Subgroups — #30598033

Order 15616.

446 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×#92420	C₂	1	2	yes	maximal
2	C₂×#16840624	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#92420	C₂	1	1	yes	maximal, characteristic
4	#21460655	C₂	1	2	yes	maximal
5	#21445707	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	C₂²	1	2	yes	—
7	C₂×C₄×#92420	—	2	2	no	—
8	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
10	#16840628	—	2	2	no	—
11	#16840624	C₄	1	4	yes	complemented
12	C₈×#92420	C₄	1	4	yes	complemented
13	C₂×#14519708	C₄	1	2	yes	—
14	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
16	#16839525	C₄	1	1	yes	characteristic
17	C₆₁×#340	C₄	1	1	yes	characteristic
18	C₂²×#92420	—	2	1	no	—
19	C₂²×#92420	—	2	2	no	—
20	C₂²×#92420	D₄	1	1	yes	characteristic
21	C₂²×#92420	D₄	1	1	yes	characteristic
22	C₄⋊#92420	—	2	2	no	—
23	C₄⋊#92420	C₂×C₄	1	4	yes	—
24	C₄×#92420	—	2	1	no	—
25	C₄×#92420	—	4	4	no	—
26	C₄×#92420	C₂×C₄	1	2	yes	—
27	C₄×#92420	C₂×C₄	1	2	yes	—
28	C₄×#92420	Q₈	1	2	yes	—
29	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₆₁	—	2	2	no	—
32	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
33	#14519708	C₈	1	8	yes	complemented
34	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
35	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
36	#14519684	—	2	2	no	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₆₁	—	2	2	no	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	—	2	2	no	—
42	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	#41	1	1	yes	characteristic
44	C₂×#92420	—	2	2	no	—
45	C₂×#92420	—	2	2	no	—
46	C₂×#92420	—	2	2	no	—
47	C₂×#92420	—	2	2	no	—
48	C₂×#92420	—	2	4	no	—
49	C₂×#92420	—	4	2	no	—
50	C₂×#92420	—	4	2	no	—
51	C₂×#92420	C₄⋊C₄	1	2	yes	—
52	C₂×#92420	C₄⋊C₄	1	2	yes	—
53	C₂×#92420	SD₁₆	1	4	yes	—
54	C₂×C₈×C₆₁	—	2	2	no	—
55	C₂×C₈×C₆₁	C₄²	1	2	yes	—
56	C₄⋊C₄×C₆₁	—	2	2	no	—
57	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
60	C₄×Dic₆₁	—	4	2	no	—
61	C₄×Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₆₁	#41	1	1	yes	characteristic
66	C₂×#92418	C₄²	1	2	yes	—
67	C₂×#92418	C₂×C₈	1	4	yes	—
68	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	2	4	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₆₁	#131	1	2	yes	—
76	C₂×Dic₆₁	—	2	1	no	—
77	C₂×Dic₆₁	—	2	1	no	—
78	C₂×Dic₆₁	—	2	4	no	—
79	C₂×Dic₆₁	—	4	2	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₆₁	#117	1	1	yes	characteristic
83	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
84	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
85	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
86	C₂×Dic₆₁	#135	1	2	yes	—
87	#92420	—	2	4	no	—
88	#92420	—	2	2	no	—
89	#92420	—	4	4	no	—
90	#92420	#106	1	4	yes	complemented
91	C₈×C₆₁	—	2	2	no	—
92	#92418	C₄×C₈	1	4	yes	complemented
93	C₈×#106	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₆₁	C₈×D₄	1	2	yes	—
96	C₂²×C₆₁	C₄×SD₁₆	1	2	yes	—
97	C₂²×C₆₁	#443	1	1	yes	characteristic
98	C₂²×C₆₁	#443	1	1	yes	characteristic
99	C₄×C₆₁	—	4	4	no	—
100	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
101	C₄×C₆₁	#405	1	1	yes	characteristic
102	C₄×C₆₁	C₈×Q₈	1	2	yes	—
103	Dic₆₁	—	2	1	no	—
104	Dic₆₁	—	2	1	no	—
105	Dic₆₁	—	4	4	no	—
106	Dic₆₁	C₂×#106	1	1	yes	characteristic
107	Dic₆₁	#427	1	1	yes	characteristic
108	Dic₆₁	C₄×SD₁₆	1	2	yes	—
109	C₄×#106	—	61	1	no	—
110	C₈×C₄⋊C₄	—	61	2	no	—
111	C₂×#334	—	61	1	no	—
112	C₂×C₈²	—	61	1	no	—
113	#1279	—	61	2	no	—
114	C₂×C₆₁	C₈×SD₁₆	1	2	yes	—
115	C₂×C₆₁	C₈×SD₁₆	1	2	yes	—
