Subgroups — #29598617

Order 15616.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840167	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
3	#21443910	C₂	1	4	yes	maximal
4	C₆₁×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519777	—	2	1	no	—
6	C₂²×#11758185	C₂²	1	2	yes	—
7	#16840167	—	2	2	no	—
8	#16840167	C₄	1	4	yes	complemented
9	C₂×#14519711	—	2	1	no	—
10	C₂×#14519711	C₂²	1	2	yes	—
11	#16839538	—	2	2	no	—
12	C₂×#107×C₆₁	C₂²	1	2	yes	—
13	C₂×#106×C₆₁	—	2	1	no	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#382	—	2	1	no	—
16	C₂²×Dic₁₂₂	—	4	1	no	—
17	#14519777	—	2	4	no	—
18	#14519777	—	4	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	4	1	no	—
21	C₂×#11758185	—	2	2	no	—
22	C₂×#11758185	—	2	2	no	—
23	C₂×#11758185	Q₈	1	4	yes	complemented
24	#14519711	—	2	4	no	—
25	#14519711	—	2	2	no	—
26	#14519711	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
31	#107×C₆₁	—	2	4	no	—
32	#106×C₆₁	—	4	2	no	—
33	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₁	D₄	1	2	yes	—
35	#95×C₆₁	—	4	1	no	—
36	#95×C₆₁	—	4	1	no	—
37	C₂²×C₄×C₆₁	—	4	2	no	—
38	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	—	4	2	no	—
42	C₂×Dic₁₂₂	—	4	4	no	—
43	C₂×Dic₁₂₂	—	8	4	no	—
44	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
46	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
47	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₆₁	#41	1	2	yes	—
51	C₄⋊C₄×C₆₁	—	4	2	no	—
52	C₄⋊C₄×C₆₁	—	4	2	no	—
53	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₆₁	SD₁₆	1	2	yes	—
55	C₄⋊Dic₆₁	—	2	4	no	—
56	C₄⋊Dic₆₁	—	4	4	no	—
57	C₄⋊Dic₆₁	—	4	2	no	—
58	#11758185	—	2	4	no	—
59	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₆₁	—	2	2	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	8	2	no	—
63	C₂×C₄×C₆₁	—	8	2	no	—
64	C₂×C₄×C₆₁	—	8	2	no	—
65	C₂×C₄×C₆₁	#102	1	2	yes	—
66	C₂×C₄×C₆₁	#102	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×Dic₆₁	—	4	4	no	—
74	C₂×Dic₆₁	—	8	4	no	—
75	Dic₁₂₂	—	8	8	no	—
76	C₈×C₆₁	#103	1	4	yes	—
77	C₈×C₆₁	C₄×Q₈	1	2	yes	—
78	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
79	#13283	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
81	C₂²×C₆₁	#486	1	1	yes	characteristic
82	C₂²×C₆₁	#500	1	1	yes	characteristic
83	C₂²×C₆₁	#509	1	2	yes	—
84	C₂²×C₆₁	#510	1	2	yes	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	2	2	no	—
87	C₄×C₆₁	—	16	2	no	—
88	C₄×C₆₁	#391	1	1	yes	characteristic
89	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
90	C₄×C₆₁	#487	1	1	yes	characteristic
91	C₄×C₆₁	#498	1	1	yes	characteristic
92	Dic₆₁	—	8	4	no	—
93	C₂×#500	—	61	1	no	—
94	C₂×#346	—	61	1	no	—
95	#1846	D₆₁	1	1	yes	characteristic
96	#1380	—	61	4	no	—
97	C₂×C₆₁	#1990	1	1	yes	characteristic
98	C₂×C₆₁	#2140	1	1	yes	characteristic
99	C₂×C₆₁	#2141	1	1	yes	characteristic
100	C₂×C₆₁	#2243	1	2	yes	—
101	C₂×C₆₁	#2255	1	2	yes	—
102	C₂×C₄⋊Q₈	—	122	1	no	—
103	C₂²×C₁₆	—	61	2	no	—
104	#500	—	61	4	no	—
105	#500	—	122	2	no	—
106	C₂×#107	—	61	2	no	—
107	C₂×#107	—	122	1	no	—
108	C₂×#107	C₂×D₆₁	1	2	yes	—
