Subgroups — #29566953

Order 15616.

274 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839583	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1772	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	—	2	1	no	—
5	C₂×C₄×#92418	—	2	1	no	—
6	#16839583	—	2	1	no	—
7	#16839583	C₄	1	2	yes	complemented
8	C₈×#92418	—	2	1	no	—
9	C₈×#92418	C₄	1	2	yes	complemented
10	C₂×#108×C₆₁	C₄	1	1	yes	characteristic
11	C₆₁×#427	C₄	1	1	yes	characteristic
12	C₆₁×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#339	—	2	1	no	—
15	C₂²×#92418	—	2	2	no	—
16	#14519717	—	2	2	no	—
17	#14519717	—	4	1	no	—
18	C₂×#11758185	—	4	1	no	—
19	C₄×#92418	—	2	2	no	—
20	C₄×#92418	—	2	2	no	—
21	C₂×M₁₆×C₆₁	—	2	1	no	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
24	#117×C₆₁	—	2	1	no	—
25	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
26	#108×C₆₁	—	2	1	no	—
27	#108×C₆₁	C₈	1	2	yes	complemented
28	#107×C₆₁	C₈	1	2	yes	complemented
29	#106×C₆₁	—	2	1	no	—
30	#97×C₆₁	—	4	1	no	—
31	C₄×C₈×C₆₁	—	4	1	no	—
32	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
35	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	—	2	1	no	—
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	M₁₆×C₆₁	—	2	2	no	—
39	M₁₆×C₆₁	—	4	1	no	—
40	C₂×C₈×C₆₁	—	4	1	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	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
48	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
50	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
53	C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₂²⋊C₄×C₆₁	—	4	1	no	—
55	C₄²×C₆₁	—	2	1	no	—
56	C₄²×C₆₁	—	4	1	no	—
57	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂×#92418	—	2	4	no	—
60	C₂×#92418	—	2	8	no	—
61	#11758185	—	4	2	no	—
62	C₂³×C₆₁	#104	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	4	no	—
64	C₂×C₄×C₆₁	—	2	1	no	—
65	C₂×C₄×C₆₁	—	2	1	no	—
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#104	1	1	yes	characteristic
73	C₈×C₆₁	—	4	2	no	—
74	C₈×C₆₁	C₄×C₈	1	1	yes	characteristic
75	C₈×C₆₁	#97	1	1	yes	characteristic
76	C₈×C₆₁	#102	1	1	yes	characteristic
77	C₈×C₆₁	#103	1	1	yes	characteristic
78	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₆₁	C₄⋊C₈	1	1	yes	characteristic
80	C₈×C₆₁	#106	1	1	yes	characteristic
81	C₈×C₆₁	#107	1	1	yes	characteristic
82	#92418	—	2	8	no	—
83	#7179	—	61	1	no	maximal, Sylow
84	C₂²×C₆₁	—	2	2	no	—
85	C₂²×C₆₁	D₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₆₁	Q₈⋊C₈	1	1	yes	characteristic
87	C₂²×C₆₁	#339	1	1	yes	characteristic
88	C₄×C₆₁	—	2	2	no	—
89	C₄×C₆₁	—	2	2	no	—
90	C₄×C₆₁	—	8	2	no	—
91	C₄×C₆₁	D₄⋊C₈	1	1	yes	characteristic
92	C₄×C₆₁	Q₈⋊C₈	1	1	yes	characteristic
93	C₄×C₆₁	#336	1	1	yes	characteristic
94	C₄×C₆₁	#340	1	1	yes	characteristic
95	C₂×#364	—	61	1	no	—
96	#1772	D₆₁	1	1	yes	characteristic
97	C₂×C₈²	—	61	1	no	—
98	C₂×C₆₁	—	2	1	no	—
99	C₂×C₆₁	—	2	1	no	—
100	C₂×C₆₁	#1279	1	1	yes	characteristic
101	C₂×C₆₁	#1335	1	2	yes	—
102	C₂×M₃₂	—	122	1	no	—
103	C₂×#108	Dic₆₁	1	1	yes	characteristic
104	#427	Dic₆₁	1	1	yes	characteristic
105	#405	—	2	1	no	—
106	C₂×C₄×C₈	—	122	1	no	—
107	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
108	#364	—	61	2	no	—
109	#364	—	122	1	no	—
110	#339	—	2	1	no	—
111	C₈²	—	61	2	no	—
112	C₈²	—	122	1	no	—
113	C₆₁	#7179	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	2	1	no	—
115	C₂²×C₈	—	122	2	no	—
