Subgroups — #347×#6923 #30606115

Order 15616.

418 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462036	C₂	1	1	yes	maximal, characteristic
2	#21462034	C₂	1	2	yes	maximal
3	M₃₂×#6923	C₂	1	2	yes	maximal
4	#97×#6923	C₂	1	1	yes	maximal, characteristic
5	#347×D₆₁	C₂	1	1	yes	maximal, characteristic
6	D₆₁⋊M₃₂	C₂²	1	2	yes	—
7	#16840721	C₄	1	1	yes	characteristic
8	C₂×#14519831	C₄	1	1	yes	characteristic
9	C₂×#14519831	C₂²	1	1	yes	characteristic
10	C₂×C₈×#6923	C₄	1	1	yes	characteristic
11	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
12	C₄²×#6923	C₄	1	1	yes	characteristic
13	#16840655	C₄	1	2	yes	complemented
14	#16840654	C₄	1	2	yes	complemented
15	#16840653	C₄	1	4	yes	complemented
16	#16840652	C₄	1	4	yes	complemented
17	#16840649	C₄	1	4	yes	complemented
18	C₁₆×#6923	C₄	1	4	yes	complemented
19	M₃₂×D₆₁	C₂²	1	2	yes	—
20	#97×D₆₁	C₂²	1	1	yes	characteristic
21	#16839540	C₄	1	1	yes	characteristic, complemented
22	#16839539	C₄	1	2	yes	complemented
23	C₆₁×#347	C₄	1	1	yes	characteristic, complemented
24	C₂×D₆₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×D₆₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#6923	—	2	2	no	—
27	C₂×C₄×#6923	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×#6923	C₂×C₄	1	1	yes	characteristic
29	#14519852	C₂×C₄	1	4	yes	—
30	#14519851	C₂×C₄	1	4	yes	—
31	D₆₁⋊C₁₆	C₂×C₄	1	4	yes	—
32	#14519832	—	2	2	no	—
33	#14519831	—	2	2	no	—
34	#14519831	C₂×C₄	1	2	yes	—
35	#14519831	C₂×C₄	1	2	yes	—
36	C₈×#6923	—	2	2	no	—
37	C₈×#6923	C₂×C₄	1	2	yes	—
38	C₈×#6923	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	4	yes	—
44	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
45	#14519688	C₂×C₄	1	1	yes	characteristic
46	#14519687	C₂×C₄	1	1	yes	characteristic
47	#14519687	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₆₁	C₂×C₄	1	2	yes	—
49	#97×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	1	no	—
57	C₄×#6923	—	4	4	no	—
58	C₄×#6923	C₄²	1	2	yes	—
59	C₄×#6923	C₄²	1	1	yes	characteristic
60	C₄×#6923	C₄²	1	1	yes	characteristic
61	C₄×#6923	M₁₆	1	2	yes	—
62	D₆₁⋊C₈	—	2	1	no	—
63	D₆₁⋊C₈	C₄²	1	1	yes	characteristic
64	D₆₁⋊C₈	C₄²	1	1	yes	characteristic
65	D₆₁⋊C₈	C₄²	1	2	yes	—
66	D₆₁⋊C₈	M₁₆	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	—	2	1	no	—
75	C₈×D₆₁	—	2	1	no	—
76	C₈×D₆₁	C₂²×C₄	1	2	yes	—
77	C₈×D₆₁	C₂²×C₄	1	2	yes	—
78	#11758187	C₄²	1	8	yes	complemented
79	C₁₆×C₆₁	C₄²	1	4	yes	complemented
80	#11758185	C₄²	1	4	yes	complemented
81	C₂²×D₆₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×#6923	—	2	4	no	—
83	C₂×#6923	—	2	2	no	—
84	C₂×#6923	—	4	2	no	—
