Subgroups — #30594309

Order 15616.

274 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840651	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840623	C₂	1	1	yes	maximal, characteristic
3	#107×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758187	—	2	1	no	—
5	C₂×C₄⋊#92420	—	2	1	no	—
6	#16840651	—	2	1	no	—
7	#16840651	C₄	1	2	yes	complemented
8	#16840623	—	2	1	no	—
9	#16840623	C₄	1	2	yes	complemented
10	C₂×#14519711	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×Dic₆₁	—	2	1	no	—
13	#16839524	—	2	1	no	—
14	C₂×#107×C₆₁	C₄	1	1	yes	characteristic
15	C₂²×#92420	—	4	1	no	—
16	C₂×#11758187	—	2	2	no	—
17	C₂×#11758187	—	2	2	no	—
18	C₂×#11758187	—	2	2	no	—
19	C₄⋊#92420	—	2	2	no	—
20	C₄⋊#92420	—	4	1	no	—
21	C₂²×#92418	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₆₁	—	2	1	no	—
23	C₂×C₄×Dic₆₁	—	4	1	no	—
24	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
25	#14519711	—	2	1	no	—
26	#14519711	C₈	1	2	yes	complemented
27	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₆₁	D₄	1	1	yes	characteristic
30	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
31	#14519684	—	4	1	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
34	#107×C₆₁	—	2	1	no	—
35	#107×C₆₁	C₈	1	2	yes	complemented
36	C₂²×C₄×C₆₁	—	4	1	no	—
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×Dic₆₁	—	4	1	no	—
39	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
40	C₂²×Dic₆₁	SD₁₆	1	1	yes	characteristic
41	C₂×#92420	—	4	2	no	—
42	C₂×#92420	—	8	1	no	—
43	C₂×#92420	—	8	1	no	—
44	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
50	C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₄⋊C₄×C₆₁	—	4	1	no	—
52	C₄⋊Dic₆₁	—	2	2	no	—
53	C₄⋊Dic₆₁	—	4	1	no	—
54	C₄×Dic₆₁	—	8	2	no	—
55	C₄×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₄×Dic₆₁	SD₁₆	1	1	yes	characteristic
58	C₄×Dic₆₁	Q₁₆	1	1	yes	characteristic
59	C₂×#92418	—	2	1	no	—
60	C₂×#92418	—	2	1	no	—
61	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
63	#11758187	—	2	4	no	—
64	C₂³×C₆₁	#102	1	1	yes	characteristic
65	C₂×C₄×C₆₁	—	4	2	no	—
66	C₂×C₄×C₆₁	—	8	1	no	—
67	C₂×C₄×C₆₁	—	8	1	no	—
68	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#104	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#104	1	1	yes	characteristic
74	C₂×Dic₆₁	—	2	1	no	—
75	C₂×Dic₆₁	—	2	1	no	—
76	C₂×Dic₆₁	—	2	1	no	—
77	C₂×Dic₆₁	—	2	1	no	—
78	C₂×Dic₆₁	—	4	2	no	—
79	C₂×Dic₆₁	—	8	1	no	—
80	C₂×Dic₆₁	—	8	1	no	—
81	C₂×Dic₆₁	#102	1	1	yes	characteristic
82	C₂×Dic₆₁	#103	1	1	yes	characteristic
83	C₂×Dic₆₁	#106	1	1	yes	characteristic
84	C₂×Dic₆₁	#107	1	1	yes	characteristic
85	#92420	—	8	2	no	—
86	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
87	C₈×C₆₁	#97	1	1	yes	characteristic
88	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
89	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
90	#92418	—	2	1	no	—
91	#92418	—	2	1	no	—
92	#7178	—	61	1	no	maximal, Sylow
93	C₂²×C₆₁	D₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₆₁	D₄⋊C₈	1	1	yes	characteristic
95	C₂²×C₆₁	#340	1	1	yes	characteristic
96	C₂²×C₆₁	#360	1	1	yes	characteristic
97	C₂²×C₆₁	#360	1	1	yes	characteristic
98	C₂²×C₆₁	#361	1	1	yes	characteristic
99	C₂²×C₆₁	#362	1	1	yes	characteristic
100	C₄×C₆₁	—	8	2	no	—
101	C₄×C₆₁	Q₈⋊C₈	1	1	yes	characteristic
102	C₄×C₆₁	Q₈⋊C₈	1	1	yes	characteristic
103	C₄×C₆₁	#336	1	1	yes	characteristic
104	C₄×C₆₁	#339	1	1	yes	characteristic
105	Dic₆₁	—	2	1	no	—
106	Dic₆₁	—	2	1	no	—
107	Dic₆₁	—	2	1	no	—
108	Dic₆₁	—	2	1	no	—
109	Dic₆₁	—	8	2	no	—
110	C₂×#346	—	61	1	no	—
111	C₄×#107	—	61	1	no	—
112	C₂×#334	—	61	1	no	—
113	C₂×C₆₁	#1279	1	1	yes	characteristic
114	C₂×C₆₁	#1333	1	1	yes	characteristic
115	C₂×C₆₁	#1333	1	1	yes	characteristic
116	C₂×C₆₁	#1334	1	1	yes	characteristic
117	C₂×C₆₁	#1334	1	1	yes	characteristic
118	C₂×C₆₁	#1386	1	1	yes	characteristic
119	C₂×C₆₁	#1387	1	1	yes	characteristic
120	C₂²×C₁₆	—	122	1	no	—
121	C₂×#107	—	61	1	no	—
