Subgroups — #30599456

Order 15616.

378 conjugacy classes of subgroups.

The subgroup lattice has 244 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	#21461736	C₂	1	1	yes	maximal, characteristic
3	#21460649	C₂	1	2	yes	maximal
4	#21460649	C₂	1	2	yes	maximal
5	#97×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519844	—	2	2	no	—
7	C₂×C₄⋊#92420	C₂²	1	2	yes	—
8	C₂×C₄⋊#92420	C₂²	1	2	yes	—
9	#16840628	—	2	2	no	—
10	C₂×#14519688	C₄	1	1	yes	characteristic
11	C₂×#14519687	C₄	1	1	yes	characteristic
12	C₂×#14519687	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
16	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
17	C₂²×#92420	—	2	2	no	—
18	C₂²×#92420	—	2	2	no	—
19	#14519844	—	4	4	no	—
20	C₄⋊#92420	—	2	2	no	—
21	C₄⋊#92420	—	2	2	no	—
22	C₄⋊#92420	—	2	2	no	—
23	C₄⋊#92420	—	2	2	no	—
24	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
25	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
28	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
29	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
30	#14519688	—	2	2	no	—
31	#14519687	—	2	2	no	—
32	#14519687	C₈	1	4	yes	complemented
33	C₈×Dic₆₁	—	2	1	no	—
34	C₈×Dic₆₁	—	2	1	no	—
35	C₈×Dic₆₁	—	2	1	no	—
36	C₈×Dic₆₁	—	2	1	no	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
40	#97×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
44	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×#92420	—	4	2	no	—
47	C₂×#92420	—	4	2	no	—
48	C₂×#92420	—	4	4	no	—
49	C₂×#92420	—	4	2	no	—
50	C₂×#92420	—	4	2	no	—
51	C₂×C₈×C₆₁	—	2	1	no	—
52	C₂×C₈×C₆₁	—	2	1	no	—
53	C₂×C₈×C₆₁	—	2	1	no	—
54	C₂×C₈×C₆₁	—	2	1	no	—
55	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
57	C₄²×C₆₁	—	2	1	no	—
58	C₄²×C₆₁	—	2	1	no	—
59	C₄×Dic₆₁	—	2	2	no	—
60	C₄×Dic₆₁	—	2	2	no	—
61	C₄×Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
63	C₄×Dic₆₁	M₁₆	1	2	yes	—
64	C₄×Dic₆₁	#41	1	1	yes	characteristic
65	C₄×Dic₆₁	#41	1	1	yes	characteristic
66	C₄×Dic₆₁	#41	1	1	yes	characteristic
67	C₄×Dic₆₁	#41	1	1	yes	characteristic
68	C₂×#92418	—	2	1	no	—
69	C₂×#92418	—	2	1	no	—
70	C₂×#92418	—	2	1	no	—
71	C₂×#92418	—	2	1	no	—
72	C₂×#92418	C₂×C₈	1	2	yes	—
73	C₂×#92418	M₁₆	1	2	yes	—
74	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	—	2	2	no	—
77	C₂×C₄×C₆₁	—	2	2	no	—
78	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
84	C₂×Dic₆₁	—	2	2	no	—
85	C₂×Dic₆₁	—	2	2	no	—
86	C₂×Dic₆₁	—	2	2	no	—
87	C₂×Dic₆₁	C₄⋊C₈	1	4	yes	—
88	C₂×Dic₆₁	#124	1	1	yes	characteristic
89	C₂×Dic₆₁	#124	1	1	yes	characteristic
90	C₂×Dic₆₁	#125	1	1	yes	characteristic
91	C₂×Dic₆₁	#125	1	1	yes	characteristic
92	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×Dic₆₁	#137	1	2	yes	—
94	#92420	—	8	2	no	—
95	#92420	—	8	2	no	—
96	C₈×C₆₁	—	2	2	no	—
97	C₈×C₆₁	—	2	2	no	—
98	#92418	—	2	2	no	—
99	#92418	—	2	2	no	—
100	#8786	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	#382	1	1	yes	characteristic
102	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₂²×C₆₁	#423	1	1	yes	characteristic
104	C₂²×C₆₁	#428	1	2	yes	—
105	C₂²×C₆₁	#430	1	2	yes	—
106	C₄×C₆₁	—	4	2	no	—
107	C₄×C₆₁	#431	1	1	yes	characteristic
108	C₄×C₆₁	#431	1	1	yes	characteristic
109	C₄×C₆₁	#432	1	1	yes	characteristic
110	C₄×C₆₁	#432	1	1	yes	characteristic
111	Dic₆₁	—	2	4	no	—
112	Dic₆₁	—	2	1	no	—
113	Dic₆₁	—	2	1	no	—
114	Dic₆₁	#489	1	2	yes	—
115	Dic₆₁	#490	1	1	yes	characteristic
116	Dic₆₁	#491	1	1	yes	characteristic
117	#1842	—	61	1	no	—
118	#1831	—	61	1	no	—
119	C₄×#97	—	61	1	no	—
120	#1277	—	61	2	no	—
121	#1277	—	61	2	no	—
122	C₂×C₆₁	#1565	1	2	yes	—
123	C₂×C₆₁	#1569	1	2	yes	—
124	C₂×C₆₁	#1831	1	1	yes	characteristic
