Subgroups — C₂²×C₁₃×#392 #15720870

Order 3328.

13646 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	C₂⁴×D₄×C₁₃	C₂	1	3	yes	maximal, complemented
2	C₂³×C₁₃×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂×C₁₃×#392	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₁₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁶×C₁₃	C₂²	1	3	yes	—
6	C₂³×D₄×C₁₃	—	2	48	no	—
7	C₂³×D₄×C₁₃	—	2	24	no	—
8	C₂³×D₄×C₁₃	C₂²	1	36	yes	complemented
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
11	C₂²×C₁₃×C₂²⋊C₄	—	2	24	no	—
12	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	36	yes	complemented
13	C₁₃×#392	C₂²	1	64	yes	direct product factor
14	C₂×C₁₃×#95	C₂²	1	12	yes	complemented
15	C₂⁵×C₁₃	—	2	12	no	—
16	C₂⁵×C₁₃	—	2	24	no	—
17	C₂⁵×C₁₃	—	2	24	no	—
18	C₂⁵×C₁₃	—	2	12	no	—
19	C₂⁵×C₁₃	D₄	1	24	yes	complemented
20	C₂⁵×C₁₃	C₂³	1	18	yes	—
21	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×D₄×C₁₃	—	2	384	no	—
23	C₂²×D₄×C₁₃	—	2	288	no	—
24	C₂²×D₄×C₁₃	—	2	144	no	—
25	C₂²×D₄×C₁₃	C₂³	1	48	yes	complemented
26	C₂³×C₄×C₁₃	—	2	24	no	—
27	C₂³×C₄×C₁₃	—	2	12	no	—
28	C₂³×C₄×C₁₃	D₄	1	24	yes	complemented
29	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	18	yes	—
31	C₂×C₁₃×C₂²⋊C₄	—	2	144	no	—
32	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	complemented
33	C₁₃×#95	C₂³	1	16	yes	complemented
34	C₂²×#392	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂⁴×C₁₃	—	2	192	no	—
36	C₂⁴×C₁₃	—	2	96	no	—
37	C₂⁴×C₁₃	—	2	36	no	—
38	C₂⁴×C₁₃	—	2	72	no	—
39	C₂⁴×C₁₃	—	2	72	no	—
40	C₂⁴×C₁₃	—	2	36	no	—
41	C₂⁴×C₁₃	—	4	48	no	—
42	C₂⁴×C₁₃	—	4	96	no	—
43	C₂⁴×C₁₃	—	4	48	no	—
44	C₂⁴×C₁₃	C₂×D₄	1	12	yes	—
45	C₂⁴×C₁₃	C₂×D₄	1	12	yes	—
46	C₂⁴×C₁₃	C₂×D₄	1	72	yes	complemented
47	C₂⁴×C₁₃	#41	1	4	yes	—
48	C₂⁴×C₁₃	C₂⁴	1	12	yes	—
49	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
50	C₂×D₄×C₁₃	—	2	1152	no	—
51	C₂×D₄×C₁₃	—	2	192	no	—
52	C₂×D₄×C₁₃	—	2	96	no	—
53	C₂²×C₄×C₁₃	—	2	96	no	—
54	C₂²×C₄×C₁₃	—	2	72	no	—
55	C₂²×C₄×C₁₃	—	2	36	no	—
56	C₂²×C₄×C₁₃	—	4	16	no	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	72	yes	complemented
58	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
60	C₁₃×C₂²⋊C₄	—	2	96	no	—
61	C₂⁴×D₄	C₂×C₁₃	1	3	yes	complemented
62	C₂³×C₂²⋊C₄	C₂×C₁₃	1	3	yes	complemented
63	C₂×#392	C₂×C₁₃	1	24	yes	direct product factor
64	C₂²×#95	C₂×C₁₃	1	1	yes	characteristic, complemented
65	C₂³×C₁₃	—	2	288	no	—
66	C₂³×C₁₃	—	2	144	no	—
67	C₂³×C₁₃	—	2	12	no	—
68	C₂³×C₁₃	—	2	24	no	—
69	C₂³×C₁₃	—	2	24	no	—
70	C₂³×C₁₃	—	2	12	no	—
71	C₂³×C₁₃	—	4	72	no	—
72	C₂³×C₁₃	—	4	144	no	—
73	C₂³×C₁₃	—	4	72	no	—
74	C₂³×C₁₃	—	4	384	no	—
75	C₂³×C₁₃	#120	1	48	yes	—
76	C₂³×C₁₃	#121	1	48	yes	—
77	C₂³×C₁₃	#127	1	16	yes	—
78	C₂³×C₁₃	C₂²×D₄	1	18	yes	—
79	C₂³×C₁₃	C₂²×D₄	1	18	yes	—
80	C₂³×C₁₃	C₂²×D₄	1	24	yes	complemented
81	C₂³×C₁₃	C₂×#41	1	6	yes	—