122	C₂×#107	#6923	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	122	1	no	—
124	C₂×C₄×C₈	—	61	1	no	—
125	C₄×C₄⋊C₄	—	122	1	no	—
126	#346	—	61	2	no	—
127	#346	—	122	1	no	—
128	#340	—	122	1	no	—
129	#334	—	61	2	no	—
130	#334	—	122	1	no	—
131	C₆₁	#7178	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	61	1	no	—
133	C₂²×C₈	—	244	1	no	—
134	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	122	1	no	—
137	C₂×C₄²	—	61	1	no	—
138	C₂×C₄²	—	244	1	no	—
139	C₂×C₁₆	—	122	2	no	—
140	C₂×C₁₆	—	122	2	no	—
141	C₂×C₁₆	—	122	2	no	—
142	#107	—	2	1	no	—
143	#107	—	61	2	no	—
144	#107	—	122	1	no	—
145	#107	#92420	1	2	yes	complemented
146	C₄⋊C₈	—	122	2	no	—
147	C₄⋊C₈	—	244	1	no	—
148	C₄×C₈	—	61	1	no	—
149	C₄×C₈	—	61	1	no	—
150	C₄×C₈	—	61	1	no	—
151	C₄×C₈	—	61	1	no	—
152	#95	—	244	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	—	61	1	no	—
155	C₂²×C₄	—	61	1	no	—
156	C₂²×C₄	—	244	1	no	—
157	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
158	C₂×C₈	—	61	1	no	—
159	C₂×C₈	—	61	1	no	—
160	C₂×C₈	—	122	1	no	—
161	C₂×C₈	—	122	1	no	—
162	C₂×C₈	—	488	1	no	—
163	C₂×C₈	—	488	1	no	—
164	C₂×C₈	—	244	2	no	—
165	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
166	C₂×C₈	#11758213	1	1	yes	characteristic
167	C₂×C₈	C₄×#6923	1	1	yes	characteristic
168	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
169	C₂×C₈	C₂×#92420	1	1	yes	characteristic
170	C₂×C₈	#11758217	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₄⋊C₄	—	122	2	no	—
174	C₄⋊C₄	—	244	1	no	—
175	C₄²	—	61	1	no	—
176	C₄²	—	61	1	no	—
177	C₄²	—	61	1	no	—
178	C₄²	—	61	1	no	—
179	C₄²	—	488	2	no	—
180	C₁₆	—	122	4	no	—
181	C₂³	#14519839	1	1	yes	characteristic
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	1	no	—
184	C₂×C₄	—	8	1	no	—
185	C₂×C₄	—	61	1	no	—
186	C₂×C₄	—	61	1	no	—
187	C₂×C₄	—	61	1	no	—
188	C₂×C₄	—	61	1	no	—
189	C₂×C₄	—	122	1	no	—
190	C₂×C₄	—	122	1	no	—
191	C₂×C₄	—	122	1	no	—
192	C₂×C₄	—	122	1	no	—
193	C₂×C₄	—	488	1	no	—
194	C₂×C₄	—	488	1	no	—
195	C₂×C₄	—	244	2	no	—
196	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊#92420	1	1	yes	characteristic
198	C₂×C₄	#14519834	1	1	yes	characteristic
199	C₂×C₄	#14519840	1	1	yes	characteristic
200	C₂×C₄	#14519841	1	1	yes	characteristic
201	C₂×C₄	#14519842	1	1	yes	characteristic
202	C₈	—	122	1	no	—
203	C₈	—	122	1	no	—
204	C₈	—	488	2	no	—
205	C₈	C₄×#92420	1	1	yes	characteristic
206	C₈	#14519832	1	1	yes	characteristic
207	C₈	C₄⋊#92420	1	1	yes	characteristic
208	C₈	#14519844	1	1	yes	characteristic
209	C₂²	D₄⋊#92420	1	1	yes	characteristic
210	C₂²	#16840610	1	1	yes	characteristic
211	C₂²	#16840635	1	1	yes	characteristic
212	C₂²	#16840678	1	1	yes	characteristic
213	C₂²	#16840679	1	1	yes	characteristic
214	C₂²	#16840680	1	1	yes	characteristic
215	C₂²	#16840681	1	1	yes	characteristic
216	C₄	—	8	2	no	—
217	C₄	—	122	1	no	—
218	C₄	—	122	1	no	—
219	C₄	—	122	1	no	—
220	C₄	—	122	1	no	—
221	C₄	—	488	2	no	—
222	C₄	Q₈⋊#92420	1	1	yes	characteristic
223	C₄	#16840612	1	1	yes	characteristic
224	C₄	#16840628	1	1	yes	characteristic
225	C₄	#16840632	1	1	yes	characteristic
226	C₂	#21460655	1	1	yes	characteristic
227	C₂	#21460777	1	1	yes	characteristic
228	C₂	#21460778	1	1	yes	characteristic
229	C₂	#21460779	1	1	yes	characteristic
230	C₂	#21460780	1	1	yes	characteristic
231	C₂	#21460890	1	1	yes	characteristic
232	C₂	#21460893	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519839
commutator	C₈×C₆₁	C₄×C₈
Frattini	C₂²×C₈	C₂×#6923
Fitting	C₂×#107×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	#102
cosocle	C₂×C₈×Dic₆₁	C₂²
hypercenter	C₂×#107	#6923
hypocenter	C₆₁	#7178
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
7. #16840651	C₄	semidirect
9. #16840623	C₄	semidirect
26. #14519711	C₈	semidirect
35. #107×C₆₁	C₈	semidirect
131. C₆₁	#7178	semidirect
145. #107	#92420	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840651	2
C₂×#16840623	2
#107×Dic₆₁	2
#7178	61