Subgroups — #30624208

Order 15616.

528 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462193	C₂	1	4	yes	maximal, complemented
2	#21462189	C₂	1	4	yes	maximal, complemented
3	C₂×#16840684	C₂	1	2	yes	maximal, complemented
4	#21462055	C₂	1	2	yes	maximal
5	#21462042	C₂	1	2	yes	maximal, complemented
6	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#14519851	C₂²	1	4	yes	—
8	#16840684	C₂²	1	4	yes	—
9	#16840684	C₂²	1	4	yes	—
10	#16840658	C₂²	1	8	yes	—
11	#16840655	C₂²	1	8	yes	—
12	C₂×#14519784	C₂²	1	1	yes	characteristic
13	#41×Dic₆₁	C₄	1	1	yes	characteristic
14	#16839720	C₄	1	2	yes	—
15	#16839720	C₂²	1	1	yes	characteristic
16	#16839717	C₄	1	1	yes	characteristic
17	#16839717	C₂²	1	2	yes	—
18	M₁₆×Dic₆₁	C₄	1	1	yes	characteristic
19	M₁₆×Dic₆₁	C₂²	1	2	yes	—
20	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
21	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	C₂×#131×C₆₁	C₄	1	1	yes	characteristic
23	#14519851	—	2	4	no	—
24	#14519851	C₂³	1	8	yes	—
25	C₂×#11758187	—	2	4	no	—
26	#14519784	—	2	2	no	—
27	#14519784	—	2	2	no	—
28	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
29	C₂×#11758194	C₂³	1	2	yes	—
30	C₂²×#92418	C₂×C₄	1	2	yes	—
31	C₂²×#92418	C₂³	1	1	yes	characteristic
32	Q₈×Dic₆₁	—	2	1	no	—
33	D₄×Dic₆₁	—	2	2	no	—
34	D₄×Dic₆₁	—	2	1	no	—
35	#14519736	C₂×C₄	1	1	yes	characteristic
36	#14519736	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
39	#14519687	C₂×C₄	1	2	yes	—
40	#14519687	C₂×C₄	1	2	yes	—
41	#14519687	C₂³	1	2	yes	—
42	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
44	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
45	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
46	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
47	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
48	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
49	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
50	#131×C₆₁	—	2	4	no	—
51	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
55	#41×C₆₁	—	2	2	no	—
56	#41×C₆₁	—	2	2	no	—
57	C₂×Q₈×C₆₁	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₆₁	M₁₆	1	2	yes	—
59	C₂×D₄×C₆₁	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₆₁	—	2	2	no	—
63	C₂²×Dic₆₁	—	2	1	no	—
64	M₁₆×C₆₁	—	2	2	no	—
65	M₁₆×C₆₁	—	2	4	no	—
66	C₂×C₈×C₆₁	—	2	4	no	—
67	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
68	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²⋊Dic₆₁	—	2	2	no	—
72	C₂²⋊Dic₆₁	—	2	1	no	—
73	C₄⋊Dic₆₁	—	2	2	no	—
74	C₄⋊Dic₆₁	—	2	1	no	—
75	C₄×Dic₆₁	—	2	2	no	—
76	C₄×Dic₆₁	—	2	1	no	—
77	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
78	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
79	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
80	#11758194	—	2	2	no	—
81	#11758194	—	2	4	no	—
82	C₂×#92418	—	2	2	no	—
83	C₂×#92418	—	2	2	no	—
84	C₂×#92418	C₂²×C₄	1	2	yes	—
85	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
86	C₂×#92418	#41	1	2	yes	—
87	C₂×#92418	#41	1	2	yes	—
88	C₂×#92418	C₂⁴	1	1	yes	characteristic
89	#11758187	—	2	8	no	—
90	C₂³×C₆₁	C₂×M₁₆	1	2	yes	—
91	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
92	Q₈×C₆₁	—	2	2	no	—
93	D₄×C₆₁	—	2	4	no	—
94	D₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	—	2	2	no	—
97	C₂×C₄×C₆₁	#117	1	2	yes	—
98	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×Dic₆₁	—	2	1	no	—
103	C₂×Dic₆₁	—	2	2	no	—
104	C₂×Dic₆₁	—	2	1	no	—
105	C₂×Dic₆₁	—	4	2	no	—
106	C₂×Dic₆₁	—	4	4	no	—
107	C₈×C₆₁	—	2	2	no	—
108	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
109	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
110	#92418	—	2	2	no	—
111	#92418	#117	1	1	yes	characteristic
112	#92418	#117	1	1	yes	characteristic
113	#92418	C₂×#41	1	2	yes	—
114	#33428	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	—	2	2	no	—
116	C₂²×C₆₁	—	4	1	no	—
117	C₂²×C₆₁	—	4	2	no	—
118	C₂²×C₆₁	#432	1	2	yes	—
119	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
120	C₄×C₆₁	—	2	2	no	—
121	C₄×C₆₁	#432	1	2	yes	—
122	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
123	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
