	no	—
200	C₄	—	122	3	no	—
201	C₄	—	122	1	no	—
202	C₄	C₂×#14519846	1	6	yes	—
203	C₄	C₂×#14519865	1	1	yes	characteristic
204	C₄	#16840841	1	1	yes	characteristic
205	C₂	—	122	6	no	—
206	C₂	#21462465	1	1	yes	characteristic
207	C₂	C₂²×#14519846	1	1	yes	characteristic
208	C₂	#21462481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₈×Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#11758226	D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₈×Q₈	#6923
hypocenter	C₆₁	#17828
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #21461389	C₂	semidirect
3. #21461377	C₂	semidirect
4. D₆₁⋊#335	C₂	semidirect
85. C₈×C₆₁	#131	semidirect
125. C₆₁	#17828	semidirect
192. C₈	#14519865	semidirect

Maximal subgroups

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

subgroup	index
#21462465	2
#21461389	2
#21461377	2
D₆₁⋊#335	2
C₈×#11758226	2
#17828	61