82	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
83	D₄×C₁₃	—	2	384	no	—
84	C₂×C₄×C₁₃	—	2	144	no	—
85	C₂×C₄×C₁₃	—	2	24	no	—
86	C₂×C₄×C₁₃	—	2	12	no	—
87	C₂×C₄×C₁₃	—	4	24	no	—
88	C₂×C₄×C₁₃	C₂²×D₄	1	24	yes	complemented
89	C₂⁶	C₂²×C₁₃	1	3	yes	—
90	C₂³×D₄	—	2	48	no	—
91	C₂³×D₄	—	2	24	no	—
92	C₂³×D₄	C₂²×C₁₃	1	36	yes	complemented
93	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
94	C₂⁴×C₄	C₂²×C₁₃	1	3	yes	—
95	C₂²×C₂²⋊C₄	—	2	24	no	—
96	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	36	yes	complemented
97	#392	C₂²×C₁₃	1	64	yes	direct product factor
98	C₂×#95	C₂²×C₁₃	1	12	yes	complemented
99	C₂²×C₁₃	—	2	48	no	—
100	C₂²×C₁₃	—	2	24	no	—
101	C₂²×C₁₃	—	4	288	no	—
102	C₂²×C₁₃	—	4	12	no	—
103	C₂²×C₁₃	—	4	24	no	—
104	C₂²×C₁₃	—	4	12	no	—
105	C₂²×C₁₃	#392	1	64	yes	direct product factor
106	C₂²×C₁₃	C₂×#120	1	36	yes	—
107	C₂²×C₁₃	C₂×#121	1	36	yes	—
108	C₂²×C₁₃	C₂×#127	1	12	yes	—
109	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
110	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
111	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
112	C₄×C₁₃	—	2	24	no	—
113	C₄×C₁₃	—	4	4	no	—
114	C₂⁵	—	2	12	no	—
115	C₂⁵	—	2	24	no	—
116	C₂⁵	—	2	24	no	—
117	C₂⁵	—	2	12	no	—
118	C₂⁵	D₄×C₁₃	1	24	yes	complemented
119	C₂⁵	C₂³×C₁₃	1	18	yes	—
120	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
121	C₂²×D₄	—	2	384	no	—
122	C₂²×D₄	—	2	288	no	—
123	C₂²×D₄	—	2	144	no	—
124	C₂²×D₄	C₂³×C₁₃	1	48	yes	complemented
125	C₂³×C₄	—	2	24	no	—
126	C₂³×C₄	—	2	12	no	—
127	C₂³×C₄	D₄×C₁₃	1	24	yes	complemented
128	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
129	C₂³×C₄	C₂³×C₁₃	1	18	yes	—
130	C₂×C₂²⋊C₄	—	2	144	no	—
131	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	complemented
132	#95	C₂³×C₁₃	1	16	yes	complemented
133	C₂×C₁₃	—	4	24	no	—
134	C₂×C₁₃	C₂×#392	1	24	yes	direct product factor
135	C₂×C₁₃	C₂²×#120	1	3	yes	—
136	C₂×C₁₃	C₂²×#121	1	3	yes	—
137	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
138	C₂⁴	—	2	192	no	—
139	C₂⁴	—	2	96	no	—
140	C₂⁴	—	2	36	no	—
141	C₂⁴	—	2	72	no	—
142	C₂⁴	—	2	72	no	—
143	C₂⁴	—	2	36	no	—
144	C₂⁴	—	4	48	no	—
145	C₂⁴	—	4	96	no	—
146	C₂⁴	—	4	48	no	—
147	C₂⁴	C₂×D₄×C₁₃	1	12	yes	—
148	C₂⁴	C₂×D₄×C₁₃	1	12	yes	—
149	C₂⁴	C₂×D₄×C₁₃	1	72	yes	complemented
150	C₂⁴	C₁₃×#41	1	4	yes	—
151	C₂⁴	C₂⁴×C₁₃	1	12	yes	—
152	C₂⁴	C₂⁴×C₁₃	1	3	yes	—
153	C₂×D₄	—	2	1152	no	—
154	C₂×D₄	—	2	192	no	—
155	C₂×D₄	—	2	96	no	—
156	C₂²×C₄	—	2	96	no	—
157	C₂²×C₄	—	2	72	no	—
158	C₂²×C₄	—	2	36	no	—
159	C₂²×C₄	—	4	16	no	—
160	C₂²×C₄	C₂×D₄×C₁₃	1	72	yes	complemented
161	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
162	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
163	C₂²⋊C₄	—	2	96	no	—
164	C₁₃	C₂²×#392	1	1	yes	characteristic, direct product factor, Sylow
165	C₂³	—	2	288	no	—
166	C₂³	—	2	144	no	—
167	C₂³	—	2	12	no	—
