Subgroups — #364×#6923 #30606666

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊#364	C₂	1	1	yes	maximal, characteristic
2	M₃₂×#6923	C₂	1	2	yes	maximal
3	C₄×C₈×#6923	C₂	1	1	yes	maximal, characteristic
4	#21460862	C₂	1	2	yes	maximal
5	#364×D₆₁	C₂	1	1	yes	maximal, characteristic
6	D₆₁⋊M₃₂	C₂²	1	2	yes	—
7	C₄×D₆₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×#6923	C₄	1	1	yes	characteristic
9	C₂×C₈×#6923	C₄	1	1	yes	characteristic
10	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
11	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
12	C₄²×#6923	C₄	1	1	yes	characteristic
13	#16840685	C₄	1	2	yes	complemented
14	#16840684	C₄	1	2	yes	complemented
15	#16840649	—	2	2	no	—
16	C₁₆×#6923	—	2	2	no	—
17	M₃₂×D₆₁	C₂²	1	2	yes	—
18	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
19	#16839583	C₄	1	1	yes	characteristic, complemented
20	#16839582	C₄	1	2	yes	complemented
21	C₆₁×#364	C₄	1	1	yes	characteristic, complemented
22	C₂×D₆₁⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×D₆₁⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#6923	—	2	2	no	—
25	C₂×C₄×#6923	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#6923	C₂×C₄	1	1	yes	characteristic
27	#14519852	—	2	2	no	—
28	#14519851	C₂×C₄	1	4	yes	—
29	D₆₁⋊C₁₆	—	2	2	no	—
30	C₄×#92420	—	2	2	no	—
31	C₈×#6923	—	2	1	no	—
32	C₈×#6923	—	2	1	no	—
33	C₈×#6923	C₈	1	4	yes	—
34	C₈×#6923	C₈	1	4	yes	—
35	C₈×#6923	D₄	1	1	yes	characteristic
36	C₈×#6923	D₄	1	1	yes	characteristic
37	C₈×#6923	Q₈	1	1	yes	characteristic
38	C₈×#6923	Q₈	1	1	yes	characteristic
39	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
41	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
42	#14519717	C₂×C₄	1	2	yes	—
43	#14519716	—	2	2	no	—
44	C₁₆×D₆₁	—	2	2	no	—
45	C₄×#92418	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
47	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₆₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×#6923	—	2	1	no	—
51	C₂²×#6923	M₁₆	1	2	yes	—
52	C₂×C₄×D₆₁	—	2	1	no	—
53	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×#92420	—	2	1	no	—
55	C₂×#92420	M₁₆	1	2	yes	—
56	C₄×#6923	—	2	8	no	—
57	C₄×#6923	—	2	1	no	—
58	C₄×#6923	C₄⋊C₄	1	1	yes	characteristic
59	C₄×#6923	C₄⋊C₄	1	1	yes	characteristic
60	C₄×#6923	C₂×C₈	1	2	yes	—
61	C₄×#6923	C₂×C₈	1	2	yes	—
62	D₆₁⋊C₈	—	2	1	no	—
63	D₆₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
64	D₆₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
65	D₆₁⋊C₈	C₂×C₈	1	2	yes	—
66	D₆₁⋊C₈	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₆₁	C₄²	1	1	yes	characteristic
70	C₄×Dic₆₁	—	2	1	no	—
71	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
72	C₂×#92418	C₄²	1	1	yes	characteristic
73	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