116	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₆₁	#1772	1	1	yes	characteristic
118	C₂×C₆₁	C₄×#106	1	1	yes	characteristic
119	C₂×#106	—	61	2	no	—
120	C₂×C₄⋊C₈	—	61	2	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	C₂×C₄×C₈	—	61	1	no	—
123	C₂×C₄×C₈	—	61	1	no	—
124	C₂×C₄×C₈	—	122	2	no	—
125	C₄×C₄⋊C₄	—	61	2	no	—
126	#340	—	61	1	no	—
127	#340	#6923	1	1	yes	characteristic
128	#336	—	122	2	no	—
129	#334	—	61	4	no	—
130	C₈²	—	61	4	no	—
131	C₆₁	C₈×#106	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	61	1	no	—
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	122	1	no	—
136	C₂²×C₈	—	122	2	no	—
137	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	61	2	no	—
139	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
140	C₂×C₄²	—	61	1	no	—
141	C₂×C₄²	—	122	2	no	—
142	#106	—	61	8	no	—
143	C₄⋊C₈	—	61	4	no	—
144	C₄⋊C₈	—	122	2	no	—
145	C₄×C₈	—	61	2	no	—
146	C₄×C₈	—	61	2	no	—
147	C₄×C₈	—	61	2	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	61	2	no	—
150	C₄×C₈	—	122	1	no	—
151	C₄×C₈	—	244	4	no	—
152	#95	—	122	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	61	1	no	—
155	C₂²×C₄	—	61	1	no	—
156	C₂²×C₄	—	122	2	no	—
157	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	61	2	no	—
160	C₂×C₈	—	61	4	no	—
161	C₂×C₈	—	61	2	no	—
162	C₂×C₈	—	61	4	no	—
163	C₂×C₈	—	61	2	no	—
164	C₂×C₈	—	122	2	no	—
165	C₂×C₈	—	122	2	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	122	2	no	—
168	C₂×C₈	—	122	4	no	—
169	C₂×C₈	—	244	2	no	—
170	C₂×C₈	—	244	2	no	—
171	C₂×C₈	C₄×#6923	1	2	yes	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	61	4	no	—
174	C₄⋊C₄	—	122	2	no	—
175	C₄⋊C₄	D₆₁⋊C₈	1	4	yes	—
176	C₄²	—	61	1	no	—
177	C₄²	—	61	1	no	—
178	C₄²	—	61	1	no	—
179	C₄²	—	61	1	no	—
180	C₄²	—	244	2	no	—
181	C₄²	—	244	2	no	—
182	C₂³	D₄×#6923	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	61	2	no	—
187	C₂×C₄	—	61	2	no	—
188	C₂×C₄	—	61	1	no	—
189	C₂×C₄	—	61	1	no	—
190	C₂×C₄	—	61	1	no	—
191	C₂×C₄	—	61	1	no	—
192	C₂×C₄	—	122	1	no	—
193	C₂×C₄	—	122	1	no	—
194	C₂×C₄	—	122	4	no	—
195	C₂×C₄	—	244	2	no	—
196	C₂×C₄	—	244	2	no	—
197	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
198	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
199	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
200	C₂×C₄	#14519864	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	—	61	4	no	—
203	C₈	—	61	4	no	—
204	C₈	—	122	2	no	—
205	C₈	—	122	4	no	—
206	C₈	—	244	4	no	—
207	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
208	C₂²	#16840789	1	2	yes	—
209	C₂²	SD₁₆×#6923	1	2	yes	—
210	C₂²	#16840804	1	1	yes	characteristic
211	C₂²	#16840806	1	1	yes	characteristic
212	C₄	—	4	4	no	—
213	C₄	—	61	1	no	—
214	C₄	—	61	1	no	—
215	C₄	—	61	2	no	—
216	C₄	—	122	1	no	—
217	C₄	—	122	1	no	—
218	C₄	—	244	4	no	—
219	C₄	C₄×D₆₁⋊C₈	1	1	yes	characteristic
220	C₄	#16840726	1	1	yes	characteristic
221	C₄	#16840811	1	2	yes	—
222	C₂	#21461402	1	2	yes	—
223	C₂	#21461403	1	2	yes	—
224	C₂	#21461611	1	1	yes	characteristic
225	C₂	#21461623	1	1	yes	characteristic
226	C₂	#106×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#340	C₄
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#340	#6923
hypocenter	C₆₁	C₈×#106
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
11. #16840624	C₄	semidirect
12. C₈×#92420	C₄	semidirect
33. #14519708	C₈	semidirect
90. #92420	#106	semidirect
92. #92418	C₄×C₈	semidirect
131. C₆₁	C₈×#106	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×#92420	2
C₂×#16840624	2
C₂×C₈×#92420	2
#21460655	2
#21445707	2
C₈×#106	61