125	C₂×C₆₁	#1836	1	1	yes	characteristic
126	C₂×C₆₁	#1838	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	61	2	no	—
128	C₂×C₄⋊C₈	—	61	2	no	—
129	C₂×C₄⋊C₈	—	122	2	no	—
130	C₂×#97	—	61	1	no	—
131	C₂×#97	—	61	1	no	—
132	C₂×#97	—	61	1	no	—
133	C₂×#97	#6923	1	1	yes	characteristic
134	C₂×C₄×C₈	—	61	1	no	—
135	C₂×C₄×C₈	—	61	1	no	—
136	C₄³	—	61	1	no	—
137	#336	—	122	2	no	—
138	C₆₁	#8786	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₈	—	61	1	no	—
140	C₂²×C₈	—	61	1	no	—
141	C₂²×C₈	—	122	2	no	—
142	C₂²×C₈	—	122	2	no	—
143	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
144	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
145	C₂×C₄²	—	61	1	no	—
146	C₂×C₄²	—	61	1	no	—
147	C₂×C₄²	—	61	2	no	—
148	C₂×C₄²	—	61	2	no	—
149	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
150	C₄⋊C₈	—	122	2	no	—
151	C₄⋊C₈	—	122	2	no	—
152	C₄⋊C₈	—	122	2	no	—
153	C₄⋊C₈	—	122	2	no	—
154	C₄⋊C₈	—	244	4	no	—
155	#97	—	2	2	no	—
156	#97	—	61	4	no	—
157	#97	—	122	2	no	—
158	#97	—	122	2	no	—
159	C₄×C₈	—	122	1	no	—
160	C₄×C₈	—	122	1	no	—
161	C₄×C₈	—	122	1	no	—
162	C₄×C₈	—	122	1	no	—
163	C₂²×C₄	—	61	1	no	—
164	C₂²×C₄	—	61	2	no	—
165	C₂²×C₄	—	61	1	no	—
166	C₂²×C₄	C₄⋊#6923	1	2	yes	—
167	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	61	2	no	—
173	C₂×C₈	—	61	2	no	—
174	C₂×C₈	—	122	1	no	—
175	C₂×C₈	—	122	1	no	—
176	C₂×C₈	—	122	1	no	—
177	C₂×C₈	—	122	1	no	—
178	C₂×C₈	—	244	2	no	—
179	C₂×C₈	—	244	2	no	—
180	C₂×C₈	—	244	4	no	—
181	C₂×C₈	—	244	2	no	—
182	C₂×C₈	—	244	2	no	—
183	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
184	C₂×C₈	#11758217	1	2	yes	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	61	1	no	—
188	C₄²	—	61	1	no	—
189	C₄²	—	61	1	no	—
190	C₄²	—	61	1	no	—
191	C₄²	—	61	2	no	—
192	C₄²	—	61	2	no	—
193	C₄²	—	122	2	no	—
194	C₄²	—	122	2	no	—
195	C₄²	—	244	2	no	—
196	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	61	1	no	—
201	C₂×C₄	—	61	1	no	—
202	C₂×C₄	—	61	2	no	—
203	C₂×C₄	—	61	2	no	—
204	C₂×C₄	—	61	1	no	—
205	C₂×C₄	—	61	1	no	—
206	C₂×C₄	—	61	4	no	—
207	C₂×C₄	—	122	2	no	—
208	C₂×C₄	—	122	2	no	—
209	C₂×C₄	—	122	2	no	—
210	C₂×C₄	#14519856	1	1	yes	characteristic
211	C₂×C₄	#14519856	1	1	yes	characteristic
212	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
213	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
214	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
215	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	2	no	—
218	C₈	—	122	2	no	—
219	C₈	—	122	2	no	—
220	C₈	—	488	2	no	—
221	C₈	—	488	2	no	—
222	C₂²	#16840706	1	1	yes	characteristic
223	C₂²	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
224	C₂²	#16840765	1	1	yes	characteristic
225	C₂²	#16840773	1	2	yes	—
226	C₂²	#16840777	1	2	yes	—
227	C₄	—	4	2	no	—
228	C₄	—	61	2	no	—
229	C₄	—	61	1	no	—
230	C₄	—	61	1	no	—
231	C₄	—	122	4	no	—
232	C₄	—	122	1	no	—
233	C₄	—	122	1	no	—
234	C₄	#16840779	1	1	yes	characteristic
235	C₄	#16840780	1	1	yes	characteristic
236	C₄	#16840782	1	1	yes	characteristic
237	C₄	#16840785	1	1	yes	characteristic
238	C₂	#21461371	1	2	yes	—
239	C₂	#21461383	1	2	yes	—
240	C₂	#21461735	1	1	yes	characteristic
241	C₂	#21461742	1	1	yes	characteristic
242	C₂	#21461745	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#97×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×#97	#6923
hypocenter	C₆₁	#8786
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
32. #14519687	C₈	semidirect
138. C₆₁	#8786	semidirect

Maximal subgroups

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

subgroup	index
#21461752	2
#21461736	2
#21460649	2
#97×Dic₆₁	2
#8786	61