74	C₈×D₆₁	C₂×C₈	1	2	yes	—
75	C₈×D₆₁	C₂×C₈	1	2	yes	—
76	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
77	C₈×D₆₁	C₂×Q₈	1	1	yes	characteristic
78	C₈×D₆₁	#41	1	1	yes	characteristic
79	C₈×D₆₁	#41	1	1	yes	characteristic
80	#11758187	—	2	4	no	—
81	C₁₆×C₆₁	—	2	2	no	—
82	#11758185	—	2	2	no	—
83	C₂²×D₆₁	C₂×M₁₆	1	1	yes	characteristic
84	C₂×#6923	—	2	4	no	—
85	C₂×#6923	—	2	2	no	—
86	C₂×#6923	—	2	4	no	—
87	C₂×#6923	C₄⋊C₈	1	2	yes	—
88	C₂×#6923	C₄⋊C₈	1	2	yes	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
91	C₂×Dic₆₁	—	2	1	no	—
92	C₂×Dic₆₁	C₂×M₁₆	1	1	yes	characteristic
93	C₄×D₆₁	—	2	4	no	—
94	C₄×D₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄×D₆₁	#117	1	1	yes	characteristic
96	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
97	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
98	#92420	—	2	2	no	—
99	#92420	C₄⋊C₈	1	2	yes	—
100	#92420	C₄⋊C₈	1	2	yes	—
101	C₈×C₆₁	C₄×C₈	1	2	yes	—
102	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
103	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
104	#92418	C₄×C₈	1	2	yes	—
105	#92418	C₄×D₄	1	1	yes	characteristic
106	#92418	C₄×Q₈	1	1	yes	characteristic
107	C₄×#364	—	61	1	no	maximal, Sylow
108	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
109	C₂×D₆₁	—	2	2	no	—
110	C₂×D₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₂×D₆₁	#431	1	1	yes	characteristic
112	#6923	—	2	4	no	—
113	#6923	—	2	2	no	—
114	#6923	#364	1	4	yes	direct product factor
115	C₄×C₆₁	—	2	2	no	—
116	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
118	Dic₆₁	—	2	2	no	—
119	Dic₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
120	Dic₆₁	#431	1	1	yes	characteristic
121	C₂×#364	—	61	1	no	—
122	C₂×#364	—	61	1	no	—
123	C₄×M₃₂	—	61	2	no	—
124	C₄²×C₈	—	61	1	no	—
125	#1376	—	61	2	no	—
126	C₂×C₆₁	—	2	1	no	—
127	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
128	D₆₁	—	2	1	no	—
129	D₆₁	C₂×#364	1	2	yes	—
130	C₂×M₃₂	—	61	2	no	—
131	C₂×M₃₂	—	61	2	no	—
132	C₂×C₄×C₈	—	61	1	no	—
133	C₂×C₄×C₈	—	61	1	no	—
134	C₂×C₄×C₈	—	61	1	no	—
135	C₂×C₄×C₈	—	61	1	no	—
136	C₂×C₄×C₈	—	61	1	no	—
137	C₂×C₄×C₈	—	61	1	no	—
138	C₄³	—	61	1	no	—
139	#364	—	61	2	no	—
140	#364	—	61	2	no	—
141	#364	—	61	2	no	—
142	#364	—	61	1	no	—
143	#364	#6923	1	1	yes	characteristic, direct product factor
144	#346	—	122	2	no	—
145	C₄×C₁₆	—	122	2	no	—
146	C₆₁	C₄×#364	1	1	yes	characteristic, complemented, Sylow
147	C₂²×C₈	—	61	1	no	—
148	C₂²×C₈	—	61	1	no	—
149	C₂²×C₈	—	61	1	no	—
150	C₂²×C₈	—	61	1	no	—
151	C₂×C₄²	—	61	1	no	—
152	C₂×C₄²	—	61	1	no	—
153	C₂×C₄²	—	61	1	no	—
154	C₂×C₄²	—	122	2	no	—
155	M₃₂	—	61	4	no	—
156	M₃₂	—	61	2	no	—
157	M₃₂	—	122	2	no	—
158	M₃₂	—	122	2	no	—
159	M₃₂	C₂×#6923	1	2	yes	—
160	C₂×C₁₆	—	122	2	no	—
161	C₂×C₁₆	—	122	2	no	—
