Subgroups — #30598061

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461615	C₂	1	2	yes	maximal
2	C₂×#16840623	C₂	1	1	yes	maximal, characteristic
3	C₂×#16840603	C₂	1	1	yes	maximal, characteristic
4	#21460655	C₂	1	2	yes	maximal
5	#107×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	C₂²	1	2	yes	—
7	C₂×#14519832	—	2	2	no	—
8	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92420	C₂²	1	1	yes	characteristic
10	#16840628	—	2	2	no	—
11	#16840623	C₄	1	4	yes	complemented
12	#16840603	C₄	1	4	yes	complemented
13	C₂×#14519711	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₆₁	C₂²	1	2	yes	—
16	#16839524	C₄	1	2	yes	—
17	C₂×#107×C₆₁	C₄	1	1	yes	characteristic
18	C₂²×#92420	—	2	2	no	—
19	C₂²×#92420	—	2	1	no	—
20	C₂²×#92420	D₄	1	2	yes	—
21	C₄⋊#92420	—	2	2	no	—
22	C₄⋊#92420	C₂×C₄	1	4	yes	—
23	#14519832	—	4	4	no	—
24	C₄×#92420	—	2	1	no	—
25	C₄×#92420	C₂×C₄	1	2	yes	—
26	C₄×#92420	C₂×C₄	1	2	yes	—
27	C₄×#92420	Q₈	1	2	yes	—
28	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
32	#14519711	—	2	2	no	—
33	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
34	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
35	#14519684	—	2	2	no	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
38	#107×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	—	2	2	no	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	—	2	2	no	—
42	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	#41	1	1	yes	characteristic
44	C₂×#92420	—	2	2	no	—
45	C₂×#92420	—	2	2	no	—
46	C₂×#92420	—	2	2	no	—
47	C₂×#92420	—	2	4	no	—
48	C₂×#92420	—	4	1	no	—
49	C₂×#92420	—	4	1	no	—
50	C₂×#92420	—	4	2	no	—
51	C₂×#92420	—	4	2	no	—
52	C₂×#92420	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₆₁	C₄²	1	2	yes	—
54	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
55	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
56	C₄⋊C₄×C₆₁	—	2	2	no	—
57	C₄⋊C₄×C₆₁	—	2	2	no	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄⋊Dic₆₁	—	2	2	no	—
60	C₄×Dic₆₁	—	4	2	no	—
61	C₄×Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₆₁	#41	1	1	yes	characteristic
66	C₂×#92418	—	2	1	no	—
67	C₂×#92418	—	2	1	no	—
68	C₂×#92418	C₄²	1	2	yes	—
69	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
79	C₂×Dic₆₁	—	2	1	no	—
80	C₂×Dic₆₁	—	2	1	no	—
81	C₂×Dic₆₁	—	4	2	no	—
82	C₂×Dic₆₁	—	4	2	no	—
83	C₂×Dic₆₁	—	4	2	no	—
84	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×Dic₆₁	#117	1	1	yes	characteristic
86	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
87	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
88	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×Dic₆₁	#137	1	2	yes	—
90	#92420	—	2	4	no	—
91	#92420	—	4	2	no	—
92	#92420	—	4	4	no	—
93	C₈×C₆₁	#97	1	4	yes	complemented
94	#92418	—	2	2	no	—
95	#8287	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₆₁	#436	1	1	yes	characteristic
98	C₂²×C₆₁	#436	1	1	yes	characteristic
99	C₂²×C₆₁	#441	1	1	yes	characteristic
100	C₂²×C₆₁	#442	1	1	yes	characteristic
101	C₂²×C₆₁	#444	1	1	yes	characteristic
102	C₂²×C₆₁	#444	1	1	yes	characteristic
103	C₄×C₆₁	—	8	2	no	—
104	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
105	C₄×C₆₁	#405	1	1	yes	characteristic
106	C₄×C₆₁	#446	1	1	yes	characteristic
107	C₄×C₆₁	#446	1	1	yes	characteristic
108	Dic₆₁	—	2	1	no	—
109	Dic₆₁	—	2	1	no	—
110	Dic₆₁	—	8	2	no	—
111	Dic₆₁	#428	1	2	yes	—
112	Dic₆₁	#440	1	2	yes	—
113	C₄×#107	—	61	1	no	—
114	#1770	—	61	2	no	—
115	C₂×#334	—	61	1	no	—
116	C₂×#322	—	61	1	no	—
117	#1279	—	61	2	no	—
118	C₂×C₆₁	#1593	1	1	yes	characteristic
119	C₂×C₆₁	#1593	1	1	yes	characteristic
120	C₂×C₆₁	#1594	1	1	yes	characteristic
121	C₂×C₆₁	#1594	1	1	yes	characteristic
122	C₂×C₆₁	#1770	1	1	yes	characteristic
123	C₂×C₆₁	#1773	1	1	yes	characteristic
124	C₂×C₆₁	#1776	1	1	yes	characteristic
125	C₂×#107	—	61	1	no	—
