Subgroups — #30623855

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462788	C₂	1	2	yes	maximal
2	#21462041	C₂	1	4	yes	maximal, complemented
3	#21462040	C₂	1	2	yes	maximal, complemented
4	C₂×#16840655	C₂	1	2	yes	maximal, complemented
5	#21462036	C₂	1	4	yes	maximal, complemented
6	#21454246	C₂	1	1	yes	maximal, characteristic
7	#16840821	C₄	1	16	yes	complemented
8	D₆₁⋊M₃₂	C₂²	1	8	yes	—
9	C₂×#14519851	C₂²	1	4	yes	—
10	#16840655	C₂²	1	4	yes	—
11	#16840655	C₂²	1	4	yes	—
12	#16840654	C₂²	1	8	yes	—
13	C₂×#14519785	C₂²	1	1	yes	characteristic
14	C₂×#14519783	C₄	1	1	yes	characteristic
15	#16840230	C₄	1	1	yes	characteristic
16	#16839719	C₄	1	2	yes	—
17	#16839718	C₂²	1	1	yes	characteristic
18	#16839716	C₂²	1	2	yes	—
19	M₁₆×Dic₆₁	C₄	1	1	yes	characteristic
20	C₄×#11758189	C₄	1	2	yes	—
21	C₂×#14519687	C₂²	1	1	yes	characteristic
22	#97×D₆₁	C₂²	1	2	yes	—
23	C₆₁×#405	C₄	1	1	yes	characteristic
24	#14519852	C₂×C₄	1	16	yes	complemented
25	#14519851	C₂×C₄	1	8	yes	complemented
26	#14519851	C₂³	1	8	yes	—
27	D₆₁⋊C₁₆	C₂×C₄	1	16	yes	complemented
28	C₂×#11758187	C₂×C₄	1	8	yes	complemented
29	C₂×#11758222	C₂×C₄	1	1	yes	characteristic
30	#14519785	C₂×C₄	1	8	yes	—
31	#14519783	—	2	4	no	—
32	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
33	C₂×#11758189	C₂×C₄	1	2	yes	—
34	C₂×#11758189	C₂×C₄	1	2	yes	—
35	C₂²×#92418	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₆₁	C₂³	1	2	yes	—
38	#14519742	—	2	1	no	—
39	#14519739	—	2	1	no	—
40	#14519737	—	2	2	no	—
41	#14519734	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₆₁	C₂×C₄	1	2	yes	—
44	#14519688	C₂³	1	2	yes	—
45	#14519687	C₂×C₄	1	2	yes	—
46	#14519687	C₂×C₄	1	1	yes	characteristic
47	#14519687	C₂×C₄	1	1	yes	characteristic
48	#14519687	C₂³	1	1	yes	characteristic
49	#14519687	C₂³	1	1	yes	characteristic
50	C₄×#92418	C₂×C₄	1	2	yes	—
51	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
54	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
55	#97×C₆₁	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₆₁	M₁₆	1	2	yes	—
59	C₂²×Dic₆₁	—	2	1	no	—
60	#11758222	C₄²	1	4	yes	—
61	#11758222	M₁₆	1	4	yes	—
62	C₂×D₂₄₄	M₁₆	1	1	yes	characteristic
63	C₂×C₄×D₆₁	—	2	2	no	—
64	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₁₂₂	M₁₆	1	1	yes	characteristic
66	M₁₆×C₆₁	C₄²	1	4	yes	—
67	C₂×C₈×C₆₁	—	2	2	no	—
68	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
71	C₄⋊C₄×C₆₁	—	2	1	no	—
72	C₂²⋊C₄×C₆₁	—	2	1	no	—
73	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
74	#11758198	—	2	2	no	—
75	#11758196	—	2	2	no	—
76	C₄×Dic₆₁	—	2	1	no	—
77	C₄×Dic₆₁	—	2	2	no	—
78	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
79	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
80	#11758194	—	2	2	no	—
81	C₂×#92418	C₂²×C₄	1	2	yes	—
82	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
