Subgroups — C₂³×#14519697 #30543520

Order 15616.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#92418	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14519697	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₆₁	—	2	1	no	—
6	C₂³×D₂₄₄	—	2	7	no	—
7	C₂³×D₂₄₄	—	2	8	no	—
8	C₂³×D₂₄₄	C₄	1	16	yes	complemented
9	C₂³×D₂₄₄	C₂²	1	14	yes	—
10	C₂³×#92418	—	2	8	no	—
11	C₂³×#92418	C₂²	1	14	yes	complemented
12	C₂×#14519697	C₂²	1	112	yes	direct product factor
13	C₂⁴×C₄×C₆₁	—	2	1	no	—
14	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₁	—	2	7	no	—
16	C₂²×C₄⋊C₄×C₆₁	C₂²	1	14	yes	complemented
17	C₂⁴×D₆₁	—	2	16	no	—
18	C₂⁴×D₆₁	—	2	14	no	—
19	C₂⁴×D₆₁	—	4	8	no	—
20	C₂⁴×D₆₁	—	4	8	no	—
21	C₂²×D₂₄₄	—	2	112	no	—
22	C₂²×D₂₄₄	—	2	56	no	—
23	C₂²×D₂₄₄	—	2	42	no	—
24	C₂²×D₂₄₄	C₂×C₄	1	112	yes	complemented
25	C₂²×D₂₄₄	C₂³	1	28	yes	—
26	C₂²×#92418	—	2	56	no	—
27	C₂²×#92418	C₂³	1	28	yes	complemented
28	#14519697	C₂³	1	64	yes	direct product factor
29	C₂⁵×C₆₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₁	—	2	14	no	—
31	C₂³×C₄×C₆₁	—	4	8	no	—
32	C₂³×C₄×C₆₁	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₆₁	D₄	1	8	yes	—
34	C₂³×C₄×C₆₁	D₄	1	7	yes	—
35	C₂³×C₄×C₆₁	C₂³	1	7	yes	—
36	C₂×C₄⋊C₄×C₆₁	—	2	42	no	—
37	C₂×C₄⋊C₄×C₆₁	C₂³	1	28	yes	complemented
38	C₂⁴×C₆₁	C₂²⋊C₄	1	8	yes	—
39	C₂⁴×C₆₁	D₈	1	8	yes	—
40	C₂⁴×C₆₁	SD₁₆	1	8	yes	—
41	C₂⁴×C₆₁	C₂×D₄	1	7	yes	—
42	C₂³×D₆₁	—	2	112	no	—
43	C₂³×D₆₁	—	2	28	no	—
44	C₂³×D₆₁	—	4	128	no	—
45	C₂³×D₆₁	—	4	56	no	—
46	C₂³×D₆₁	—	4	56	no	—
47	C₂²×C₄×C₆₁	—	2	28	no	—
48	C₂²×C₄×C₆₁	—	4	56	no	—
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	56	yes	—
50	C₂²×C₄×C₆₁	C₂²×C₄	1	28	yes	—
51	C₂²×C₄×C₆₁	C₂×D₄	1	21	yes	—
52	C₂²×C₄×C₆₁	C₂×D₄	1	28	yes	—
53	C₂²×C₄×C₆₁	C₂⁴	1	7	yes	—
54	C₂×D₂₄₄	—	2	336	no	—
55	C₂×D₂₄₄	—	2	28	no	—
56	C₂×D₂₄₄	—	2	56	no	—
57	C₂×D₂₄₄	C₂²×C₄	1	112	yes	complemented
58	C₂×D₂₄₄	C₂⁴	1	8	yes	—
59	C₄⋊C₄×C₆₁	—	2	28	no	—
60	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	complemented
61	C₂×#92418	—	2	56	no	—
62	C₂×#92418	C₂⁴	1	8	yes	complemented
63	C₂³×C₆₁	#102	1	64	yes	complemented
64	C₂³×C₆₁	C₂×C₂²⋊C₄	1	28	yes	—
65	C₂³×C₆₁	C₂×D₈	1	28	yes	—
66	C₂³×C₆₁	C₂×SD₁₆	1	28	yes	—
67	C₂³×C₆₁	C₂²×D₄	1	7	yes	—
68	C₂²×D₆₁	—	2	112	no	—
69	C₂²×D₆₁	—	2	8	no	—
70	C₂²×D₆₁	—	4	56	no	—
71	C₂²×D₆₁	—	4	56	no	—
72	C₂²×D₆₁	—	4	448	no	—
73	C₂×C₄×C₆₁	—	2	8	no	—
74	C₂×C₄×C₆₁	—	4	56	no	—
75	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	84	yes	—
76	C₂×C₄×C₆₁	C₂³×C₄	1	14	yes	—
77	C₂×C₄×C₆₁	C₂²×D₄	1	7	yes	—
78	C₂×C₄×C₆₁	C₂²×D₄	1	14	yes	—
79	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
