Subgroups — #30599458

Order 15616.

362 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	#21461752	C₂	1	1	yes	maximal, characteristic
2	#21461751	C₂	1	1	yes	maximal, characteristic
3	#21460649	C₂	1	4	yes	maximal
4	#21445563	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519844	—	2	1	no	—
6	C₂×#14519844	—	2	1	no	—
7	C₂×C₄⋊#92420	—	2	1	no	—
8	C₂×C₄⋊#92420	—	2	1	no	—
9	C₂×C₄⋊#92420	C₂²	1	2	yes	—
10	C₂×C₄⋊#92420	C₂²	1	2	yes	—
11	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
12	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
13	#16839517	C₄	1	1	yes	characteristic
14	#16839516	C₄	1	2	yes	—
15	C₆₁×#336	C₄	1	1	yes	characteristic
16	C₂²×#92420	—	2	2	no	—
17	C₂²×#92420	—	2	2	no	—
18	#14519844	—	4	2	no	—
19	#14519844	—	4	2	no	—
20	C₄⋊#92420	—	2	4	no	—
21	C₄⋊#92420	—	2	4	no	—
22	C₄⋊#92420	—	4	2	no	—
23	C₄⋊#92420	—	4	2	no	—
24	C₂²×#92418	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
28	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
29	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
30	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
31	C₈×Dic₆₁	—	2	2	no	—
32	C₈×Dic₆₁	—	2	2	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×#92420	—	4	2	no	—
43	C₂×#92420	—	4	2	no	—
44	C₂×#92420	—	4	2	no	—
45	C₂×#92420	—	4	2	no	—
46	C₂×#92420	—	4	2	no	—
47	C₂×#92420	—	4	2	no	—
48	C₂×C₈×C₆₁	—	2	2	no	—
49	C₂×C₈×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₄²×C₆₁	—	2	2	no	—
52	C₄×Dic₆₁	—	2	1	no	—
53	C₄×Dic₆₁	—	2	1	no	—
54	C₄×Dic₆₁	—	4	1	no	—
55	C₄×Dic₆₁	—	4	1	no	—
56	C₄×Dic₆₁	D₈	1	2	yes	—
57	C₄×Dic₆₁	SD₁₆	1	2	yes	—
58	C₄×Dic₆₁	SD₁₆	1	2	yes	—
59	C₄×Dic₆₁	Q₁₆	1	2	yes	—
60	C₄×Dic₆₁	#41	1	2	yes	—
61	C₄×Dic₆₁	#41	1	2	yes	—
62	C₂×#92418	—	2	2	no	—
63	C₂×#92418	—	2	2	no	—
64	C₂×#92418	—	2	2	no	—
65	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₁	—	2	1	no	—
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	#106	1	2	yes	—
71	C₂×C₄×C₆₁	#107	1	2	yes	—
72	C₂×C₄×C₆₁	#117	1	2	yes	—
73	C₂×C₄×C₆₁	#131	1	2	yes	—
74	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
76	C₂×Dic₆₁	—	2	1	no	—
77	C₂×Dic₆₁	—	2	1	no	—
78	C₂×Dic₆₁	—	2	1	no	—
79	C₂×Dic₆₁	—	2	1	no	—
80	C₂×Dic₆₁	—	2	1	no	—
81	C₂×Dic₆₁	—	2	1	no	—
82	C₂×Dic₆₁	#106	1	2	yes	—
83	C₂×Dic₆₁	#107	1	2	yes	—
84	C₂×Dic₆₁	#124	1	1	yes	characteristic
85	C₂×Dic₆₁	#124	1	1	yes	characteristic
86	C₂×Dic₆₁	#125	1	2	yes	—
87	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
88	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
91	#92420	—	8	2	no	—
92	#92420	—	8	2	no	—
93	C₈×C₆₁	—	4	2	no	—
94	#92418	—	4	2	no	—
95	#8787	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	#382	1	1	yes	characteristic
97	C₂²×C₆₁	#424	1	1	yes	characteristic
98	C₂²×C₆₁	#424	1	1	yes	characteristic
99	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
100	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
101	C₂²×C₆₁	#430	1	1	yes	characteristic
102	C₂²×C₆₁	#430	1	1	yes	characteristic
103	C₄×C₆₁	—	2	2	no	—
104	C₄×C₆₁	—	2	2	no	—
105	C₄×C₆₁	#433	1	2	yes	—
106	C₄×C₆₁	#433	1	2	yes	—
107	Dic₆₁	—	2	2	no	—
108	Dic₆₁	—	2	2	no	—
109	Dic₆₁	—	2	2	no	—
110	Dic₆₁	#486	1	1	yes	characteristic
111	Dic₆₁	#486	1	1	yes	characteristic
112	Dic₆₁	#487	1	1	yes	characteristic
113	Dic₆₁	#487	1	1	yes	characteristic
114	#1842	—	61	1	no	—
115	#1842	—	61	1	no	—
116	#1744	—	61	1	no	—
117	#1277	—	61	4	no	—
118	C₂×C₆₁	#1570	1	1	yes	characteristic
119	C₂×C₆₁	#1570	1	1	yes	characteristic
120	C₂×C₆₁	#1571	1	1	yes	characteristic
121	C₂×C₆₁	#1571	1	1	yes	characteristic