83	C₂×#92418	C₂²×C₄	1	4	yes	—
84	C₂×#92418	C₂⁴	1	1	yes	characteristic
85	#11758189	—	2	4	no	—
86	#11758189	C₄²	1	8	yes	—
87	C₈×D₆₁	—	2	4	no	—
88	C₈×D₆₁	C₂²×C₄	1	8	yes	—
89	#11758187	C₂²×C₄	1	16	yes	—
90	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
91	C₂²×D₆₁	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
94	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
97	C₂²⋊D₆₁	#97	1	8	yes	—
98	C₂×Dic₆₁	—	2	1	no	—
99	C₂×Dic₆₁	—	2	1	no	—
100	C₂×Dic₆₁	—	8	1	no	—
101	C₂×Dic₆₁	C₂×M₁₆	1	2	yes	—
102	D₂₄₄	#97	1	4	yes	—
103	C₄×D₆₁	—	8	2	no	—
104	C₄×D₆₁	C₂×C₄²	1	4	yes	—
105	C₄×D₆₁	C₂×M₁₆	1	4	yes	—
106	Dic₁₂₂	#97	1	4	yes	—
107	C₈×C₆₁	—	2	1	no	—
108	C₈×C₆₁	—	2	1	no	—
109	C₈×C₆₁	C₂×C₄²	1	4	yes	—
110	#92418	—	2	2	no	—
111	#92418	C₂×C₄²	1	2	yes	—
112	#92418	C₂³×C₄	1	2	yes	—
113	#33329	—	61	1	no	maximal, Sylow
114	C₂²×C₆₁	—	4	1	no	—
115	C₂²×C₆₁	C₂×#97	1	2	yes	—
116	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
117	C₂×D₆₁	—	4	2	no	—
118	C₂×D₆₁	C₂×#97	1	4	yes	—
119	C₄×C₆₁	—	4	2	no	—
120	C₄×C₆₁	C₂×#97	1	2	yes	—
121	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
122	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
123	Dic₆₁	—	4	1	no	—
124	Dic₆₁	—	4	1	no	—
125	Dic₆₁	C₂×#97	1	4	yes	—
126	#3407	—	61	2	no	—
127	#2875	—	61	1	no	—
128	#2110	—	61	2	no	—
129	#2110	—	61	4	no	—
130	C₂×#347	—	61	4	no	—
131	C₂×#347	—	61	2	no	—
132	C₂×C₆₁	—	2	1	no	—
133	C₂×C₆₁	—	2	2	no	—
134	C₂×C₆₁	C₂²×#97	1	1	yes	characteristic
135	D₆₁	—	2	4	no	—
136	C₂×#131	—	61	1	no	—
137	C₂×#131	—	61	1	no	—
138	C₄×#41	—	61	1	no	—
139	#504	—	61	16	no	—
140	C₂×M₃₂	—	61	4	no	—
141	C₂×M₃₂	—	61	8	no	—
142	#405	—	61	1	no	—
143	#405	—	61	2	no	—
144	#405	—	61	2	no	—
145	#405	#6923	1	1	yes	characteristic
146	C₄×M₁₆	—	61	1	no	—
147	C₄×M₁₆	—	61	2	no	—
148	C₂×#97	—	61	1	no	—
149	C₂×#97	—	61	2	no	—
150	#347	—	61	4	no	—
151	#347	—	61	8	no	—
152	#347	—	61	4	no	—
153	C₆₁	#33329	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	—	61	1	no	—
155	#131	—	61	8	no	—
156	#131	—	122	4	no	—
157	C₂×M₁₆	—	61	1	no	—
158	C₂×M₁₆	—	61	2	no	—
159	C₂×M₁₆	—	61	2	no	—
160	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
161	C₂²×C₈	—	61	1	no	—
162	C₂²×C₈	—	61	2	no	—
163	C₂²×C₈	—	61	2	no	—
164	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
165	C₄×Q₈	—	122	1	no	—
166	C₄×D₄	—	122	2	no	—
167	C₄×D₄	—	122	1	no	—
168	#117	—	61	2	no	—
169	#117	C₂×#6923	1	1	yes	characteristic
170	C₂×C₄²	—	61	1	no	—
171	C₂×C₄²	—	61	2	no	—
172	M₃₂	—	61	8	no	—
173	M₃₂	—	61	8	no	—
174	M₃₂	—	61	16	no	—
175	C₂×C₁₆	—	61	8	no	—
176	C₂×C₁₆	—	61	16	no	—
177	#97	—	61	1	no	—
178	#97	—	61	1	no	—
179	#97	—	61	1	no	—
180	#97	—	61	2	no	—
181	#97	—	61	1	no	—
182	#97	—	61	2	no	—
183	#97	C₂×#6923	1	2	yes	—
184	C₄×C₈	—	61	2	no	—
185	C₄×C₈	—	61	2	no	—