109	C₂×#106	—	2	1	no	—
110	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
111	#382	—	2	1	no	—
112	#346	—	122	2	no	—
113	C₆₁	#13283	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	244	1	no	—
115	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
117	C₄⋊Q₈	—	122	4	no	—
118	C₄⋊Q₈	—	244	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	122	2	no	—
121	C₂×C₄⋊C₄	—	244	1	no	—
122	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂×C₁₆	—	61	4	no	—
124	C₂×C₁₆	—	122	2	no	—
125	C₂×C₁₆	—	122	2	no	—
126	#107	—	2	4	no	—
127	#107	—	61	4	no	—
128	#107	—	122	4	no	—
129	#107	—	122	2	no	—
130	#106	—	4	2	no	—
131	C₄×C₈	C₄×D₆₁	1	2	yes	—
132	C₄×C₈	D₂₄₄	1	2	yes	—
133	#95	—	4	1	no	—
134	#95	—	4	1	no	—
135	C₂×Q₈	—	488	4	no	—
136	C₂×Q₈	—	244	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	244	2	no	—
139	C₂²×C₄	#11758199	1	1	yes	characteristic
140	C₂²×C₄	#11758199	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
142	C₂×C₈	#11758198	1	2	yes	—
143	C₂×C₈	#11758200	1	2	yes	—
144	C₂×C₈	#11758202	1	2	yes	—
145	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
146	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
147	C₂×C₈	#11758224	1	2	yes	—
148	C₂×C₈	Q₈×D₆₁	1	2	yes	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	122	4	no	—
152	C₄⋊C₄	—	244	4	no	—
153	C₄⋊C₄	—	244	2	no	—
154	C₄²	#11758198	1	2	yes	—
155	C₄²	#11758201	1	2	yes	—
156	C₁₆	—	122	4	no	—
157	C₂³	C₂×#11758199	1	1	yes	characteristic
158	Q₈	—	488	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	488	4	no	—
165	C₂×C₄	—	244	4	no	—
166	C₂×C₄	#14519697	1	2	yes	—
167	C₂×C₄	#14519697	1	2	yes	—
168	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
169	C₂×C₄	#14519764	1	1	yes	characteristic
170	C₂×C₄	#14519775	1	1	yes	characteristic
171	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
172	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
173	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
174	C₈	#14519700	1	4	yes	—
175	C₈	#14519742	1	2	yes	—
176	C₈	#14519757	1	2	yes	—
177	C₂²	C₂×#14519697	1	1	yes	characteristic
178	C₂²	#16840108	1	1	yes	characteristic
179	C₂²	#16840165	1	1	yes	characteristic
180	C₂²	#16840197	1	2	yes	—
181	C₂²	#16840202	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	16	2	no	—
185	C₄	—	488	4	no	—
186	C₄	#16839667	1	1	yes	characteristic
187	C₄	C₂×#14519700	1	1	yes	characteristic
188	C₄	#16840112	1	1	yes	characteristic
189	C₄	#16840158	1	1	yes	characteristic
190	C₂	#21446779	1	1	yes	characteristic
191	C₂	#21447534	1	1	yes	characteristic
192	C₂	#21447538	1	1	yes	characteristic
193	C₂	#21448055	1	2	yes	—
194	C₂	#21448106	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1846	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1846	D₆₁
hypocenter	C₆₁	#13283
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
8. #16840167	C₄	semidirect
23. C₂×#11758185	Q₈	semidirect
113. C₆₁	#13283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840167	2
C₂×#16839538	2
#21443910	2
C₆₁×#1846	2
#13283	61