Subgroups — C₂²×#16839517 #29677364

Order 15616.

3086 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#92418	C₂	1	2	yes	maximal
2	C₂×#16839517	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄²×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×#92418	—	2	8	no	—
5	C₂³×#92418	—	2	8	no	—
6	C₂³×#92418	C₄	1	16	yes	complemented
7	C₂³×#92418	C₂²	1	12	yes	—
8	#16839517	C₂²	1	16	yes	direct product factor
9	C₂⁴×C₄×C₆₁	C₄	1	2	yes	—
10	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₁	—	2	8	no	—
12	C₂²×C₄²×C₆₁	C₄	1	6	yes	—
13	C₂²×C₄²×C₆₁	C₂²	1	6	yes	—
14	C₂²×#92418	—	2	64	no	—
15	C₂²×#92418	—	2	24	no	—
16	C₂²×#92418	—	2	24	no	—
17	C₂²×#92418	C₂×C₄	1	48	yes	complemented
18	C₂²×#92418	C₂³	1	8	yes	—
19	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₆₁	—	2	8	no	—
21	C₂³×C₄×C₆₁	—	2	8	no	—
22	C₂³×C₄×C₆₁	C₈	1	16	yes	complemented
23	C₂³×C₄×C₆₁	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₁	C₂×C₄	1	12	yes	—
25	C₂³×C₄×C₆₁	C₂×C₄	1	3	yes	—
26	C₂³×C₄×C₆₁	D₄	1	4	yes	—
27	C₂³×C₄×C₆₁	D₄	1	8	yes	—
28	C₂³×C₄×C₆₁	Q₈	1	4	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
30	C₂×C₄²×C₆₁	—	2	24	no	—
31	C₂×C₄²×C₆₁	—	2	24	no	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	12	yes	—
33	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
34	C₂⁴×C₆₁	C₄²	1	4	yes	—
35	C₂⁴×C₆₁	C₂²⋊C₄	1	4	yes	—
36	C₂⁴×C₆₁	C₄⋊C₄	1	4	yes	—
37	C₂⁴×C₆₁	C₂×C₈	1	8	yes	—
38	C₂⁴×C₆₁	M₁₆	1	8	yes	—
39	C₂⁴×C₆₁	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₆₁	—	2	64	no	—
41	C₂²×C₄×C₆₁	—	2	24	no	—
42	C₂²×C₄×C₆₁	—	2	24	no	—
43	C₂²×C₄×C₆₁	C₄²	1	12	yes	—
44	C₂²×C₄×C₆₁	C₂²⋊C₄	1	12	yes	—
45	C₂²×C₄×C₆₁	C₂²⋊C₄	1	32	yes	—
46	C₂²×C₄×C₆₁	C₄⋊C₄	1	12	yes	—
47	C₂²×C₄×C₆₁	C₄⋊C₄	1	32	yes	—
48	C₂²×C₄×C₆₁	C₂×C₈	1	48	yes	complemented
49	C₂²×C₄×C₆₁	C₂²×C₄	1	8	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₆₁	C₂×D₄	1	12	yes	—
54	C₂²×C₄×C₆₁	C₂×Q₈	1	6	yes	—
55	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₄²×C₆₁	—	2	24	no	—
57	C₄²×C₆₁	—	2	8	no	—
58	C₂×#92418	—	2	96	no	—
59	C₂×#92418	—	2	8	no	—
60	C₂×#92418	—	2	8	no	—
61	C₂×#92418	C₂²×C₄	1	16	yes	complemented
62	C₂³×C₆₁	#95	1	16	yes	—
63	C₂³×C₆₁	C₄×C₈	1	16	yes	—
64	C₂³×C₆₁	#97	1	16	yes	—
65	C₂³×C₆₁	C₂²⋊C₈	1	32	yes	—
66	C₂³×C₆₁	C₄⋊C₈	1	32	yes	—
67	C₂³×C₆₁	C₂×C₄²	1	6	yes	—
68	C₂³×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
69	C₂³×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
70	C₂³×C₆₁	C₂²×C₈	1	12	yes	—
71	C₂³×C₆₁	C₂×M₁₆	1	12	yes	—
72	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	—	2	8	no	—
74	C₂×C₄×C₆₁	—	2	8	no	—
75	C₂×C₄×C₆₁	—	2	96	no	—
76	C₂×C₄×C₆₁	#95	1	48	yes	—
77	C₂×C₄×C₆₁	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	24	yes	—
79	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
