Subgroups — #29566950

Order 15616.

274 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839513	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758185	—	2	1	no	—
5	C₂×C₄⋊#92418	—	2	1	no	—
6	#16839538	—	2	1	no	—
7	#16839538	C₄	1	2	yes	complemented
8	#16839513	—	2	1	no	—
9	#16839513	C₄	1	2	yes	complemented
10	C₂×#107×C₆₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₆₁	—	2	1	no	—
13	C₆₁×#340	—	2	1	no	—
14	C₂²×#92418	—	4	1	no	—
15	C₂×#11758185	—	2	4	no	—
16	C₂×#11758185	—	2	2	no	—
17	C₄⋊#92418	—	2	2	no	—
18	C₄⋊#92418	—	4	1	no	—
19	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
22	C₂×C₄²×C₆₁	—	4	1	no	—
23	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
24	#107×C₆₁	—	2	2	no	—
25	#107×C₆₁	C₈	1	4	yes	complemented
26	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
30	#95×C₆₁	—	4	1	no	—
31	C₂²×C₄×C₆₁	—	4	2	no	—
32	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₆₁	—	2	2	no	—
36	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
42	C₄⋊C₄×C₆₁	—	2	4	no	—
43	C₄⋊C₄×C₆₁	—	4	2	no	—
44	C₄²×C₆₁	—	8	2	no	—
45	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
48	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
49	C₂×#92418	—	4	2	no	—
50	C₂×#92418	—	8	2	no	—
51	#11758185	—	2	4	no	—
52	C₂³×C₆₁	#102	1	1	yes	characteristic
53	C₂×C₄×C₆₁	—	2	2	no	—
54	C₂×C₄×C₆₁	—	2	1	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	—	4	4	no	—
57	C₂×C₄×C₆₁	—	8	2	no	—
58	C₂×C₄×C₆₁	—	8	2	no	—
59	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
62	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
63	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
64	C₂×C₄×C₆₁	#104	1	2	yes	—
65	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
67	C₈×C₆₁	—	2	1	no	—
68	C₈×C₆₁	—	2	1	no	—
69	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
70	C₈×C₆₁	#97	1	1	yes	characteristic
71	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
72	#92418	—	8	2	no	—
73	#7178	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	D₄⋊C₈	1	2	yes	—
75	C₂²×C₆₁	#340	1	1	yes	characteristic
76	C₂²×C₆₁	#360	1	2	yes	—
77	C₂²×C₆₁	#361	1	1	yes	characteristic
78	C₂²×C₆₁	#362	1	1	yes	characteristic
79	C₄×C₆₁	—	2	1	no	—
80	C₄×C₆₁	—	2	1	no	—
81	C₄×C₆₁	—	2	1	no	—
82	C₄×C₆₁	—	2	1	no	—
83	C₄×C₆₁	—	8	4	no	—
84	C₄×C₆₁	Q₈⋊C₈	1	2	yes	—
85	C₄×C₆₁	#336	1	1	yes	characteristic
86	C₄×C₆₁	#339	1	1	yes	characteristic
87	C₂×#346	—	61	1	no	—
88	C₄×#107	D₆₁	1	1	yes	characteristic
89	C₂×#334	—	61	1	no	—
90	C₂×C₆₁	#1279	1	1	yes	characteristic
91	C₂×C₆₁	#1333	1	2	yes	—
92	C₂×C₆₁	#1334	1	2	yes	—
93	C₂×C₆₁	#1386	1	1	yes	characteristic
94	C₂×C₆₁	#1387	1	1	yes	characteristic
95	C₂²×C₁₆	—	122	1	no	—
96	C₂×#107	Dic₆₁	1	2	yes	—
97	C₂×C₄⋊C₈	—	122	1	no	—
98	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	2	1	no	—
100	#346	—	61	2	no	—
101	#346	—	122	1	no	—
102	#340	—	2	1	no	—
103	#334	—	61	2	no	—
104	#334	—	122	1	no	—
105	C₆₁	#7178	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	244	1	no	—
107	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
108	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	—	4	1	no	—
111	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
112	C₂×C₁₆	—	122	4	no	—
113	C₂×C₁₆	—	122	2	no	—
114	#107	—	2	2	no	—
115	#107	#92418	1	4	yes	complemented
116	C₄⋊C₈	—	122	2	no	—
117	C₄⋊C₈	—	244	1	no	—
118	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
119	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
120	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
121	C₄×C₈	D₂₄₄	1	1	yes	characteristic
122	#95	—	4	1	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	#11758199	1	1	yes	characteristic
125	C₂²×C₄	#11758200	1	1	yes	characteristic
126	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	488	2	no	—
129	C₂×C₈	—	244	2	no	—
130	C₂×C₈	C₂×#92418	1	2	yes	—
131	C₂×C₈	#11758194	1	2	yes	—
132	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
133	C₂×C₈	#11758196	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
135	C₂×C₈	#11758198	1	1	yes	characteristic
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄²	—	8	2	no	—
139	C₄²	#11758196	1	1	yes	characteristic
140	C₄²	#11758198	1	1	yes	characteristic
141	C₄²	#11758201	1	1	yes	characteristic
142	C₄²	#11758202	1	1	yes	characteristic
143	C₁₆	—	122	4	no	—
144	C₂³	#14519698	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	#14519684	1	1	yes	characteristic
152	C₂×C₄	C₂²⋊#92418	1	2	yes	—
153	C₂×C₄	#14519697	1	1	yes	characteristic
154	C₂×C₄	#14519700	1	1	yes	characteristic
155	C₂×C₄	#14519701	1	1	yes	characteristic
156	C₂×C₄	#14519704	1	2	yes	—
157	C₂×C₄	#14519708	1	1	yes	characteristic
158	C₂×C₄	#14519710	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	488	2	no	—
162	C₈	C₄×#92418	1	1	yes	characteristic
163	C₈	#14519688	1	1	yes	characteristic
164	C₈	C₄⋊#92418	1	2	yes	—
165	C₂²	D₄⋊#92418	1	2	yes	—
166	C₂²	#16839524	1	1	yes	characteristic
167	C₂²	#16839572	1	2	yes	—
168	C₂²	#16839575	1	1	yes	characteristic
169	C₂²	#16839578	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	8	4	no	—
175	C₄	Q₈⋊#92418	1	2	yes	—
176	C₄	#16839517	1	1	yes	characteristic
177	C₄	#16839521	1	1	yes	characteristic
178	C₂	#21443660	1	1	yes	characteristic
179	C₂	#21443786	1	2	yes	—
180	C₂	#21443789	1	2	yes	—
181	C₂	#21443924	1	1	yes	characteristic
182	C₂	#21443927	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519698
commutator	C₈×C₆₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₄×#107×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	#102
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₄×#107	D₆₁
hypocenter	C₆₁	#7178
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. #16839538	C₄	semidirect
9. #16839513	C₄	semidirect
25. #107×C₆₁	C₈	semidirect
105. C₆₁	#7178	semidirect
115. #107	#92418	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839538	2
C₂×#16839513	2
C₄×#107×C₆₁	2
#7178	61