Subgroups — #29566923

Order 15616.

266 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447575	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519717	—	2	1	no	—
5	C₂×#14519717	—	2	1	no	—
6	C₄⋊#11758185	—	2	1	no	—
7	C₄⋊#11758185	C₄	1	2	yes	complemented
8	#16839538	C₄	1	2	yes	complemented
9	C₄×#11758185	—	2	1	no	—
10	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	—	2	1	no	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
14	C₆₁×#336	—	2	1	no	—
15	#14519717	—	2	2	no	—
16	#14519717	—	2	2	no	—
17	#14519717	—	4	1	no	—
18	#14519717	—	4	1	no	—
19	C₂×#11758185	—	2	2	no	—
20	C₂×#11758185	—	2	2	no	—
21	C₂²×C₈×C₆₁	—	2	1	no	—
22	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
25	C₂×C₄²×C₆₁	—	2	1	no	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₆₁	—	2	1	no	—
28	C₄⋊C₈×C₆₁	C₈	1	2	yes	—
29	C₄×C₈×C₆₁	—	4	1	no	—
30	C₄×C₈×C₆₁	—	4	1	no	—
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
35	#95×C₆₁	C₈	1	2	yes	—
36	C₂²×C₄×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₆₁	—	2	1	no	—
41	C₂×C₈×C₆₁	—	2	1	no	—
42	C₂×C₈×C₆₁	—	2	2	no	—
43	C₂×C₈×C₆₁	—	4	1	no	—
44	C₂×C₈×C₆₁	—	4	1	no	—
45	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₄⋊C₄	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	4	no	—
54	C₄²×C₆₁	—	4	1	no	—
55	C₄²×C₆₁	—	4	1	no	—
56	C₄²×C₆₁	C₄²	1	1	yes	characteristic
57	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
60	#11758185	—	2	4	no	—
61	#11758185	—	4	2	no	—
62	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	2	4	no	—
65	C₂×C₄×C₆₁	—	4	1	no	—
66	C₂×C₄×C₆₁	—	4	2	no	—
67	C₂×C₄×C₆₁	—	4	2	no	—
68	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#103	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#104	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#104	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₆₁	—	2	1	no	—
79	C₈×C₆₁	—	2	1	no	—
80	C₈×C₆₁	—	4	2	no	—
81	C₈×C₆₁	#102	1	1	yes	characteristic
82	C₈×C₆₁	#103	1	1	yes	characteristic
83	C₈×C₆₁	#106	1	1	yes	characteristic
84	C₈×C₆₁	#107	1	1	yes	characteristic
85	#7169	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	—	2	1	no	—
87	C₂²×C₆₁	—	2	1	no	—
88	C₂²×C₆₁	D₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₆₁	Q₈⋊C₈	1	1	yes	characteristic
90	C₂²×C₆₁	#336	1	1	yes	characteristic
91	C₄×C₆₁	—	2	1	no	—
92	C₄×C₆₁	—	2	1	no	—
93	C₄×C₆₁	—	2	1	no	—
94	C₄×C₆₁	—	4	4	no	—
95	C₄×C₆₁	D₄⋊C₈	1	1	yes	characteristic
96	C₄×C₆₁	Q₈⋊C₈	1	1	yes	characteristic
97	C₄×C₆₁	#339	1	1	yes	characteristic
98	C₄×C₆₁	#340	1	1	yes	characteristic
99	C₄×C₆₁	#364	1	2	yes	—
100	#2150	—	61	1	no	—
101	C₄×M₃₂	—	61	1	no	—
102	C₈×C₄⋊C₄	D₆₁	1	1	yes	characteristic
103	C₂×C₆₁	—	2	1	no	—
104	C₂×C₆₁	—	2	1	no	—
105	C₂×C₆₁	#1279	1	1	yes	characteristic
106	C₂×C₆₁	#1376	1	1	yes	characteristic
107	C₂×C₆₁	#1378	1	1	yes	characteristic
108	C₂×M₃₂	—	122	1	no	—
109	C₂×M₃₂	—	122	1	no	—
110	C₂×C₄⋊C₈	Dic₆₁	1	1	yes	characteristic
111	C₂×C₄×C₈	—	2	1	no	—
112	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
113	C₄×C₄⋊C₄	Dic₆₁	1	1	yes	characteristic
114	C₄⋊C₁₆	—	61	2	no	—
115	C₄⋊C₁₆	—	122	1	no	—
116	#346	—	61	2	no	—
117	C₄×C₁₆	—	122	1	no	—
118	#336	—	2	1	no	—