80	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	24	yes	—
81	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×C₄×C₆₁	C₂²×C₈	1	16	yes	complemented
83	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
87	#92418	—	2	16	no	—
88	C₂²×#336	—	61	1	no	maximal, Sylow
89	C₂²×C₆₁	#336	1	64	yes	complemented
90	C₂²×C₆₁	C₂×#95	1	12	yes	—
91	C₂²×C₆₁	C₂×C₄×C₈	1	12	yes	—
92	C₂²×C₆₁	C₂×#97	1	12	yes	—
93	C₂²×C₆₁	C₂×C₂²⋊C₈	1	24	yes	—
94	C₂²×C₆₁	C₂×C₄⋊C₈	1	24	yes	—
95	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
96	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₆₁	C₂³×C₈	1	2	yes	—
99	C₂²×C₆₁	C₂²×M₁₆	1	2	yes	—
100	C₄×C₆₁	—	2	16	no	—
101	C₄×C₆₁	C₂×#95	1	12	yes	—
102	C₄×C₆₁	C₂²×C₂²⋊C₄	1	2	yes	—
103	C₄×C₆₁	C₂²×C₄⋊C₄	1	2	yes	—
104	C₂⁴×C₈	—	61	2	no	—
105	C₂³×C₄²	D₆₁	1	1	yes	characteristic
106	C₂×#336	—	61	12	no	—
107	C₂×C₆₁	C₂×#336	1	24	yes	complemented
108	C₂×C₆₁	C₂²×#95	1	1	yes	characteristic
109	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
110	C₂×C₆₁	C₂²×#97	1	1	yes	characteristic
111	C₂×C₆₁	C₂²×C₂²⋊C₈	1	2	yes	—
112	C₂×C₆₁	C₂²×C₄⋊C₈	1	2	yes	—
113	C₂⁴×C₄	Dic₆₁	1	2	yes	—
114	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
115	C₂³×C₈	—	61	16	no	—
116	C₂³×C₈	—	61	12	no	—
117	C₂³×C₈	—	122	8	no	—
118	C₂³×C₈	—	122	8	no	—
119	C₂²×C₄²	—	2	8	no	—
120	C₂²×C₄²	Dic₆₁	1	6	yes	—
121	C₂²×C₄²	C₂×D₆₁	1	6	yes	—
122	#336	—	61	16	no	—
123	C₆₁	C₂²×#336	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	C₂×Dic₆₁	1	1	yes	characteristic
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	—	2	8	no	—
127	C₂³×C₄	#92418	1	16	yes	complemented
128	C₂³×C₄	Dic₁₂₂	1	4	yes	—
129	C₂³×C₄	C₄×D₆₁	1	8	yes	—
130	C₂³×C₄	D₂₄₄	1	4	yes	—
131	C₂³×C₄	C₂×Dic₆₁	1	12	yes	—
132	C₂³×C₄	C₂×Dic₆₁	1	3	yes	—
133	C₂³×C₄	C₂²⋊D₆₁	1	8	yes	—
134	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
135	C₂²×C₈	—	61	48	no	—
136	C₂²×C₈	—	61	8	no	—
137	C₂²×C₈	—	122	64	no	—
138	C₂²×C₈	—	122	24	no	—
139	C₂²×C₈	—	122	24	no	—
140	C₂×C₄²	—	2	24	no	—
141	C₂×C₄²	—	2	24	no	—
142	C₂×C₄²	C₂×Dic₆₁	1	12	yes	—
143	C₂×C₄²	C₂²×D₆₁	1	4	yes	—
144	C₂⁴	C₂×#92418	1	8	yes	—
145	C₂⁴	#11758194	1	8	yes	—
146	C₂⁴	C₄×Dic₆₁	1	4	yes	—
147	C₂⁴	C₄⋊Dic₆₁	1	4	yes	—
148	C₂⁴	C₂²⋊Dic₆₁	1	4	yes	—
149	C₂⁴	C₂²×Dic₆₁	1	3	yes	—
150	C₂²×C₄	—	2	64	no	—
151	C₂²×C₄	—	2	24	no	—
152	C₂²×C₄	—	2	24	no	—
153	C₂²×C₄	C₂×#92418	1	48	yes	complemented
154	C₂²×C₄	C₄×Dic₆₁	1	12	yes	—
155	C₂²×C₄	#11758196	1	32	yes	—
156	C₂²×C₄	C₄⋊Dic₆₁	1	12	yes	—
157	C₂²×C₄	#11758198	1	32	yes	—
158	C₂²×C₄	C₂²⋊Dic₆₁	1	12	yes	—
159	C₂²×C₄	C₂×Dic₁₂₂	1	6	yes	—
160	C₂²×C₄	C₂×C₄×D₆₁	1	12	yes	—
161	C₂²×C₄	C₂×D₂₄₄	1	6	yes	—
162	C₂²×C₄	C₂²×Dic₆₁	1	8	yes	—
163	C₂²×C₄	C₂²×Dic₆₁	1	3	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₆₁	1	12	yes	—
165	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
166	C₂×C₈	—	61	16	no	—