85	C₂×#6923	#97	1	2	yes	—
86	C₂×#6923	#97	1	2	yes	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
89	C₂×Dic₆₁	—	2	1	no	—
90	C₂×Dic₆₁	C₂×M₁₆	1	1	yes	characteristic
91	C₄×D₆₁	—	4	2	no	—
92	C₄×D₆₁	C₂×C₄²	1	1	yes	characteristic
93	C₄×D₆₁	C₂×C₄²	1	1	yes	characteristic
94	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
95	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
96	#92420	—	2	1	no	—
97	#92420	—	2	1	no	—
98	#92420	#97	1	4	yes	—
99	C₈×C₆₁	—	2	1	no	—
100	C₈×C₆₁	C₂×C₄²	1	2	yes	—
101	#92418	—	2	1	no	—
102	#92418	C₂×C₄²	1	2	yes	—
103	C₄×#347	—	61	1	no	maximal, Sylow
104	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
105	C₂×D₆₁	—	2	2	no	—
106	C₂×D₆₁	C₂×#97	1	1	yes	characteristic
107	C₂×D₆₁	C₄×M₁₆	1	1	yes	characteristic
108	#6923	—	2	2	no	—
109	#6923	—	4	2	no	—
110	#6923	#347	1	4	yes	direct product factor
111	C₄×C₆₁	—	4	1	no	—
112	C₄×C₆₁	C₄³	1	1	yes	characteristic
113	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
114	Dic₆₁	—	4	1	no	—
115	Dic₆₁	C₂×#97	1	1	yes	characteristic
116	Dic₆₁	C₄×M₁₆	1	1	yes	characteristic
117	C₂×#347	—	61	2	no	—
118	C₂×#347	—	61	1	no	—
119	C₂×#347	—	61	1	no	—
120	C₄×M₃₂	—	61	2	no	—
121	C₄×#97	—	61	1	no	—
122	C₂×C₆₁	—	2	1	no	—
123	C₂×C₆₁	C₄×#97	1	1	yes	characteristic
124	D₆₁	—	2	1	no	—
125	D₆₁	C₂×#347	1	2	yes	—
126	C₂×M₃₂	—	61	2	no	—
127	C₂×M₃₂	—	61	2	no	—
128	C₂×#97	—	61	1	no	—
129	C₂×#97	—	61	1	no	—
130	C₂×#97	—	61	1	no	—
131	C₂×#97	—	61	1	no	—
132	C₂×C₄×C₈	—	61	1	no	—
133	C₂×C₄×C₈	—	61	1	no	—
134	C₄³	—	61	1	no	—
135	#347	—	61	4	no	—
136	#347	—	61	4	no	—
137	#347	—	61	2	no	—
138	#347	—	61	2	no	—
139	#347	—	61	2	no	—
140	#347	—	61	1	no	—
141	#347	#6923	1	1	yes	characteristic, direct product factor
142	#346	—	61	4	no	—
143	C₄×C₁₆	—	61	4	no	—
144	C₆₁	C₄×#347	1	1	yes	characteristic, complemented, Sylow
145	C₂²×C₈	—	61	1	no	—
146	C₂²×C₈	—	61	1	no	—
147	C₂²×C₈	—	61	1	no	—
148	C₂²×C₈	—	61	1	no	—
149	C₂×C₄²	—	61	1	no	—
150	C₂×C₄²	—	61	1	no	—
151	C₂×C₄²	—	61	1	no	—
152	C₂×C₄²	—	122	2	no	—
153	M₃₂	—	61	4	no	—
154	M₃₂	—	61	4	no	—
155	M₃₂	—	61	2	no	—
156	M₃₂	—	61	4	no	—
157	M₃₂	C₂×#6923	1	2	yes	—
158	C₂×C₁₆	—	61	4	no	—
159	C₂×C₁₆	—	61	4	no	—
160	#97	—	61	2	no	—
161	#97	—	61	2	no	—
162	#97	—	61	1	no	—
163	#97	—	61	1	no	—
164	#97	—	61	1	no	—
165	#97	—	122	2	no	—
166	#97	—	122	2	no	—
167	#97	C₂×#6923	1	1	yes	characteristic
168	C₄×C₈	—	61	2	no	—
169	C₄×C₈	—	61	2	no	—
170	C₄×C₈	—	122	2	no	—
171	C₂²×C₄	—	61	1	no	—
172	C₂²×C₄	—	61	2	no	—
173	C₂²×C₄	—	122	1	no	—
174	C₂²×C₄	—	122	1	no	—
175	C₂×C₈	—	61	2	no	—
176	C₂×C₈	—	61	2	no	—
177	C₂×C₈	—	61	2	no	—