122	C₂×C₆₁	#1832	1	1	yes	characteristic
123	C₂×C₆₁	#1834	1	1	yes	characteristic
124	C₂×C₆₁	#1838	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	61	2	no	—
126	C₂×C₄⋊C₈	—	61	2	no	—
127	C₂×C₄⋊C₈	—	122	1	no	—
128	C₂×C₄⋊C₈	—	122	1	no	—
129	C₂×C₄⋊C₈	—	122	1	no	—
130	C₂×C₄⋊C₈	—	122	1	no	—
131	C₂×C₄×C₈	—	61	2	no	—
132	C₄³	—	61	1	no	—
133	#336	—	61	2	no	—
134	#336	—	61	1	no	—
135	#336	#6923	1	1	yes	characteristic
136	C₆₁	#8787	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	61	2	no	—
138	C₂²×C₈	—	122	2	no	—
139	C₂²×C₈	—	122	2	no	—
140	C₂²×C₈	C₂×#6923	1	2	yes	—
141	C₂×C₄²	—	61	1	no	—
142	C₂×C₄²	—	61	1	no	—
143	C₂×C₄²	—	61	1	no	—
144	C₂×C₄²	—	61	1	no	—
145	C₂×C₄²	—	61	1	no	—
146	C₂×C₄²	—	61	1	no	—
147	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
148	C₄⋊C₈	—	122	4	no	—
149	C₄⋊C₈	—	122	4	no	—
150	C₄⋊C₈	—	244	2	no	—
151	C₄⋊C₈	—	244	2	no	—
152	C₄⋊C₈	—	244	2	no	—
153	C₄⋊C₈	—	244	2	no	—
154	C₄×C₈	—	122	2	no	—
155	C₄×C₈	—	122	2	no	—
156	C₂²×C₄	—	61	1	no	—
157	C₂²×C₄	—	61	1	no	—
158	C₂²×C₄	—	61	1	no	—
159	C₂²×C₄	—	61	1	no	—
160	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
162	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	122	2	no	—
168	C₂×C₈	—	122	2	no	—
169	C₂×C₈	—	244	2	no	—
170	C₂×C₈	—	244	2	no	—
171	C₂×C₈	—	244	2	no	—
172	C₂×C₈	—	244	2	no	—
173	C₂×C₈	—	244	2	no	—
174	C₂×C₈	—	244	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	61	2	no	—
177	C₄²	—	61	2	no	—
178	C₄²	—	61	2	no	—
179	C₄²	—	61	2	no	—
180	C₄²	—	61	2	no	—
181	C₄²	—	61	2	no	—
182	C₄²	—	122	1	no	—
183	C₄²	—	122	1	no	—
184	C₄²	—	244	1	no	—
185	C₄²	—	244	1	no	—
186	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	61	1	no	—
192	C₂×C₄	—	61	1	no	—
193	C₂×C₄	—	61	1	no	—
194	C₂×C₄	—	61	1	no	—
195	C₂×C₄	—	61	1	no	—
196	C₂×C₄	—	61	1	no	—
197	C₂×C₄	—	61	2	no	—
198	C₂×C₄	—	61	2	no	—
199	C₂×C₄	—	61	2	no	—
200	C₂×C₄	—	122	1	no	—
201	C₂×C₄	—	122	1	no	—
202	C₂×C₄	—	122	1	no	—
203	C₂×C₄	—	122	1	no	—
204	C₂×C₄	—	122	1	no	—
205	C₂×C₄	—	122	1	no	—
206	C₂×C₄	#14519845	1	2	yes	—
207	C₂×C₄	#14519846	1	2	yes	—
208	C₂×C₄	#14519856	1	2	yes	—
209	C₂×C₄	#14519864	1	1	yes	characteristic
210	C₂×C₄	#14519864	1	2	yes	—
211	C₂×C₄	#14519865	1	1	yes	characteristic
212	C₈	—	4	2	no	—
213	C₈	—	488	2	no	—
214	C₈	—	488	2	no	—
215	C₈	—	244	2	no	—
216	C₂²	#16840706	1	1	yes	characteristic
217	C₂²	#16840768	1	1	yes	characteristic
218	C₂²	#16840769	1	1	yes	characteristic
219	C₂²	C₂×#14519845	1	1	yes	characteristic
220	C₂²	C₂×#14519846	1	1	yes	characteristic
221	C₂²	#16840777	1	1	yes	characteristic
222	C₂²	#16840777	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	61	1	no	—
226	C₄	—	61	1	no	—
227	C₄	—	61	1	no	—
228	C₄	—	61	1	no	—
229	C₄	—	122	2	no	—
230	C₄	—	122	2	no	—
231	C₄	—	122	2	no	—
232	C₄	#16840786	1	2	yes	—
233	C₄	#16840787	1	2	yes	—
234	C₂	#21461386	1	1	yes	characteristic
235	C₂	#21461387	1	1	yes	characteristic
236	C₂	#21461388	1	1	yes	characteristic
237	C₂	#21461389	1	1	yes	characteristic
238	C₂	#21461737	1	1	yes	characteristic
239	C₂	#21461740	1	1	yes	characteristic
240	C₂	#21461745	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#336	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	#336	#6923
hypocenter	C₆₁	#8787
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

Complements

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

N	H ≅ G/N	type
136. C₆₁	#8787	semidirect

Maximal subgroups

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

subgroup	index
#21461752	2
#21461751	2
#21460649	2
#21445563	2
#8787	61