Subgroups — #29567207

Order 15616.

222 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic
2	#21444567	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758185	—	2	1	no	—
5	#16839815	—	2	1	no	—
6	C₄⋊#11758185	—	2	1	no	—
7	C₄⋊#11758185	C₄	1	2	yes	—
8	#16839515	C₄	1	2	yes	—
9	#16839513	—	2	1	no	—
10	C₂×#108×C₆₁	—	2	1	no	—
11	C₂×#108×C₆₁	C₄	1	2	yes	—
12	C₆₁×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#11758185	—	2	2	no	—
15	C₂×#11758185	—	4	1	no	—
16	C₂×#11758185	—	4	1	no	—
17	C₄⋊#92418	—	2	2	no	—
18	C₂²⋊#92418	—	4	1	no	—
19	C₄×#92418	—	4	1	no	—
20	C₂×M₁₆×C₆₁	—	2	2	no	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
24	#108×C₆₁	—	4	2	no	—
25	#108×C₆₁	—	4	1	no	—
26	#108×C₆₁	—	4	1	no	—
27	C₄⋊C₈×C₆₁	—	4	1	no	—
28	C₄⋊C₈×C₆₁	—	4	1	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
36	M₁₆×C₆₁	—	4	2	no	—
37	M₁₆×C₆₁	—	4	2	no	—
38	C₂×C₈×C₆₁	—	2	1	no	—
39	C₂×C₈×C₆₁	—	2	1	no	—
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
46	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
49	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₆₁	D₈	1	1	yes	characteristic
52	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
53	C₂×#92418	—	4	2	no	—
54	#11758185	—	4	2	no	—
55	C₂³×C₆₁	#103	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	2	2	no	—
57	C₂×C₄×C₆₁	—	4	1	no	—
58	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#100	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
68	C₈×C₆₁	—	2	1	no	—
69	C₈×C₆₁	—	2	1	no	—
70	C₈×C₆₁	—	8	2	no	—
71	C₈×C₆₁	#102	1	1	yes	characteristic
72	C₈×C₆₁	#103	1	1	yes	characteristic
73	C₈×C₆₁	#106	1	1	yes	characteristic
74	C₈×C₆₁	#107	1	1	yes	characteristic
75	#92418	—	8	2	no	—
76	#7277	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	—	2	1	no	—
78	C₂²×C₆₁	—	2	1	no	—
79	C₂²×C₆₁	#332	1	1	yes	characteristic
80	C₂²×C₆₁	#333	1	1	yes	characteristic
81	C₂²×C₆₁	#340	1	1	yes	characteristic
82	C₄×C₆₁	—	2	1	no	—
83	C₄×C₆₁	—	2	1	no	—
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	#331	1	1	yes	characteristic
86	C₄×C₆₁	#332	1	1	yes	characteristic
87	C₄×C₆₁	#340	1	1	yes	characteristic
88	C₄×C₆₁	#343	1	1	yes	characteristic
89	C₄×C₆₁	#359	1	2	yes	—
90	C₂×C₄⋊C₁₆	—	61	1	no	—
91	#1848	D₆₁	1	1	yes	characteristic
92	#1564	—	61	1	no	—
93	C₂×C₆₁	—	2	1	no	—
94	C₂×C₆₁	—	2	1	no	—
95	C₂×C₆₁	#1296	1	1	yes	characteristic
96	C₂×C₆₁	#1382	1	1	yes	characteristic
97	C₂×C₆₁	#1388	1	1	yes	characteristic
98	C₂²×C₁₆	—	122	1	no	—
99	#431	—	122	1	no	—
100	C₂×#108	—	2	1	no	—
101	C₂×#108	Dic₆₁	1	2	yes	—
102	#423	—	2	1	no	—
103	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
104	C₄⋊C₁₆	—	61	2	no	—
105	C₄⋊C₁₆	—	122	1	no	—
106	#335	—	61	2	no	—
107	#334	—	122	1	no	—
108	C₆₁	#7277	1	1	yes	characteristic, complemented, Sylow
109	C₂×M₁₆	—	2	2	no	—
110	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
111	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
113	C₂×C₁₆	—	122	2	no	—
114	C₂×C₁₆	—	244	1	no	—
115	C₂×C₁₆	—	244	1	no	—
116	#108	—	4	2	no	—
117	#108	—	4	1	no	—
118	#108	—	4	1	no	—
119	C₄⋊C₈	—	4	1	no	—
120	C₄⋊C₈	—	4	1	no	—
121	C₄⋊C₈	—	122	2	no	—
122	C₂²⋊C₈	—	244	1	no	—
123	C₄×C₈	—	244	1	no	—
124	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
125	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
126	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
127	C₄×C₈	D₂₄₄	1	1	yes	characteristic
128	C₂²×C₄	#11758201	1	1	yes	characteristic
129	C₂²×C₄	#11758202	1	1	yes	characteristic
130	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
131	M₁₆	—	4	2	no	—
132	M₁₆	—	4	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	244	2	no	—
137	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
138	C₂×C₈	#11758196	1	1	yes	characteristic
139	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
140	C₂×C₈	#11758198	1	1	yes	characteristic
141	C₂×C₈	#11758199	1	1	yes	characteristic
142	C₂×C₈	#11758200	1	1	yes	characteristic
143	C₂×C₈	#11758201	1	1	yes	characteristic
144	C₂×C₈	#11758202	1	1	yes	characteristic
145	C₄²	#11758196	1	1	yes	characteristic
146	C₄²	#11758198	1	1	yes	characteristic
147	C₄²	#11758199	1	1	yes	characteristic
148	C₄²	#11758200	1	1	yes	characteristic
149	C₁₆	—	244	2	no	—
150	C₂³	#14519701	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	#14519684	1	1	yes	characteristic
154	C₂×C₄	#14519694	1	1	yes	characteristic
155	C₂×C₄	#14519696	1	1	yes	characteristic
156	C₂×C₄	#14519697	1	1	yes	characteristic
157	C₂×C₄	#14519698	1	1	yes	characteristic
158	C₂×C₄	#14519698	1	1	yes	characteristic
159	C₂×C₄	#14519700	1	1	yes	characteristic
160	C₂×C₄	#14519701	1	1	yes	characteristic
161	C₂×C₄	#14519708	1	1	yes	characteristic
162	C₂×C₄	#14519710	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	488	2	no	—
167	C₈	#14519697	1	1	yes	characteristic
168	C₈	#14519700	1	1	yes	characteristic
169	C₈	#14519708	1	1	yes	characteristic
170	C₈	#14519710	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#16839508	1	1	yes	characteristic
174	C₂²	#16839511	1	1	yes	characteristic
175	C₂²	#16839524	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#16839506	1	1	yes	characteristic
180	C₄	#16839508	1	1	yes	characteristic
181	C₄	#16839524	1	1	yes	characteristic
182	C₄	#16839531	1	1	yes	characteristic
183	C₄	#16839568	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#21443702	1	1	yes	characteristic
187	C₂	#21443916	1	1	yes	characteristic
188	C₂	#21443930	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519698
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#1848	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#103
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#1848	D₆₁
hypocenter	C₆₁	#7277
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×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
108. C₆₁	#7277	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11758185	2
#21444567	2
C₆₁×#1848	2
#7277	61