124	Dic₆₁	—	2	2	no	—
125	Dic₆₁	—	4	2	no	—
126	Dic₆₁	—	4	1	no	—
127	C₄×#131	—	61	1	no	—
128	#2166	—	61	4	no	—
129	#2165	—	61	4	no	—
130	C₂×#364	—	61	2	no	—
131	#2115	—	61	2	no	—
132	#2110	—	61	2	no	—
133	C₂×C₆₁	—	2	1	no	—
134	C₂×C₆₁	—	2	2	no	—
135	C₂×C₆₁	—	4	2	no	—
136	C₂×C₆₁	C₂×#432	1	1	yes	characteristic
137	C₂×#131	—	61	1	no	—
138	C₂×#131	#6923	1	1	yes	characteristic
139	C₄×#41	—	61	1	no	—
140	C₂×M₃₂	—	61	4	no	—
141	#405	—	61	2	no	—
142	#405	—	61	1	no	—
143	#405	—	61	1	no	—
144	#405	—	61	2	no	—
145	C₄×M₁₆	—	61	2	no	—
146	C₄×M₁₆	—	61	1	no	—
147	C₂×C₄×C₈	—	61	2	no	—
148	C₂×C₄×C₈	—	61	1	no	—
149	#364	—	61	4	no	—
150	#364	—	61	4	no	—
151	#349	—	61	8	no	—
152	#347	—	61	8	no	—
153	C₆₁	#33428	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	C₂×#6923	1	1	yes	characteristic
155	#131	—	2	4	no	—
156	#131	—	122	2	no	—
157	#131	—	122	2	no	—
158	C₂×M₁₆	—	61	2	no	—
159	C₂×M₁₆	—	61	1	no	—
160	C₂×M₁₆	C₂×#6923	1	2	yes	—
161	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
162	C₂²×C₈	—	61	1	no	—
163	C₂²×C₈	—	61	2	no	—
164	C₂²×C₈	C₂×#6923	1	2	yes	—
165	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
166	C₄×Q₈	—	122	1	no	—
167	C₄×D₄	—	122	2	no	—
168	C₄×D₄	—	122	1	no	—
169	#117	—	61	2	no	—
170	#117	—	61	1	no	—
171	C₂×C₄²	—	61	2	no	—
172	C₂×C₄²	—	61	1	no	—
173	M₃₂	—	61	8	no	—
174	M₃₂	—	122	4	no	—
175	C₂×C₁₆	—	122	4	no	—
176	#97	—	61	2	no	—
177	#97	—	61	2	no	—
178	#97	—	61	2	no	—
179	C₄×C₈	—	61	1	no	—
180	C₄×C₈	—	61	1	no	—
181	C₄×C₈	—	61	1	no	—
182	C₄×C₈	—	61	2	no	—
183	C₄×C₈	—	61	2	no	—
184	C₄×C₈	—	61	1	no	—
185	C₄×C₈	—	61	2	no	—
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	C₂×Q₈	#11758213	1	1	yes	characteristic
189	C₂×D₄	#11758213	1	2	yes	—
190	C₂×D₄	#11758213	1	1	yes	characteristic
191	C₂²×C₄	—	122	2	no	—
192	C₂²×C₄	—	122	1	no	—
193	C₂²×C₄	C₂²×#6923	1	2	yes	—
194	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	M₁₆	—	122	4	no	—
198	M₁₆	—	122	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	61	2	no	—
201	C₂×C₈	—	61	2	no	—
202	C₂×C₈	—	61	1	no	—
203	C₂×C₈	—	61	1	no	—
204	C₂×C₈	—	61	2	no	—
205	C₂×C₈	—	122	2	no	—
206	C₂×C₈	—	122	2	no	—
207	C₂×C₈	#11758217	1	4	yes	—
208	C₂×C₈	C₂²×#6923	1	2	yes	—
209	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
210	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
211	C₄⋊C₄	—	122	2	no	—
212	C₄⋊C₄	—	122	1	no	—
213	C₂²⋊C₄	—	122	2	no	—
214	C₂²⋊C₄	—	122	1	no	—
215	C₄²	—	61	1	no	—
216	C₄²	—	61	1	no	—
217	C₄²	—	61	2	no	—
218	C₄²	—	122	2	no	—
219	C₄²	—	122	1	no	—
220	C₁₆	—	122	8	no	—
221	C₂³	C₂×#11758213	1	2	yes	—
222	C₂³	C₂×#11758213	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	122	1	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	—	122	1	no	—
231	C₂×C₄	—	244	2	no	—
232	C₂×C₄	—	244	4	no	—
233	C₂×C₄	#14519856	1	2	yes	—
234	C₂×C₄	C₂×#11758217	1	2	yes	—
235	C₂×C₄	C₂×#11758213	1	2	yes	—
236	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
237	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	61	1	no	—
240	C₈	—	61	1	no	—
241	C₈	—	61	2	no	—
242	C₈	—	122	2	no	—
243	C₈	C₂×#11758217	1	2	yes	—
244	C₈	C₂×#11758217	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#16840783	1	2	yes	—
249	C₂²	C₂²×#11758213	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	122	2	no	—
252	C₄	—	244	1	no	—
253	C₄	—	244	2	no	—
254	C₄	#16840783	1	2	yes	—
255	C₄	C₂×#14519856	1	1	yes	characteristic
256	C₄	C₂²×#11758217	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#16840783	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14519856
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂×#131×C₆₁	C₄
socle	C₂×C₆₁	C₂×#432
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×#131	#6923
hypocenter	C₆₁	#33428
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. #21462193	C₂	semidirect
2. #21462189	C₂	semidirect
3. C₂×#16840684	C₂	semidirect
5. #21462042	C₂	semidirect
153. C₆₁	#33428	semidirect

Maximal subgroups

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

subgroup	index
#21462193	2
#21462189	2
C₂×#16840684	2
#21462055	2
#21462042	2
#131×Dic₆₁	2
#33428	61