119	C₆₁	#7169	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
126	M₃₂	—	122	2	no	—
127	M₃₂	—	122	2	no	—
128	M₃₂	—	244	1	no	—
129	M₃₂	—	244	1	no	—
130	C₂×C₁₆	—	122	2	no	—
131	C₂×C₁₆	—	122	2	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	#92418	1	2	yes	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
137	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
138	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
139	C₄×C₈	D₂₄₄	1	1	yes	characteristic
140	#95	#92418	1	2	yes	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₂×#92418	1	1	yes	characteristic
143	C₂²×C₄	#11758194	1	1	yes	characteristic
144	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	#11758196	1	1	yes	characteristic
151	C₂×C₈	#11758196	1	1	yes	characteristic
152	C₂×C₈	#11758198	1	1	yes	characteristic
153	C₂×C₈	#11758198	1	1	yes	characteristic
154	C₂×C₈	#11758199	1	1	yes	characteristic
155	C₂×C₈	#11758200	1	1	yes	characteristic
156	C₂×C₈	#11758201	1	1	yes	characteristic
157	C₂×C₈	#11758202	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₂×#92418	1	1	yes	characteristic
162	C₄²	#11758194	1	1	yes	characteristic
163	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
164	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
165	C₁₆	—	122	4	no	—
166	C₁₆	—	244	2	no	—
167	C₂³	C₂²⋊#92418	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	#14519684	1	1	yes	characteristic
174	C₂×C₄	C₄×#92418	1	1	yes	characteristic
175	C₂×C₄	#14519688	1	1	yes	characteristic
176	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
177	C₂×C₄	#14519698	1	1	yes	characteristic
178	C₂×C₄	#14519701	1	1	yes	characteristic
179	C₂×C₄	#14519704	1	1	yes	characteristic
180	C₂×C₄	#14519704	1	1	yes	characteristic
181	C₂×C₄	C₄⋊#92418	1	1	yes	characteristic
182	C₂×C₄	C₄⋊#92418	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	2	no	—
186	C₈	#14519697	1	1	yes	characteristic
187	C₈	#14519700	1	1	yes	characteristic
188	C₈	#14519708	1	1	yes	characteristic
189	C₈	#14519710	1	1	yes	characteristic
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	D₄⋊#92418	1	1	yes	characteristic
193	C₂²	Q₈⋊#92418	1	1	yes	characteristic
194	C₂²	#16839517	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	4	no	—
199	C₄	D₄⋊#92418	1	1	yes	characteristic
200	C₄	Q₈⋊#92418	1	1	yes	characteristic
201	C₄	#16839521	1	1	yes	characteristic
202	C₄	#16839524	1	1	yes	characteristic
203	C₄	#16839583	1	2	yes	—
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	#21443660	1	1	yes	characteristic
207	C₂	#21443901	1	1	yes	characteristic
208	C₂	#21443905	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#11758198
commutator	C₂×C₄×C₆₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₈×C₄⋊C₄×C₆₁	C₂
socle	C₂²×C₆₁	#336
prime radical	C₂³×C₆₁	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₈×C₄⋊C₄	D₆₁
hypocenter	C₆₁	#7169
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₄⋊#11758185	C₄	semidirect
8. #16839538	C₄	semidirect
119. C₆₁	#7169	semidirect

Maximal subgroups

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

subgroup	index
#21447575	2
C₄×#14519717	2
C₈×C₄⋊C₄×C₆₁	2
#7169	61