116	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
117	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
118	#117	—	2	1	no	—
119	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
120	M₃₂	—	122	2	no	—
121	M₃₂	—	244	1	no	—
122	C₂×C₁₆	—	244	1	no	—
123	#108	—	2	1	no	—
124	#108	#92418	1	2	yes	complemented
125	#107	#92418	1	2	yes	complemented
126	#106	—	2	1	no	—
127	#97	—	4	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	—	122	2	no	—
130	C₄×C₈	—	122	2	no	—
131	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
132	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
133	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
134	C₄×C₈	D₂₄₄	1	1	yes	characteristic
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
137	M₁₆	—	2	2	no	—
138	M₁₆	—	4	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	122	4	no	—
141	C₂×C₈	—	122	8	no	—
142	C₂×C₈	C₂×#92418	1	1	yes	characteristic
143	C₂×C₈	C₂×#92418	1	1	yes	characteristic
144	C₂×C₈	#11758194	1	1	yes	characteristic
145	C₂×C₈	#11758194	1	1	yes	characteristic
146	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
147	C₂×C₈	#11758196	1	1	yes	characteristic
148	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
149	C₂×C₈	#11758198	1	1	yes	characteristic
150	C₂×C₈	#11758199	1	1	yes	characteristic
151	C₂×C₈	#11758200	1	1	yes	characteristic
152	C₂×C₈	#11758201	1	1	yes	characteristic
153	C₂×C₈	#11758202	1	1	yes	characteristic
154	C₄⋊C₄	—	2	2	no	—
155	C₂²⋊C₄	—	4	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	#11758196	1	1	yes	characteristic
159	C₄²	#11758198	1	1	yes	characteristic
160	C₁₆	—	244	2	no	—
161	C₂³	#14519704	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	#14519684	1	1	yes	characteristic
167	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
168	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
169	C₂×C₄	#14519698	1	1	yes	characteristic
170	C₂×C₄	#14519701	1	1	yes	characteristic
171	C₂×C₄	#14519704	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	—	122	8	no	—
174	C₈	C₄×#92418	1	1	yes	characteristic
175	C₈	#14519688	1	1	yes	characteristic
176	C₈	#14519697	1	1	yes	characteristic
177	C₈	#14519700	1	1	yes	characteristic
178	C₈	C₄⋊#92418	1	1	yes	characteristic
179	C₈	C₄⋊#92418	1	1	yes	characteristic
180	C₈	#14519708	1	1	yes	characteristic
181	C₈	#14519710	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	D₄⋊#92418	1	1	yes	characteristic
184	C₂²	Q₈⋊#92418	1	1	yes	characteristic
185	C₂²	#16839521	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	8	2	no	—
189	C₄	D₄⋊#92418	1	1	yes	characteristic
190	C₄	Q₈⋊#92418	1	1	yes	characteristic
191	C₄	#16839517	1	1	yes	characteristic
192	C₄	#16839524	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#21443660	1	1	yes	characteristic
196	C₂	#21443792	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#11758199
commutator	C₈×C₆₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₆₁×#1772	C₂
socle	C₂²×C₆₁	D₄⋊C₈
prime radical	C₂³×C₆₁	#104
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	#1772	D₆₁
hypocenter	C₆₁	#7179
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
7. #16839583	C₄	semidirect
9. C₈×#92418	C₄	semidirect
27. #108×C₆₁	C₈	semidirect
28. #107×C₆₁	C₈	semidirect
113. C₆₁	#7179	semidirect
124. #108	#92418	semidirect
125. #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×#16839583	2
C₂×C₈×#92418	2
C₆₁×#1772	2
#7179	61