178	C₂×C₈	—	61	1	no	—
179	C₂×C₈	—	61	2	no	—
180	C₂×C₈	—	61	2	no	—
181	C₂×C₈	—	61	1	no	—
182	C₂×C₈	—	61	1	no	—
183	C₂×C₈	—	61	1	no	—
184	C₂×C₈	—	122	1	no	—
185	C₂×C₈	—	122	1	no	—
186	C₂×C₈	—	122	1	no	—
187	C₂×C₈	—	122	1	no	—
188	C₂×C₈	C₄×#6923	1	1	yes	characteristic
189	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
190	C₄²	—	61	1	no	—
191	C₄²	—	61	1	no	—
192	C₄²	—	61	2	no	—
193	C₄²	—	61	2	no	—
194	C₄²	—	61	1	no	—
195	C₄²	—	122	1	no	—
196	C₄²	—	122	1	no	—
197	C₄²	—	244	4	no	—
198	C₄²	C₄×#6923	1	1	yes	characteristic
199	C₁₆	—	61	8	no	—
200	C₁₆	—	61	4	no	—
201	C₁₆	C₄×#6923	1	4	yes	complemented
202	C₂³	—	61	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	61	2	no	—
205	C₂×C₄	—	61	1	no	—
206	C₂×C₄	—	61	1	no	—
207	C₂×C₄	—	61	1	no	—
208	C₂×C₄	—	61	1	no	—
209	C₂×C₄	—	61	1	no	—
210	C₂×C₄	—	61	2	no	—
211	C₂×C₄	—	122	4	no	—
212	C₂×C₄	—	122	2	no	—
213	C₂×C₄	—	122	1	no	—
214	C₂×C₄	—	244	2	no	—
215	C₂×C₄	—	244	2	no	—
216	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	61	4	no	—
219	C₈	—	61	2	no	—
220	C₈	—	122	1	no	—
221	C₈	—	122	1	no	—
222	C₈	—	122	1	no	—
223	C₈	C₂×C₄×#6923	1	2	yes	—
224	C₂²	—	61	1	no	—
225	C₂²	—	61	1	no	—
226	C₂²	—	122	2	no	—
227	C₂²	M₁₆×#6923	1	1	yes	characteristic
228	C₄	—	4	1	no	—
229	C₄	—	61	4	no	—
230	C₄	—	61	1	no	—
231	C₄	—	61	1	no	—
232	C₄	—	122	2	no	—
233	C₄	—	244	1	no	—
234	C₄	—	244	2	no	—
235	C₄	C₄²×#6923	1	1	yes	characteristic
236	C₄	M₁₆×#6923	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	61	2	no	—
239	C₂	—	122	1	no	—
240	C₂	#97×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#6923
commutator	C₄×C₆₁	C₄³
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#347	C₄
socle	C₂×C₆₁	C₄×#97
prime radical	C₂²×D₆₁	C₂×M₁₆
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#347	#6923
hypocenter	C₆₁	C₄×#347
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ #347

Complements

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

N	H ≅ G/N	type
13. #16840655	C₄	semidirect
14. #16840654	C₄	semidirect
15. #16840653	C₄	semidirect
16. #16840652	C₄	semidirect
17. #16840649	C₄	semidirect
18. C₁₆×#6923	C₄	semidirect
21. #16839540	C₄	semidirect
22. #16839539	C₄	semidirect
23. C₆₁×#347	C₄	semidirect
78. #11758187	C₄²	semidirect
79. C₁₆×C₆₁	C₄²	semidirect
80. #11758185	C₄²	semidirect
110. #6923	#347	direct (N × H)
141. #347	#6923	direct (N × H)
144. C₆₁	C₄×#347	semidirect
201. C₁₆	C₄×#6923	semidirect

Maximal subgroups

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

subgroup	index
#21462036	2
#21462034	2
M₃₂×#6923	2
#97×#6923	2
#347×D₆₁	2
C₄×#347	61