80	D₂₄₄	—	2	112	no	—
81	D₂₄₄	—	2	8	no	—
82	D₂₄₄	C₂³×C₄	1	16	yes	complemented
83	#92418	—	2	8	no	—
84	C₂³×#102	—	61	1	no	maximal, Sylow
85	C₂²×C₆₁	C₂×#102	1	112	yes	complemented
86	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	14	yes	—
87	C₂²×C₆₁	C₂²×D₈	1	14	yes	—
88	C₂²×C₆₁	C₂²×SD₁₆	1	14	yes	—
89	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
90	C₂×D₆₁	—	2	16	no	—
91	C₂×D₆₁	—	4	224	no	—
92	C₂×D₆₁	—	4	8	no	—
93	C₂×D₆₁	—	4	8	no	—
94	C₄×C₆₁	—	4	8	no	—
95	C₄×C₆₁	C₂²×C₂²⋊C₄	1	14	yes	—
96	C₄×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
97	C₄×C₆₁	C₂³×D₄	1	1	yes	characteristic
98	C₂⁴×D₄	—	61	1	no	—
99	C₂⁴×C₈	—	61	1	no	—
100	C₂³×C₄⋊C₄	D₆₁	1	1	yes	characteristic, complemented
101	C₂²×#102	—	61	28	no	—
102	C₂×C₆₁	C₂²×#102	1	28	yes	complemented
103	C₂×C₆₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₆₁	C₂³×D₈	1	1	yes	characteristic
105	C₂×C₆₁	C₂³×SD₁₆	1	1	yes	characteristic
106	D₆₁	—	4	16	no	—
107	C₂⁶	—	122	1	no	—
108	C₂³×D₄	—	61	16	no	—
109	C₂³×D₄	—	61	14	no	—
110	C₂³×D₄	—	122	8	no	—
111	C₂³×D₄	—	122	7	no	—
112	C₂⁴×C₄	—	2	1	no	—
113	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
114	C₂³×C₈	—	61	14	no	—
115	C₂³×C₈	—	122	8	no	—
116	C₂²×C₄⋊C₄	—	2	7	no	—
117	C₂²×C₄⋊C₄	C₂×D₆₁	1	14	yes	complemented
118	C₂×#102	—	61	112	no	—
119	C₆₁	C₂³×#102	1	1	yes	characteristic, complemented, Sylow
120	C₂⁵	—	122	16	no	—
121	C₂⁵	—	122	14	no	—
122	C₂⁵	—	244	8	no	—
123	C₂⁵	—	244	8	no	—
124	C₂⁵	C₂²⋊D₆₁	1	1	yes	characteristic
125	C₂²×D₄	—	61	112	no	—
126	C₂²×D₄	—	61	28	no	—
127	C₂²×D₄	—	122	112	no	—
128	C₂²×D₄	—	122	56	no	—
129	C₂²×D₄	—	122	42	no	—
130	C₂³×C₄	—	2	14	no	—
131	C₂³×C₄	—	4	8	no	—
132	C₂³×C₄	C₄×D₆₁	1	8	yes	—
133	C₂³×C₄	D₂₄₄	1	8	yes	—
134	C₂³×C₄	C₂²⋊D₆₁	1	7	yes	—
135	C₂³×C₄	C₂²×D₆₁	1	7	yes	—
136	C₂²×C₈	—	61	28	no	—
137	C₂²×C₈	—	122	56	no	—
138	C₂×C₄⋊C₄	—	2	42	no	—
139	C₂×C₄⋊C₄	C₂²×D₆₁	1	28	yes	complemented
140	#102	—	61	64	no	—
141	C₂⁴	—	122	28	no	—
142	C₂⁴	—	122	112	no	—
143	C₂⁴	—	244	128	no	—
144	C₂⁴	—	244	56	no	—
145	C₂⁴	—	244	56	no	—
146	C₂⁴	#11758198	1	8	yes	—
147	C₂⁴	#11758199	1	8	yes	—
148	C₂⁴	#11758201	1	8	yes	—
149	C₂⁴	C₂×C₂²⋊D₆₁	1	7	yes	—
150	C₂×D₄	—	61	112	no	—
151	C₂×D₄	—	61	8	no	—
152	C₂×D₄	—	122	28	no	—
153	C₂×D₄	—	122	56	no	—
154	C₂×D₄	—	122	336	no	—
155	C₂²×C₄	—	2	28	no	—
156	C₂²×C₄	—	4	56	no	—
157	C₂²×C₄	#11758198	1	56	yes	—
158	C₂²×C₄	C₂×C₄×D₆₁	1	28	yes	—
159	C₂²×C₄	C₂×D₂₄₄	1	28	yes	—
160	C₂²×C₄	C₂×C₂²⋊D₆₁	1	21	yes	—
161	C₂²×C₄	C₂³×D₆₁	1	7	yes	—
162	C₂×C₈	—	61	8	no	—
163	C₂×C₈	—	122	56	no	—
164	C₄⋊C₄	—	2	28	no	—