126	C₂×#107	#6923	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	61	2	no	—
128	C₂×#97	—	122	2	no	—
129	C₂×C₄×C₈	—	61	1	no	—
130	C₂×C₄×C₈	—	61	1	no	—
131	C₂×C₄×C₈	—	61	1	no	—
132	C₄×C₄⋊C₄	—	61	2	no	—
133	#340	—	61	2	no	—
134	#336	—	122	2	no	—
135	#334	—	61	4	no	—
136	#322	—	61	4	no	—
137	C₆₁	#8287	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	61	2	no	—
139	C₂²×C₈	—	61	1	no	—
140	C₂²×C₈	—	122	2	no	—
141	C₂²×C₈	—	122	1	no	—
142	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	61	2	no	—
144	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
145	C₂×C₄²	—	61	1	no	—
146	C₂×C₄²	—	122	2	no	—
147	#107	—	2	2	no	—
148	#107	—	122	2	no	—
149	C₄⋊C₈	—	61	4	no	—
150	C₄⋊C₈	—	122	2	no	—
151	#97	—	244	4	no	—
152	C₄×C₈	—	61	2	no	—
153	C₄×C₈	—	61	2	no	—
154	C₄×C₈	—	61	2	no	—
155	C₄×C₈	—	61	2	no	—
156	C₄×C₈	—	61	2	no	—
157	C₄×C₈	—	122	1	no	—
158	#95	—	122	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	61	1	no	—
161	C₂²×C₄	—	61	1	no	—
162	C₂²×C₄	—	122	2	no	—
163	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
164	C₂×C₈	—	61	4	no	—
165	C₂×C₈	—	61	2	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	122	2	no	—
168	C₂×C₈	—	122	1	no	—
169	C₂×C₈	—	122	1	no	—
170	C₂×C₈	—	122	2	no	—
171	C₂×C₈	—	122	4	no	—
172	C₂×C₈	—	244	2	no	—
173	C₂×C₈	—	244	2	no	—
174	C₂×C₈	—	244	1	no	—
175	C₂×C₈	—	244	1	no	—
176	C₂×C₈	#11758213	1	2	yes	—
177	C₂×C₈	C₄×#6923	1	2	yes	—
178	C₂×C₈	#11758217	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	122	2	no	—
182	C₄⋊C₄	—	122	2	no	—
183	C₄²	—	61	1	no	—
184	C₄²	—	61	1	no	—
185	C₄²	—	61	1	no	—
186	C₄²	—	61	1	no	—
187	C₄²	—	244	2	no	—
188	C₄²	—	244	2	no	—
189	C₂³	D₄×#6923	1	1	yes	characteristic
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	61	2	no	—
194	C₂×C₄	—	61	2	no	—
195	C₂×C₄	—	61	1	no	—
196	C₂×C₄	—	61	1	no	—
197	C₂×C₄	—	61	1	no	—
198	C₂×C₄	—	61	1	no	—
199	C₂×C₄	—	122	1	no	—
200	C₂×C₄	—	122	1	no	—
201	C₂×C₄	—	244	2	no	—
202	C₂×C₄	—	244	2	no	—
203	C₂×C₄	—	244	2	no	—
204	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
205	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
206	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
207	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
208	C₂×C₄	#14519864	1	1	yes	characteristic
209	C₂×C₄	#14519865	1	1	yes	characteristic
210	C₈	—	122	4	no	—
211	C₈	—	122	2	no	—
212	C₈	—	244	4	no	—
213	C₈	—	244	2	no	—
214	C₈	#14519832	1	4	yes	complemented
215	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
216	C₂²	#16840796	1	1	yes	characteristic
217	C₂²	#16840797	1	1	yes	characteristic
218	C₂²	#16840802	1	1	yes	characteristic
219	C₂²	#16840803	1	1	yes	characteristic
220	C₂²	#16840808	1	1	yes	characteristic
221	C₂²	#16840809	1	1	yes	characteristic
222	C₄	—	8	2	no	—
223	C₄	—	61	2	no	—
224	C₄	—	61	2	no	—
225	C₄	—	122	1	no	—
226	C₄	—	122	1	no	—
227	C₄	—	488	2	no	—
228	C₄	C₂×#14519832	1	1	yes	characteristic
229	C₄	#16840727	1	1	yes	characteristic
230	C₄	#16840812	1	1	yes	characteristic
231	C₄	#16840813	1	1	yes	characteristic
232	C₂	#21461465	1	1	yes	characteristic
233	C₂	#21461466	1	1	yes	characteristic
234	C₂	#21461468	1	1	yes	characteristic
235	C₂	#21461469	1	1	yes	characteristic
236	C₂	#21461615	1	1	yes	characteristic
237	C₂	#21461626	1	1	yes	characteristic
238	C₂	#21461634	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#107×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×#107	#6923
hypocenter	C₆₁	#8287
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
11. #16840623	C₄	semidirect
12. #16840603	C₄	semidirect
93. C₈×C₆₁	#97	semidirect
137. C₆₁	#8287	semidirect
214. C₈	#14519832	semidirect

Maximal subgroups

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

subgroup	index
#21461615	2
C₂×#16840623	2
C₂×#16840603	2
#21460655	2
#107×Dic₆₁	2
#8287	61