Subgroups — #29832580

Order 15616.

830 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840164	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840161	C₂	1	2	yes	maximal
3	#21454179	C₂	1	2	yes	maximal
4	#21445982	C₂	1	2	yes	maximal
5	#21445980	C₂	1	2	yes	maximal
6	#21445977	C₂	1	2	yes	maximal
7	#21445974	C₂	1	2	yes	maximal
8	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₆₁×#499	C₂	1	1	yes	maximal, characteristic
10	C₂×#14519769	C₂²	1	2	yes	—
11	#16840164	—	2	4	no	—
12	#16840161	—	2	8	no	—
13	C₂×#14519711	C₂²	1	2	yes	—
14	C₂×#14519710	C₂²	1	4	yes	—
15	C₂×#14519709	C₂²	1	2	yes	—
16	C₂×#14519708	C₂²	1	4	yes	—
17	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	#16839657	—	2	8	no	—
23	#16839631	C₂²	1	4	yes	—
24	#16839627	C₂²	1	4	yes	—
25	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
26	C₂×#125×C₆₁	C₂²	1	2	yes	—
27	C₆₁×#499	—	2	4	no	—
28	C₂×#107×C₆₁	C₂²	1	2	yes	—
29	C₂×#106×C₆₁	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₂²×#92418	C₂³	1	1	yes	characteristic
32	C₂²×#92418	C₂³	1	1	yes	characteristic
33	#14519769	—	2	4	no	—
34	#14519769	—	2	4	no	—
35	C₂×C₄⋊Dic₆₁	C₂³	1	4	yes	—
36	C₂×#11758196	—	2	4	no	—
37	C₂×#11758196	—	2	8	no	—
38	C₂×C₄×Dic₆₁	D₄	1	4	yes	—
39	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
41	#14519711	—	2	4	no	—
42	#14519710	—	2	8	no	—
43	#14519709	—	2	4	no	—
44	#14519708	—	2	8	no	—
45	C₄×#92418	D₄	1	4	yes	—
46	C₈×Dic₆₁	Q₈	1	4	yes	—
47	C₈×Dic₆₁	Q₈	1	4	yes	—
48	C₈×Dic₆₁	Q₈	1	4	yes	—
49	C₈×Dic₆₁	Q₈	1	4	yes	—
50	#14519684	—	2	8	no	—
51	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	#125×C₆₁	—	2	4	no	—
54	#125×C₆₁	—	2	4	no	—
55	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
56	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
57	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
58	#107×C₆₁	—	2	4	no	—
59	#106×C₆₁	—	2	4	no	—
60	C₄×C₈×C₆₁	D₄	1	4	yes	—
61	C₂²×C₄×C₆₁	—	2	4	no	—
62	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₆₁	—	2	4	no	—
65	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
66	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
73	C₄⋊C₄×C₆₁	—	2	4	no	—
74	C₄⋊C₄×C₆₁	—	2	4	no	—
75	C₄⋊C₄×C₆₁	—	4	8	no	—
76	C₄²×C₆₁	C₂×D₄	1	2	yes	—
77	C₄²×C₆₁	C₂×D₄	1	2	yes	—
78	C₄⋊Dic₆₁	—	2	4	no	—
79	C₄⋊Dic₆₁	—	2	4	no	—
80	#11758196	—	4	8	no	—
81	#11758196	—	4	16	no	—
82	C₄×Dic₆₁	—	2	8	no	—
83	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
84	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
85	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
86	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
87	C₂×#92418	C₂×D₄	1	2	yes	—
88	C₂×#92418	C₂×D₄	1	2	yes	—
89	C₂×#92418	C₂×Q₈	1	2	yes	—
90	C₂×#92418	C₂×Q₈	1	2	yes	—
91	C₂×#92418	C₂×Q₈	1	2	yes	—
92	C₂×#92418	C₂×Q₈	1	2	yes	—
93	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	—	2	4	no	—
95	C₂×C₄×C₆₁	—	4	4	no	—
96	C₂×C₄×C₆₁	—	4	8	no	—
97	C₂×C₄×C₆₁	#127	1	4	yes	—
98	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
104	C₂×Dic₆₁	—	2	4	no	—
105	C₂×Dic₆₁	—	4	4	no	—
106	C₂×Dic₆₁	—	4	8	no	—
107	C₂×Dic₆₁	C₄⋊Q₈	1	4	yes	—
108	C₂×Dic₆₁	C₄⋊Q₈	1	4	yes	—
109	C₂×Dic₆₁	#135	1	8	yes	—
110	C₈×C₆₁	C₄⋊Q₈	1	4	yes	—
111	C₈×C₆₁	C₄⋊Q₈	1	4	yes	—
112	#92418	C₄⋊Q₈	1	4	yes	—
113	#92418	C₄⋊Q₈	1	4	yes	—
114	#33311	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
116	C₂²×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₂²×C₆₁	C₂×#135	1	4	yes	—
119	C₄×C₆₁	—	2	4	no	—
120	C₄×C₆₁	—	8	4	no	—
121	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	Dic₆₁	—	2	4	no	—
126	Dic₆₁	—	8	4	no	—
127	Dic₆₁	#499	1	8	yes	complemented
128	C₂×#499	—	61	1	no	—
129	C₂×#499	—	61	2	no	—
130	C₂×#499	D₆₁	1	1	yes	characteristic
131	#2859	—	61	2	no	—
132	#1846	—	61	2	no	—
133	#1846	—	61	2	no	—
134	#1845	—	61	2	no	—
135	#1844	—	61	2	no	—
136	C₄²×C₈	—	61	1	no	—
137	C₂×C₆₁	#2868	1	1	yes	characteristic
138	C₂×C₆₁	#3147	1	2	yes	—
139	C₂×C₆₁	C₂×#499	1	2	yes	—
140	C₂×C₆₁	#3160	1	2	yes	—
141	C₂×#125	—	61	2	no	—
142	C₂×#125	C₂×D₆₁	1	2	yes	—
