Subgroups — #30601501

Order 15616.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461916	C₂	1	2	yes	maximal
2	C₂×#16840623	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#92420	C₂	1	1	yes	maximal, characteristic
4	#21460655	C₂	1	2	yes	maximal
5	#106×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92420	—	2	2	no	—
7	C₂×C₄⋊#92420	C₂²	1	2	yes	—
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	—	2	2	no	—
12	C₈×#92420	—	2	2	no	—
13	C₂×#14519709	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×#106×C₆₁	C₄	1	1	yes	characteristic
18	C₂²×#92420	—	2	1	no	—
19	C₂²×#92420	—	2	2	no	—
20	C₂²×#92420	D₄	1	2	yes	—
21	C₄⋊#92420	—	2	2	no	—
22	C₄⋊#92420	—	2	2	no	—
23	C₄⋊#92420	—	4	4	no	—
24	C₄×#92420	—	2	1	no	—
25	C₄×#92420	—	2	1	no	—
26	C₄×#92420	—	2	1	no	—
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	#14519709	—	2	2	no	—
33	C₈×Dic₆₁	D₄	1	2	yes	—
34	C₈×Dic₆₁	Q₈	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	#106×C₆₁	C₈	1	4	yes	complemented
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	4	no	—
45	C₂×#92420	—	2	2	no	—
46	C₂×#92420	—	2	2	no	—
47	C₂×#92420	—	2	2	no	—
48	C₂×#92420	—	4	2	no	—
49	C₂×#92420	—	4	2	no	—
50	C₂×#92420	—	4	2	no	—
51	C₂×#92420	SD₁₆	1	4	yes	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
54	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
55	C₄⋊C₄×C₆₁	—	2	2	no	—
56	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
57	C₄⋊Dic₆₁	—	2	2	no	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄×Dic₆₁	—	4	2	no	—
60	C₄×Dic₆₁	—	4	2	no	—
61	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
62	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
63	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
64	C₄×Dic₆₁	#41	1	1	yes	characteristic
65	C₂×#92418	—	2	1	no	—
66	C₂×#92418	—	2	1	no	—
67	C₂×#92418	C₄⋊C₄	1	2	yes	—
68	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	—	2	4	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
78	C₂×Dic₆₁	—	2	2	no	—
79	C₂×Dic₆₁	—	4	2	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₂×Dic₆₁	—	4	2	no	—
82	C₂×Dic₆₁	#121	1	1	yes	characteristic
83	C₂×Dic₆₁	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×Dic₆₁	#125	1	1	yes	characteristic
85	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
87	C₂×Dic₆₁	#135	1	2	yes	—
88	#92420	—	2	4	no	—
89	#92420	—	2	4	no	—
90	#92420	—	8	2	no	—
91	C₈×C₆₁	C₄⋊C₈	1	4	yes	complemented
92	#92418	—	2	2	no	—
93	#9593	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	#389	1	1	yes	characteristic
95	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
96	C₂²×C₆₁	#436	1	1	yes	characteristic
97	C₂²×C₆₁	C₄×SD₁₆	1	2	yes	—
98	C₂²×C₆₁	#443	1	2	yes	—
99	C₄×C₆₁	—	4	4	no	—
100	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₆₁	#424	1	1	yes	characteristic
102	C₄×C₆₁	C₈×Q₈	1	1	yes	characteristic
103	C₄×C₆₁	#446	1	1	yes	characteristic
104	Dic₆₁	—	2	1	no	—
105	Dic₆₁	—	2	1	no	—
106	Dic₆₁	—	8	2	no	—
107	Dic₆₁	#465	1	2	yes	—
108	Dic₆₁	#498	1	1	yes	characteristic
109	Dic₆₁	#499	1	1	yes	characteristic
110	#1939	—	61	2	no	—
111	C₄×#106	—	61	1	no	—
112	C₂×#334	—	61	1	no	—
113	C₂×C₈²	—	61	1	no	—
114	#1279	—	61	2	no	—
115	C₂×C₆₁	C₈×SD₁₆	1	2	yes	—
116	C₂×C₆₁	#1590	1	2	yes	—
117	C₂×C₆₁	#1939	1	1	yes	characteristic
118	C₂×C₆₁	#1941	1	1	yes	characteristic
119	C₂×C₆₁	#1945	1	1	yes	characteristic
120	C₂×#106	—	61	1	no	—
121	C₂×#106	#6923	1	1	yes	characteristic
122	C₂×C₄⋊C₈	—	61	2	no	—
123	C₂×C₄⋊C₈	—	122	2	no	—