167	C₂×C₈	—	122	8	no	—
168	C₂×C₈	—	122	8	no	—
169	C₂×C₈	—	122	96	no	—
170	C₄²	—	2	8	no	—
171	C₄²	—	2	24	no	—
172	C₂³	#14519684	1	16	yes	—
173	C₂³	C₄×#92418	1	16	yes	—
174	C₂³	#14519688	1	16	yes	—
175	C₂³	C₂²⋊#92418	1	32	yes	—
176	C₂³	C₄⋊#92418	1	32	yes	—
177	C₂³	C₂×C₄×Dic₆₁	1	6	yes	—
178	C₂³	C₂×C₂²⋊Dic₆₁	1	6	yes	—
179	C₂³	C₂×C₄⋊Dic₆₁	1	6	yes	—
180	C₂³	C₂²×#92418	1	12	yes	—
181	C₂³	C₂×#11758194	1	12	yes	—
182	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	96	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	#14519684	1	48	yes	—
187	C₂×C₄	C₂×C₄×Dic₆₁	1	6	yes	—
188	C₂×C₄	C₂×#11758198	1	24	yes	—
189	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	6	yes	—
190	C₂×C₄	C₂×#11758196	1	24	yes	—
191	C₂×C₄	C₂×C₄⋊Dic₆₁	1	6	yes	—
192	C₂×C₄	C₂²×#92418	1	16	yes	complemented
193	C₂×C₄	C₂²×C₄×D₆₁	1	2	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₆₁	1	2	yes	—
195	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
197	C₈	—	122	16	no	—
198	C₂²	#16839517	1	64	yes	direct product factor
199	C₂²	C₂×#14519684	1	12	yes	—
200	C₂²	C₂×C₄×#92418	1	12	yes	—
201	C₂²	C₂×#14519688	1	12	yes	—
202	C₂²	C₂×C₂²⋊#92418	1	24	yes	—
203	C₂²	C₂×C₄⋊#92418	1	24	yes	—
204	C₂²	C₂²×C₄×Dic₆₁	1	1	yes	characteristic
205	C₂²	C₂²×C₂²⋊Dic₆₁	1	1	yes	characteristic
206	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
207	C₂²	C₂³×#92418	1	2	yes	—
208	C₂²	C₂²×#11758194	1	2	yes	—
209	C₄	—	2	16	no	—
210	C₄	C₂×#14519684	1	12	yes	—
211	C₄	C₂²×#11758198	1	2	yes	—
212	C₄	C₂²×#11758196	1	2	yes	—
213	C₂	C₂×#16839517	1	24	yes	direct product factor
214	C₂	C₂²×#14519684	1	1	yes	characteristic
215	C₂	C₂²×C₄×#92418	1	1	yes	characteristic
216	C₂	C₂²×#14519688	1	1	yes	characteristic
217	C₂	C₂²×C₂²⋊#92418	1	2	yes	—
218	C₂	C₂²×C₄⋊#92418	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₆₁
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂³×C₄²×C₆₁	C₂
socle, prime radical	C₂⁵×C₆₁	C₂×C₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂³×C₄²	D₆₁
hypocenter	C₆₁	C₂²×#336
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#16839517	C₂	direct (N × H)
6. C₂³×#92418	C₄	semidirect
8. #16839517	C₂²	direct (N × H)
17. C₂²×#92418	C₂×C₄	semidirect
22. C₂³×C₄×C₆₁	C₈	semidirect
48. C₂²×C₄×C₆₁	C₂×C₈	semidirect
61. C₂×#92418	C₂²×C₄	semidirect
82. C₂×C₄×C₆₁	C₂²×C₈	semidirect
89. C₂²×C₆₁	#336	semidirect
107. C₂×C₆₁	C₂×#336	semidirect
123. C₆₁	C₂²×#336	semidirect
127. C₂³×C₄	#92418	semidirect
153. C₂²×C₄	C₂×#92418	semidirect
192. C₂×C₄	C₂²×#92418	semidirect
198. C₂²	#16839517	direct (N × H)
213. C₂	C₂×#16839517	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#92418	2
C₂×#16839517	2
C₂³×C₄²×C₆₁	2
C₂²×#336	61

Subgroups — C22×#16839517 #29677364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16839517 #29677364