143	#499	—	2	4	no	—
144	#499	—	122	8	no	—
145	#499	—	122	4	no	—
146	C₂×#107	—	61	2	no	—
147	C₂×#107	—	61	4	no	—
148	C₂×#107	C₂×D₆₁	1	2	yes	—
149	C₂×#106	—	61	2	no	—
150	C₂×#106	—	61	4	no	—
151	C₂×#106	C₂×D₆₁	1	2	yes	—
152	C₂×C₄×C₈	—	61	1	no	—
153	C₂×C₄×C₈	—	61	1	no	—
154	C₂×C₄×C₈	—	61	1	no	—
155	C₂×C₄×C₈	—	61	1	no	—
156	C₂×C₄×C₈	—	61	1	no	—
157	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
158	#389	—	122	8	no	—
159	#384	—	61	4	no	—
160	#382	—	61	4	no	—
161	C₄³	—	61	1	no	—
162	C₆₁	#33311	1	1	yes	characteristic, complemented, Sylow
163	C₂²×C₈	—	61	1	no	—
164	C₂²×C₈	—	61	1	no	—
165	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
167	#125	—	2	4	no	—
168	#125	—	2	4	no	—
169	#125	—	122	4	no	—
170	#125	—	122	4	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	61	4	no	—
173	C₂×C₄⋊C₄	—	122	8	no	—
174	C₂×C₄⋊C₄	—	122	4	no	—
175	C₂×C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
176	C₂×C₄²	—	61	1	no	—
177	C₂×C₄²	—	61	1	no	—
178	C₂×C₄²	—	61	4	no	—
179	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
180	#107	—	2	4	no	—
181	#107	—	122	8	no	—
182	#107	—	122	4	no	—
183	#106	—	2	4	no	—
184	#106	—	122	8	no	—
185	#106	—	122	4	no	—
186	C₄×C₈	—	61	4	no	—
187	C₄×C₈	—	61	4	no	—
188	C₄×C₈	—	61	4	no	—
189	C₄×C₈	—	61	4	no	—
190	C₄×C₈	—	61	4	no	—
191	C₄×C₈	C₂²⋊D₆₁	1	4	yes	—
192	#95	—	122	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	61	2	no	—
195	C₂²×C₄	—	61	2	no	—
196	C₂²×C₄	—	122	4	no	—
197	C₂²×C₄	D₄×D₆₁	1	2	yes	—
198	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
199	C₂×C₈	—	61	2	no	—
200	C₂×C₈	—	61	2	no	—
201	C₂×C₈	—	61	2	no	—
202	C₂×C₈	—	61	2	no	—
203	C₂×C₈	—	61	2	no	—
204	C₂×C₈	—	61	2	no	—
205	C₂×C₈	Q₈×D₆₁	1	2	yes	—
206	C₂×C₈	Q₈×D₆₁	1	2	yes	—
207	C₂×C₈	Q₈×D₆₁	1	2	yes	—
208	C₂×C₈	Q₈×D₆₁	1	2	yes	—
209	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
210	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	8	no	—
214	C₄⋊C₄	—	122	4	no	—
215	C₄⋊C₄	—	122	4	no	—
216	C₄⋊C₄	—	244	16	no	—
217	C₄⋊C₄	—	244	8	no	—
218	C₄²	—	61	2	no	—
219	C₄²	—	61	2	no	—
220	C₄²	—	61	2	no	—
221	C₄²	—	61	2	no	—
222	C₄²	—	122	8	no	—
223	C₄²	D₄×D₆₁	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₆₁	1	2	yes	—
225	C₂³	C₂×D₄×D₆₁	1	1	yes	characteristic
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	61	8	no	—
230	C₂×C₄	—	61	4	no	—
231	C₂×C₄	—	61	4	no	—
232	C₂×C₄	—	122	4	no	—
233	C₂×C₄	—	244	8	no	—
234	C₂×C₄	—	244	4	no	—
235	C₂×C₄	#14519773	1	4	yes	—
236	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
238	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
239	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
240	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
241	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
242	C₈	—	61	4	no	—
243	C₈	—	61	4	no	—
244	C₈	#14519775	1	4	yes	—
245	C₈	#14519775	1	4	yes	—
246	C₂²	C₂×#14519773	1	1	yes	characteristic
247	C₂²	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
248	C₂²	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
249	C₂²	#135×D₆₁	1	4	yes	—
250	C₄	—	2	4	no	—
251	C₄	—	8	4	no	—
252	C₄	—	61	8	no	—
253	C₄	—	122	4	no	—
254	C₄	—	488	4	no	—
255	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
256	C₄	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
257	C₄	C₂×#14519775	1	1	yes	characteristic
258	C₄	C₂×#14519775	1	1	yes	characteristic
259	C₂	#21454215	1	1	yes	characteristic
260	C₂	#21456243	1	2	yes	—
261	C₂	#499×D₆₁	1	2	yes	—
262	C₂	#21456332	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₁
commutator, cosocle	C₂²×C₄×C₆₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₆₁×#499	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
hypercenter	C₂×#499	D₆₁
hypocenter	C₆₁	#33311
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
127. Dic₆₁	#499	semidirect
162. C₆₁	#33311	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840164	2
C₂×#16840161	2
#21454179	2
#21445982	2
#21445980	2
#21445977	2
#21445974	2
C₄×C₈×Dic₆₁	2
C₂×C₆₁×#499	2
#33311	61