Subgroups — #30597967

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#92420⋊C₈	C₂	1	2	yes	maximal
3	#21460641	C₂	1	1	yes	maximal, characteristic
4	#21460641	C₂	1	1	yes	maximal, characteristic
5	#97×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	C₄	1	4	yes	—
7	C₂×C₄×#92420	C₂²	1	2	yes	—
8	C₂×C₄×#92420	C₂²	1	2	yes	—
9	#16840628	C₄	1	4	yes	—
10	#92420⋊C₈	—	2	4	no	—
11	C₂×#14519688	C₄	1	1	yes	characteristic
12	C₂×#14519687	C₄	1	1	yes	characteristic
13	C₂×#14519687	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
17	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
18	C₂²×#92420	C₂×C₄	1	4	yes	—
19	C₂²×#92420	C₂×C₄	1	4	yes	—
20	C₄⋊#92420	—	2	8	no	—
21	C₄×#92420	—	2	2	no	—
22	C₄×#92420	—	2	2	no	—
23	C₄×#92420	—	2	2	no	—
24	C₄×#92420	—	2	2	no	—
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
31	#14519688	—	2	2	no	—
32	#14519687	—	2	2	no	—
33	#14519687	—	2	2	no	—
34	C₈×Dic₆₁	—	2	2	no	—
35	C₈×Dic₆₁	D₄	1	2	yes	—
36	C₈×Dic₆₁	Q₈	1	2	yes	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
40	#97×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
44	C₂²×Dic₆₁	C₄²	1	1	yes	characteristic
45	C₂²×Dic₆₁	C₂²×C₄	1	2	yes	—
46	C₂×#92420	—	2	4	no	—
47	C₂×#92420	—	2	4	no	—
48	C₂×#92420	—	2	4	no	—
49	C₂×#92420	—	2	4	no	—
50	C₂×#92420	—	2	4	no	—
51	C₂×#92420	M₁₆	1	8	yes	—
52	C₂×C₈×C₆₁	—	2	1	no	—
53	C₂×C₈×C₆₁	—	2	1	no	—
54	C₂×C₈×C₆₁	—	2	1	no	—
55	C₂×C₈×C₆₁	—	2	1	no	—
56	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
58	C₄²×C₆₁	M₁₆	1	2	yes	—
59	C₄²×C₆₁	M₁₆	1	2	yes	—
60	C₄×Dic₆₁	—	2	1	no	—
61	C₄×Dic₆₁	—	2	1	no	—
62	C₄×Dic₆₁	—	2	4	no	—
63	C₄×Dic₆₁	—	2	4	no	—
64	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
65	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₆₁	#41	1	2	yes	—
67	C₂×#92418	—	2	1	no	—
68	C₂×#92418	—	2	1	no	—
69	C₂×#92418	—	2	1	no	—
70	C₂×#92418	—	2	1	no	—
71	C₂×#92418	C₄⋊C₄	1	2	yes	—
72	C₂×#92418	M₁₆	1	2	yes	—
73	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	#97	1	4	yes	—
77	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₆₁	—	2	1	no	—
84	C₂×Dic₆₁	—	2	2	no	—
85	C₂×Dic₆₁	—	2	1	no	—
86	C₂×Dic₆₁	—	2	2	no	—
87	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
88	C₂×Dic₆₁	C₄×Q₈	1	2	yes	—
89	C₂×Dic₆₁	C₂×M₁₆	1	4	yes	—
90	C₂×Dic₆₁	#131	1	4	yes	—
91	#92420	—	2	8	no	—
92	#92420	—	4	4	no	—
93	C₈×C₆₁	—	2	2	no	—
94	C₈×C₆₁	—	2	2	no	—
95	#92418	—	2	2	no	—
96	#92418	—	2	2	no	—
97	#8263	—	61	1	no	maximal, Sylow
98	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
100	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
101	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
102	C₂²×C₆₁	#405	1	2	yes	—
103	C₄×C₆₁	—	2	4	no	—
104	C₄×C₆₁	#423	1	1	yes	characteristic
105	C₄×C₆₁	#423	1	1	yes	characteristic
106	C₄×C₆₁	#432	1	1	yes	characteristic
107	C₄×C₆₁	#432	1	1	yes	characteristic
108	Dic₆₁	—	4	1	no	—
109	Dic₆₁	—	4	1	no	—
110	Dic₆₁	#435	1	4	yes	—
111	Dic₆₁	#446	1	4	yes	—
112	C₄×C₄⋊C₈	—	61	2	no	—
113	C₄×#97	—	61	1	no	—
114	C₂×#322	—	61	2	no	—
115	#1273	—	61	1	no	—
116	#1273	—	61	1	no	—
117	C₂×C₆₁	#1451	1	2	yes	—
118	C₂×C₆₁	#1453	1	1	yes	characteristic
119	C₂×C₆₁	#1455	1	1	yes	characteristic
120	C₂×C₆₁	#1767	1	1	yes	characteristic
121	C₂×C₆₁	#1771	1	2	yes	—
122	C₂×C₄⋊C₈	—	61	4	no	—
123	C₂×#97	—	61	1	no	—
124	C₂×#97	—	61	1	no	—
