Subgroups — #29598625

Order 15616.

350 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	#21456333	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
3	#21443916	C₂	1	4	yes	maximal
4	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic
5	#16840319	—	2	1	no	—
6	C₂²×#11758185	C₂²	1	2	yes	—
7	#16840167	C₄	1	2	yes	—
8	#16840164	—	2	1	no	—
9	#16840164	C₄	1	2	yes	—
10	#16840160	—	2	1	no	—
11	#16839800	—	2	1	no	—
12	#16839800	C₂²	1	2	yes	—
13	C₄×#11758185	—	2	1	no	—
14	C₄×#11758185	—	2	1	no	—
15	C₂×#108×C₆₁	—	2	1	no	—
16	C₂×#108×C₆₁	C₂²	1	2	yes	—
17	C₆₁×#423	—	2	1	no	—
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	#14519777	—	2	2	no	—
20	#14519769	—	2	2	no	—
21	#14519756	—	4	2	no	—
22	C₄×Dic₁₂₂	—	4	2	no	—
23	#14519736	—	2	2	no	—
24	C₂×#11758185	—	2	2	no	—
25	C₂×#11758185	—	2	2	no	—
26	C₂×#11758185	Q₈	1	4	yes	complemented
27	#14519711	—	2	2	no	—
28	#14519711	C₂×C₄	1	4	yes	—
29	#14519709	—	2	2	no	—
30	#14519709	—	2	2	no	—
31	C₂×M₁₆×C₆₁	—	2	2	no	—
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
35	#108×C₆₁	—	2	4	no	—
36	#108×C₆₁	—	4	2	no	—
37	C₄⋊C₈×C₆₁	—	4	2	no	—
38	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	SD₁₆	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₁₂₂	—	4	2	no	—
45	M₁₆×C₆₁	—	4	2	no	—
46	M₁₆×C₆₁	—	4	2	no	—
47	C₂×C₈×C₆₁	—	4	2	no	—
48	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
51	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
52	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₆₁	#41	1	2	yes	—
56	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₆₁	D₈	1	2	yes	—
59	C₂²⋊Dic₆₁	—	4	2	no	—
60	C₄⋊Dic₆₁	—	2	4	no	—
61	C₄⋊Dic₆₁	—	4	2	no	—
62	#11758196	—	4	4	no	—
63	C₄×Dic₆₁	—	4	2	no	—
64	#11758185	—	2	4	no	—
65	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₁	—	2	4	no	—
67	C₂×C₄×C₆₁	#102	1	2	yes	—
68	C₂×C₄×C₆₁	#102	1	2	yes	—
69	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₆₁	—	4	4	no	—
76	Dic₁₂₂	—	8	4	no	—
77	C₈×C₆₁	—	8	2	no	—
78	C₈×C₆₁	#103	1	2	yes	—
79	C₈×C₆₁	#103	1	2	yes	—
80	C₈×C₆₁	C₄×Q₈	1	2	yes	—
81	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
82	#13285	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	—	2	2	no	—
84	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
85	C₂²×C₆₁	#487	1	1	yes	characteristic
86	C₂²×C₆₁	#498	1	1	yes	characteristic
87	C₄×C₆₁	—	2	2	no	—
88	C₄×C₆₁	#359	1	4	yes	—
89	C₄×C₆₁	#391	1	1	yes	characteristic
90	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
91	C₄×C₆₁	#486	1	1	yes	characteristic
92	C₄×C₆₁	#500	1	1	yes	characteristic
93	Dic₆₁	—	8	4	no	—
94	#3160	—	61	1	no	—
95	C₂×C₄×C₁₆	—	61	1	no	—
96	#1848	D₆₁	1	1	yes	characteristic
97	#1382	—	61	4	no	—
98	C₂×C₆₁	—	2	1	no	—
99	C₂×C₆₁	—	2	1	no	—
100	C₂×C₆₁	#1990	1	1	yes	characteristic
101	C₂×C₆₁	C₂×#359	1	2	yes	—
102	#533	—	122	1	no	—
103	C₂²×C₁₆	—	61	2	no	—
104	#500	—	61	2	no	—
105	#499	—	61	2	no	—
106	#499	—	122	1	no	—
107	#498	—	122	1	no	—
108	C₂×#108	—	2	1	no	—
109	C₂×#108	C₂×D₆₁	1	2	yes	—
110	#427	—	61	2	no	—
111	#427	—	122	1	no	—
112	#423	—	2	1	no	—
113	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