165	C₄⋊C₄	C₂³×D₆₁	1	8	yes	complemented
166	C₂³	—	122	112	no	—
167	C₂³	—	122	8	no	—
168	C₂³	—	244	448	no	—
169	C₂³	—	244	56	no	—
170	C₂³	—	244	56	no	—
171	C₂³	#14519697	1	64	yes	direct product factor
172	C₂³	C₂×#11758198	1	28	yes	—
173	C₂³	C₂×#11758199	1	28	yes	—
174	C₂³	C₂×#11758201	1	28	yes	—
175	C₂³	C₂²×C₂²⋊D₆₁	1	7	yes	—
176	D₄	—	61	16	no	—
177	D₄	—	122	112	no	—
178	D₄	—	122	8	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	4	56	no	—
181	C₂×C₄	C₂×#11758198	1	84	yes	—
182	C₂×C₄	C₂²×C₄×D₆₁	1	14	yes	—
183	C₂×C₄	C₂²×C₂²⋊D₆₁	1	7	yes	—
184	C₂×C₄	C₂²×D₂₄₄	1	14	yes	—
185	C₂×C₄	C₂⁴×D₆₁	1	1	yes	characteristic
186	C₈	—	122	8	no	—
187	C₂²	—	122	16	no	—
188	C₂²	—	244	8	no	—
189	C₂²	—	244	8	no	—
190	C₂²	—	244	224	no	—
191	C₂²	C₂×#14519697	1	112	yes	direct product factor
192	C₂²	C₂²×#11758198	1	14	yes	—
193	C₂²	C₂²×#11758199	1	14	yes	—
194	C₂²	C₂²×#11758201	1	14	yes	—
195	C₂²	C₂³×C₂²⋊D₆₁	1	1	yes	characteristic
196	C₄	—	4	8	no	—
197	C₄	C₂²×#11758198	1	14	yes	—
198	C₄	C₂³×C₄×D₆₁	1	1	yes	characteristic
199	C₄	C₂³×D₂₄₄	1	1	yes	characteristic
200	C₂	—	244	16	no	—
201	C₂	C₂²×#14519697	1	28	yes	direct product factor
202	C₂	C₂³×#11758198	1	1	yes	characteristic
203	C₂	C₂³×#11758199	1	1	yes	characteristic
204	C₂	C₂³×#11758201	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₁
Fitting	C₂³×C₄⋊C₄×C₆₁	C₂
socle	C₂⁵×C₆₁	D₄
prime radical	C₂⁴×D₂₄₄	C₂
cosocle	C₂×C₄×C₆₁	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₆₁
hypocenter	C₆₁	C₂³×#102
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×#92418	C₂	semidirect
3. C₂²×#14519697	C₂	direct (N × H)
4. C₂³×C₄⋊C₄×C₆₁	C₂	semidirect
8. C₂³×D₂₄₄	C₄	semidirect
11. C₂³×#92418	C₂²	semidirect
12. C₂×#14519697	C₂²	direct (N × H)
16. C₂²×C₄⋊C₄×C₆₁	C₂²	semidirect
24. C₂²×D₂₄₄	C₂×C₄	semidirect
27. C₂²×#92418	C₂³	semidirect
28. #14519697	C₂³	direct (N × H)
37. C₂×C₄⋊C₄×C₆₁	C₂³	semidirect
57. C₂×D₂₄₄	C₂²×C₄	semidirect
60. C₄⋊C₄×C₆₁	C₂⁴	semidirect
62. C₂×#92418	C₂⁴	semidirect
63. C₂³×C₆₁	#102	semidirect
82. D₂₄₄	C₂³×C₄	semidirect
85. C₂²×C₆₁	C₂×#102	semidirect
100. C₂³×C₄⋊C₄	D₆₁	semidirect
102. C₂×C₆₁	C₂²×#102	semidirect
117. C₂²×C₄⋊C₄	C₂×D₆₁	semidirect
119. C₆₁	C₂³×#102	semidirect
139. C₂×C₄⋊C₄	C₂²×D₆₁	semidirect
165. C₄⋊C₄	C₂³×D₆₁	semidirect
171. C₂³	#14519697	direct (N × H)
191. C₂²	C₂×#14519697	direct (N × H)
201. C₂	C₂²×#14519697	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×D₂₄₄	2
C₂⁴×#92418	2
C₂²×#14519697	2
C₂³×C₄⋊C₄×C₆₁	2
C₂³×#102	61

Subgroups — C23×#14519697 #30543520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519697 #30543520