125	C₂×#97	—	61	1	no	—
126	C₂×#97	#6923	1	1	yes	characteristic
127	C₂×C₄×C₈	—	61	1	no	—
128	C₂×C₄×C₈	—	61	2	no	—
129	C₂×C₄×C₈	—	61	2	no	—
130	C₂×C₄×C₈	—	61	1	no	—
131	C₄³	—	61	1	no	—
132	#336	—	61	4	no	—
133	#322	—	122	4	no	—
134	C₆₁	#8263	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	—	61	4	no	—
136	C₂²×C₈	—	61	1	no	—
137	C₂²×C₈	—	61	1	no	—
138	C₂²×C₈	—	61	4	no	—
139	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
140	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
141	C₂×C₄²	—	61	1	no	—
142	C₂×C₄²	—	61	2	no	—
143	C₂×C₄²	—	61	2	no	—
144	C₂×C₄²	—	61	1	no	—
145	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
146	C₄⋊C₈	—	122	8	no	—
147	#97	—	2	2	no	—
148	#97	—	122	2	no	—
149	#97	—	122	2	no	—
150	#97	—	122	2	no	—
151	C₄×C₈	—	61	2	no	—
152	C₄×C₈	—	61	2	no	—
153	C₄×C₈	—	122	2	no	—
154	C₄×C₈	—	122	2	no	—
155	C₄×C₈	—	122	2	no	—
156	C₄×C₈	—	122	2	no	—
157	C₄×C₈	—	122	2	no	—
158	C₂²×C₄	—	61	2	no	—
159	C₂²×C₄	—	61	1	no	—
160	C₂²×C₄	—	61	1	no	—
161	C₂²×C₄	C₄×#6923	1	2	yes	—
162	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	61	8	no	—
168	C₂×C₈	—	61	2	no	—
169	C₂×C₈	—	61	2	no	—
170	C₂×C₈	—	122	4	no	—
171	C₂×C₈	—	122	4	no	—
172	C₂×C₈	—	122	1	no	—
173	C₂×C₈	—	122	1	no	—
174	C₂×C₈	—	122	1	no	—
175	C₂×C₈	—	122	1	no	—
176	C₂×C₈	—	122	4	no	—
177	C₂×C₈	—	122	4	no	—
178	C₂×C₈	—	122	4	no	—
179	C₂×C₈	#11758213	1	2	yes	—
180	C₂×C₈	C₄⋊#6923	1	2	yes	—
181	C₄²	—	61	1	no	—
182	C₄²	—	61	1	no	—
183	C₄²	—	61	2	no	—
184	C₄²	—	122	4	no	—
185	C₄²	—	122	4	no	—
186	C₄²	—	122	1	no	—
187	C₄²	—	122	1	no	—
188	C₄²	#11758213	1	2	yes	—
189	C₄²	#11758217	1	2	yes	—
190	C₂³	C₂×C₄×#6923	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	61	4	no	—
194	C₂×C₄	—	61	4	no	—
195	C₂×C₄	—	61	2	no	—
196	C₂×C₄	—	61	2	no	—
197	C₂×C₄	—	122	2	no	—
198	C₂×C₄	—	122	1	no	—
199	C₂×C₄	—	122	1	no	—
200	C₂×C₄	—	122	2	no	—
201	C₂×C₄	#14519832	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
203	C₂×C₄	#14519856	1	1	yes	characteristic
204	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
205	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
206	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
207	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	122	8	no	—
211	C₈	—	122	2	no	—
212	C₈	—	122	2	no	—
213	C₈	—	244	4	no	—
214	C₂²	C₄×C₄⋊#6923	1	1	yes	characteristic
215	C₂²	C₂×#14519832	1	1	yes	characteristic
216	C₂²	M₁₆×#6923	1	2	yes	—
217	C₂²	C₄×#11758213	1	1	yes	characteristic
218	C₂²	#16840728	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	61	4	no	—
221	C₄	—	61	4	no	—
222	C₄	—	244	1	no	—
223	C₄	—	244	1	no	—
224	C₄	#16840765	1	1	yes	characteristic
225	C₄	#16840766	1	1	yes	characteristic
226	C₄	#16840783	1	1	yes	characteristic
227	C₄	#16840784	1	1	yes	characteristic
228	C₂	#21461026	1	2	yes	—
229	C₂	#21461039	1	1	yes	characteristic
230	C₂	#21461043	1	1	yes	characteristic
231	C₂	#21461605	1	1	yes	characteristic
232	C₂	#21461617	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator, socle, prime radical	C₂³×C₆₁	C₂×C₄²
Fitting	C₂×#97×C₆₁	C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×#97	#6923
hypocenter	C₆₁	#8263
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
134. C₆₁	#8263	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×#92420⋊C₈	2
#21460641	2
#97×Dic₆₁	2
#8263	61