162	C₄×C₈	—	61	4	no	—
163	C₄×C₈	—	61	4	no	—
164	C₄×C₈	—	61	1	no	—
165	C₄×C₈	—	61	1	no	—
166	C₄×C₈	—	61	1	no	—
167	C₄×C₈	—	61	1	no	—
168	C₄×C₈	—	61	1	no	—
169	C₄×C₈	—	61	1	no	—
170	C₄×C₈	—	61	1	no	—
171	C₄×C₈	—	122	1	no	—
172	C₄×C₈	—	122	1	no	—
173	C₄×C₈	—	122	2	no	—
174	C₄×C₈	C₂×#6923	1	1	yes	characteristic
175	C₂²×C₄	—	61	1	no	—
176	C₂²×C₄	—	61	2	no	—
177	C₂²×C₄	—	122	1	no	—
178	C₂²×C₄	—	122	1	no	—
179	C₂×C₈	—	61	1	no	—
180	C₂×C₈	—	61	1	no	—
181	C₂×C₈	—	61	2	no	—
182	C₂×C₈	—	61	2	no	—
183	C₂×C₈	—	61	1	no	—
184	C₂×C₈	—	61	1	no	—
185	C₂×C₈	—	61	2	no	—
186	C₂×C₈	—	61	2	no	—
187	C₂×C₈	—	61	1	no	—
188	C₂×C₈	—	61	1	no	—
189	C₂×C₈	—	61	1	no	—
190	C₂×C₈	—	61	1	no	—
191	C₂×C₈	—	61	2	no	—
192	C₂×C₈	—	122	1	no	—
193	C₂×C₈	—	122	1	no	—
194	C₂×C₈	C₄×#6923	1	1	yes	characteristic
195	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
196	C₄²	—	61	1	no	—
197	C₄²	—	61	1	no	—
198	C₄²	—	61	2	no	—
199	C₄²	—	61	2	no	—
200	C₄²	—	61	1	no	—
201	C₄²	—	122	1	no	—
202	C₄²	—	122	1	no	—
203	C₄²	—	122	8	no	—
204	C₄²	C₄×#6923	1	1	yes	characteristic
205	C₁₆	—	2	2	no	—
206	C₁₆	—	122	4	no	—
207	C₁₆	—	122	2	no	—
208	C₂³	—	61	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	61	2	no	—
211	C₂×C₄	—	61	1	no	—
212	C₂×C₄	—	61	1	no	—
213	C₂×C₄	—	61	1	no	—
214	C₂×C₄	—	61	1	no	—
215	C₂×C₄	—	61	1	no	—
216	C₂×C₄	—	61	2	no	—
217	C₂×C₄	—	122	4	no	—
218	C₂×C₄	—	122	4	no	—
219	C₂×C₄	—	122	2	no	—
220	C₂×C₄	—	122	4	no	—
221	C₂×C₄	—	122	1	no	—
222	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
223	C₈	—	61	2	no	—
224	C₈	—	61	2	no	—
225	C₈	—	61	2	no	—
226	C₈	—	61	1	no	—
227	C₈	—	61	1	no	—
228	C₈	—	122	2	no	—
229	C₈	C₈×#6923	1	2	yes	—
230	C₈	D₄×#6923	1	1	yes	characteristic
231	C₈	Q₈×#6923	1	1	yes	characteristic
232	C₂²	—	61	1	no	—
233	C₂²	—	61	1	no	—
234	C₂²	—	122	2	no	—
235	C₂²	M₁₆×#6923	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	61	4	no	—
238	C₄	—	61	1	no	—
239	C₄	—	61	1	no	—
240	C₄	—	122	2	no	—
241	C₄	—	122	2	no	—
242	C₄	—	122	4	no	—
243	C₄	C₄⋊C₄×#6923	1	1	yes	characteristic
244	C₄	C₂×C₈×#6923	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	61	2	no	—
247	C₂	—	122	1	no	—
248	C₂	C₄⋊C₈×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#364	C₄
socle	C₂×C₆₁	C₄×C₄⋊C₈
prime radical	C₂²×D₆₁	C₂×M₁₆
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#364	#6923
hypocenter	C₆₁	C₄×#364
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
13. #16840685	C₄	semidirect
14. #16840684	C₄	semidirect
19. #16839583	C₄	semidirect
20. #16839582	C₄	semidirect
21. C₆₁×#364	C₄	semidirect
114. #6923	#364	direct (N × H)
143. #364	#6923	direct (N × H)
146. C₆₁	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊#364	2
M₃₂×#6923	2
C₄×C₈×#6923	2
#21460862	2
#364×D₆₁	2
C₄×#364	61