168	C₂³	—	2	24	no	—
169	C₂³	—	2	24	no	—
170	C₂³	—	2	12	no	—
171	C₂³	—	4	384	no	—
172	C₂³	—	4	72	no	—
173	C₂³	—	4	144	no	—
174	C₂³	—	4	72	no	—
175	C₂³	C₁₃×#120	1	48	yes	—
176	C₂³	C₁₃×#121	1	48	yes	—
177	C₂³	C₁₃×#127	1	16	yes	—
178	C₂³	C₂²×D₄×C₁₃	1	18	yes	—
179	C₂³	C₂²×D₄×C₁₃	1	18	yes	—
180	C₂³	C₂²×D₄×C₁₃	1	24	yes	complemented
181	C₂³	C₂×C₁₃×#41	1	6	yes	—
182	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
183	D₄	—	2	384	no	—
184	C₂×C₄	—	2	144	no	—
185	C₂×C₄	—	2	24	no	—
186	C₂×C₄	—	2	12	no	—
187	C₂×C₄	—	4	24	no	—
188	C₂×C₄	C₂²×D₄×C₁₃	1	24	yes	complemented
189	C₂²	—	2	48	no	—
190	C₂²	—	2	24	no	—
191	C₂²	—	4	288	no	—
192	C₂²	—	4	12	no	—
193	C₂²	—	4	24	no	—
194	C₂²	—	4	12	no	—
195	C₂²	C₁₃×#392	1	64	yes	direct product factor
196	C₂²	C₂×C₁₃×#120	1	36	yes	—
197	C₂²	C₂×C₁₃×#121	1	36	yes	—
198	C₂²	C₂×C₁₃×#127	1	12	yes	—
199	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
200	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
201	C₂²	C₂²×C₁₃×#41	1	1	yes	characteristic
202	C₄	—	2	24	no	—
203	C₄	—	4	4	no	—
204	C₂	—	4	24	no	—
205	C₂	C₂×C₁₃×#392	1	24	yes	direct product factor
206	C₂	C₂²×C₁₃×#120	1	3	yes	—
207	C₂	C₂²×C₁₃×#121	1	3	yes	—
208	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₁₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₄×C₁₃	C₂	semidirect
2. C₂³×C₁₃×C₂²⋊C₄	C₂	semidirect
3. C₂×C₁₃×#392	C₂	direct (N × H)
4. C₂²×C₁₃×#95	C₂	semidirect
8. C₂³×D₄×C₁₃	C₂²	semidirect
12. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
13. C₁₃×#392	C₂²	direct (N × H)
14. C₂×C₁₃×#95	C₂²	semidirect
19. C₂⁵×C₁₃	D₄	semidirect
25. C₂²×D₄×C₁₃	C₂³	semidirect
28. C₂³×C₄×C₁₃	D₄	semidirect
32. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
33. C₁₃×#95	C₂³	semidirect
34. C₂²×#392	C₁₃	direct (N × H)
46. C₂⁴×C₁₃	C₂×D₄	semidirect
57. C₂²×C₄×C₁₃	C₂×D₄	semidirect
61. C₂⁴×D₄	C₂×C₁₃	semidirect
62. C₂³×C₂²⋊C₄	C₂×C₁₃	semidirect
63. C₂×#392	C₂×C₁₃	direct (N × H)
64. C₂²×#95	C₂×C₁₃	semidirect
80. C₂³×C₁₃	C₂²×D₄	semidirect
88. C₂×C₄×C₁₃	C₂²×D₄	semidirect
92. C₂³×D₄	C₂²×C₁₃	semidirect
96. C₂²×C₂²⋊C₄	C₂²×C₁₃	semidirect
97. #392	C₂²×C₁₃	direct (N × H)
98. C₂×#95	C₂²×C₁₃	semidirect
105. C₂²×C₁₃	#392	direct (N × H)
118. C₂⁵	D₄×C₁₃	semidirect
124. C₂²×D₄	C₂³×C₁₃	semidirect
127. C₂³×C₄	D₄×C₁₃	semidirect
131. C₂×C₂²⋊C₄	C₂³×C₁₃	semidirect
132. #95	C₂³×C₁₃	semidirect
134. C₂×C₁₃	C₂×#392	direct (N × H)
149. C₂⁴	C₂×D₄×C₁₃	semidirect
160. C₂²×C₄	C₂×D₄×C₁₃	semidirect
164. C₁₃	C₂²×#392	direct (N × H)
180. C₂³	C₂²×D₄×C₁₃	semidirect
188. C₂×C₄	C₂²×D₄×C₁₃	semidirect
195. C₂²	C₁₃×#392	direct (N × H)
205. C₂	C₂×C₁₃×#392	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×D₄×C₁₃	2
C₂³×C₁₃×C₂²⋊C₄	2
C₂×C₁₃×#392	2
C₂²×C₁₃×#95	2
C₂²×#392	13

Subgroups — C22×C13×#392 #15720870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#392 #15720870