114	C₄×C₁₆	—	122	1	no	—
115	C₄×C₁₆	—	122	1	no	—
116	C₆₁	#13285	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	2	2	no	—
118	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
120	C₄⋊Q₈	—	122	2	no	—
121	#125	—	122	2	no	—
122	C₂²⋊Q₈	—	244	2	no	—
123	C₄×Q₈	—	244	2	no	—
124	#117	—	122	2	no	—
125	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
126	C₂×C₁₆	—	61	4	no	—
127	C₂×C₁₆	—	122	2	no	—
128	C₂×C₁₆	—	122	2	no	—
129	#108	—	2	4	no	—
130	#108	—	4	2	no	—
131	#107	—	61	4	no	—
132	#107	—	122	2	no	—
133	#106	—	122	2	no	—
134	#106	—	122	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
137	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
138	C₄×C₈	D₂₄₄	1	1	yes	characteristic
139	C₄×C₈	D₂₄₄	1	1	yes	characteristic
140	C₂×Q₈	—	244	2	no	—
141	C₂²×C₄	#11758201	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
143	M₁₆	—	4	2	no	—
144	M₁₆	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	#11758198	1	1	yes	characteristic
147	C₂×C₈	#11758198	1	1	yes	characteristic
148	C₂×C₈	#11758200	1	2	yes	—
149	C₂×C₈	#11758202	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
151	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
152	C₂×C₈	#11758224	1	2	yes	—
153	C₂×C₈	Q₈×D₆₁	1	2	yes	—
154	C₄⋊C₄	—	122	4	no	—
155	C₄⋊C₄	—	244	4	no	—
156	C₄⋊C₄	—	244	2	no	—
157	C₂²⋊C₄	—	244	2	no	—
158	C₄²	—	244	2	no	—
159	C₄²	#11758198	1	1	yes	characteristic
160	C₄²	#11758198	1	1	yes	characteristic
161	C₄²	#11758199	1	2	yes	—
162	C₁₆	—	122	4	no	—
163	C₂³	C₂×#11758201	1	1	yes	characteristic
164	Q₈	—	488	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	244	4	no	—
167	C₂×C₄	#14519697	1	2	yes	—
168	C₂×C₄	#14519697	1	2	yes	—
169	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
170	C₂×C₄	#14519764	1	1	yes	characteristic
171	C₂×C₄	#14519775	1	1	yes	characteristic
172	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
173	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
174	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
175	C₈	—	8	2	no	—
176	C₈	#14519700	1	2	yes	—
177	C₈	#14519700	1	2	yes	—
178	C₈	#14519742	1	2	yes	—
179	C₈	#14519757	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14519697	1	1	yes	characteristic
182	C₂²	#16840112	1	1	yes	characteristic
183	C₂²	#16840158	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	488	4	no	—
186	C₄	#16839568	1	4	yes	—
187	C₄	#16839667	1	1	yes	characteristic
188	C₄	C₂×#14519700	1	1	yes	characteristic
189	C₄	#16840108	1	1	yes	characteristic
190	C₄	#16840165	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#21446779	1	1	yes	characteristic
194	C₂	C₂×#16839568	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1848	C₂
socle	C₂²×C₆₁	C₂×#102
prime radical	C₂³×C₆₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1848	D₆₁
hypocenter	C₆₁	#13285
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
26. C₂×#11758185	Q₈	semidirect
116. C₆₁	#13285	semidirect

Maximal subgroups

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

subgroup	index
#21456333	2
C₂×C₄×#11758185	2
#21443916	2
C₆₁×#1848	2
#13285	61