186	C₄×C₈	C₂×#6923	1	2	yes	—
187	#41	—	61	4	no	—
188	#41	—	61	4	no	—
189	C₂×Q₈	—	61	1	no	—
190	C₂×D₄	—	61	2	no	—
191	C₂×D₄	—	61	1	no	—
192	C₂²×C₄	—	61	2	no	—
193	C₂²×C₄	—	122	2	no	—
194	C₂²×C₄	—	122	1	no	—
195	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
196	M₁₆	—	61	8	no	—
197	M₁₆	—	122	4	no	—
198	M₁₆	—	122	2	no	—
199	M₁₆	C₄×#6923	1	4	yes	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	61	4	no	—
202	C₂×C₈	—	61	1	no	—
203	C₂×C₈	—	61	1	no	—
204	C₂×C₈	—	61	8	no	—
205	C₂×C₈	—	61	2	no	—
206	C₂×C₈	—	122	4	no	—
207	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
208	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
209	C₂×C₈	C₂²×#6923	1	2	yes	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	122	2	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	122	2	no	—
214	C₄²	—	61	1	no	—
215	C₄²	—	61	1	no	—
216	C₄²	—	122	2	no	—
217	C₄²	—	122	1	no	—
218	C₄²	C₂²×#6923	1	2	yes	—
219	C₁₆	—	61	16	no	—
220	C₂³	—	61	2	no	—
221	C₂³	D₆₁⋊M₁₆	1	1	yes	characteristic
222	Q₈	—	61	4	no	—
223	D₄	—	61	8	no	—
224	D₄	—	61	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	61	2	no	—
227	C₂×C₄	—	61	4	no	—
228	C₂×C₄	—	61	4	no	—
229	C₂×C₄	—	122	1	no	—
230	C₂×C₄	—	122	1	no	—
231	C₂×C₄	—	488	2	no	—
232	C₂×C₄	—	488	1	no	—
233	C₂×C₄	C₂×C₄×#6923	1	2	yes	—
234	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
235	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
236	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	61	2	no	—
240	C₈	—	61	2	no	—
241	C₈	—	122	2	no	—
242	C₈	C₂×C₄×#6923	1	4	yes	—
243	C₂²	—	4	1	no	—
244	C₂²	—	61	4	no	—
245	C₂²	—	244	2	no	—
246	C₂²	#16840721	1	2	yes	—
247	C₂²	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
248	C₄	—	4	2	no	—
249	C₄	—	61	4	no	—
250	C₄	—	244	1	no	—
251	C₄	—	244	1	no	—
252	C₄	#16840721	1	2	yes	—
253	C₄	C₂²×C₄×#6923	1	1	yes	characteristic
254	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	2	no	—
257	C₂	—	122	4	no	—
258	C₂	C₂×#16840721	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×#6923
commutator	C₄×C₆₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#405	C₄
socle	C₂×C₆₁	C₂²×#97
prime radical	C₂×#11758222	C₂×C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	#405	#6923
hypocenter	C₆₁	#33329
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #21462041	C₂	semidirect
3. #21462040	C₂	semidirect
4. C₂×#16840655	C₂	semidirect
5. #21462036	C₂	semidirect
7. #16840821	C₄	semidirect
24. #14519852	C₂×C₄	semidirect
25. #14519851	C₂×C₄	semidirect
27. D₆₁⋊C₁₆	C₂×C₄	semidirect
28. C₂×#11758187	C₂×C₄	semidirect
153. C₆₁	#33329	semidirect

Maximal subgroups

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

subgroup	index
#21462788	2
#21462041	2
#21462040	2
C₂×#16840655	2
#21462036	2
#21454246	2
#33329	61