124	C₂×C₄×C₈	—	61	1	no	—
125	C₂×C₄×C₈	—	61	1	no	—
126	C₂×C₄×C₈	—	61	1	no	—
127	C₄×C₄⋊C₄	—	61	2	no	—
128	#340	—	61	2	no	—
129	#336	—	122	2	no	—
130	#334	—	122	2	no	—
131	C₈²	—	122	2	no	—
132	C₆₁	#9593	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	61	2	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	122	1	no	—
136	C₂²×C₈	—	122	2	no	—
137	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	61	2	no	—
139	C₂×C₄⋊C₄	C₂×#6923	1	2	yes	—
140	C₂×C₄²	—	61	1	no	—
141	C₂×C₄²	—	122	2	no	—
142	#106	—	122	2	no	—
143	#106	#92420	1	4	yes	complemented
144	C₄⋊C₈	—	122	2	no	—
145	C₄⋊C₈	—	122	2	no	—
146	C₄⋊C₈	—	244	4	no	—
147	C₄×C₈	—	61	2	no	—
148	C₄×C₈	—	61	2	no	—
149	C₄×C₈	—	61	2	no	—
150	C₄×C₈	—	122	1	no	—
151	C₄×C₈	—	122	1	no	—
152	C₄×C₈	—	122	1	no	—
153	#95	—	122	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	61	1	no	—
156	C₂²×C₄	—	61	1	no	—
157	C₂²×C₄	—	122	2	no	—
158	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
159	C₂×C₈	—	61	4	no	—
160	C₂×C₈	—	61	2	no	—
161	C₂×C₈	—	122	2	no	—
162	C₂×C₈	—	122	2	no	—
163	C₂×C₈	—	122	2	no	—
164	C₂×C₈	—	122	4	no	—
165	C₂×C₈	—	122	1	no	—
166	C₂×C₈	—	122	1	no	—
167	C₂×C₈	—	244	2	no	—
168	C₂×C₈	—	244	2	no	—
169	C₂×C₈	—	244	2	no	—
170	C₂×C₈	#11758213	1	2	yes	—
171	C₂×C₈	C₄⋊#6923	1	2	yes	—
172	C₂×C₈	C₂×#92420	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	122	2	no	—
175	C₄⋊C₄	—	122	2	no	—
176	C₄⋊C₄	C₂×#92420	1	4	yes	—
177	C₄²	—	61	1	no	—
178	C₄²	—	61	1	no	—
179	C₄²	—	61	1	no	—
180	C₄²	—	61	1	no	—
181	C₄²	—	244	2	no	—
182	C₄²	—	244	2	no	—
183	C₂³	D₄×#6923	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	61	1	no	—
188	C₂×C₄	—	61	1	no	—
189	C₂×C₄	—	61	2	no	—
190	C₂×C₄	—	61	2	no	—
191	C₂×C₄	—	61	1	no	—
192	C₂×C₄	—	61	1	no	—
193	C₂×C₄	—	122	2	no	—
194	C₂×C₄	—	244	2	no	—
195	C₂×C₄	—	244	2	no	—
196	C₂×C₄	—	244	2	no	—
197	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
198	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
199	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
200	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
201	C₂×C₄	#14519864	1	1	yes	characteristic
202	C₂×C₄	#14519865	1	1	yes	characteristic
203	C₈	—	122	4	no	—
204	C₈	—	122	4	no	—
205	C₈	—	122	2	no	—
206	C₈	—	488	2	no	—
207	C₈	C₄⋊#92420	1	4	yes	complemented
208	C₂²	#16840713	1	1	yes	characteristic
209	C₂²	D₄×#92420	1	1	yes	characteristic
210	C₂²	#16840796	1	1	yes	characteristic
211	C₂²	SD₁₆×#6923	1	2	yes	—
212	C₂²	#16840805	1	2	yes	—
213	C₄	—	4	4	no	—
214	C₄	—	61	2	no	—
215	C₄	—	61	1	no	—
216	C₄	—	61	1	no	—
217	C₄	—	122	1	no	—
218	C₄	—	122	1	no	—
219	C₄	—	488	2	no	—
220	C₄	C₂×C₄⋊#92420	1	1	yes	characteristic
221	C₄	#16840769	1	1	yes	characteristic
222	C₄	Q₈×#92420	1	1	yes	characteristic
223	C₄	#16840812	1	1	yes	characteristic
224	C₂	SD₁₆×#92420	1	2	yes	—
225	C₂	#21461454	1	2	yes	—
226	C₂	#21461916	1	1	yes	characteristic
227	C₂	#21461922	1	1	yes	characteristic
228	C₂	#21461926	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#106×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×#106	#6923
hypocenter	C₆₁	#9593
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
38. #106×C₆₁	C₈	semidirect
91. C₈×C₆₁	C₄⋊C₈	semidirect
132. C₆₁	#9593	semidirect
143. #106	#92420	semidirect
207. C₈	C₄⋊#92420	semidirect

Maximal subgroups

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

subgroup	index
#21461916	2
C₂×#16840623	2
C₂×C₈×#92420	2
#21460655	2
#106×Dic₆₁	2
#9593	61