Subgroups — C₂×C₁₃×#2248 #15720050

Order 3328.

918 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₃₂	C₂	1	4	yes	maximal, complemented
2	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2248	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×Q₁₆	—	2	1	no	—
7	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₃×D₈	—	2	1	no	—
9	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₁₃×#127	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	—	2	16	no	—
13	C₂×C₁₃×SD₃₂	C₂²	1	16	yes	complemented
14	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
15	C₁₃×#494	—	2	2	no	—
16	C₁₃×#494	C₂²	1	4	yes	complemented
17	C₁₃×#493	—	2	2	no	—
18	C₁₃×#493	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₄×C₁₃×C₁₆	C₂²	1	4	yes	complemented
21	C₂²×Q₈×C₁₃	—	2	2	no	—
22	C₂²×D₄×C₁₃	—	2	2	no	—
23	C₂²×D₄×C₁₃	—	4	2	no	—
24	C₂×C₁₃×Q₁₆	—	2	8	no	—
25	C₂×C₁₃×Q₁₆	—	2	2	no	—
26	C₂×C₁₃×Q₁₆	—	2	4	no	—
27	C₂×C₁₃×Q₁₆	—	4	4	no	—
28	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
29	C₂×C₁₃×D₈	—	2	8	no	—
30	C₂×C₁₃×D₈	—	2	2	no	—
31	C₂×C₁₃×D₈	—	2	4	no	—
32	C₂×C₁₃×D₈	—	4	4	no	—
33	C₂×C₁₃×D₈	C₂³	1	4	yes	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	—	2	4	no	—
37	C₁₃×C₄⋊Q₈	—	4	2	no	—
38	C₁₃×#127	—	2	4	no	—
39	C₁₃×#127	—	4	2	no	—
40	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₁₃×SD₃₂	—	2	32	no	—
43	C₂×C₁₃×C₁₆	D₄	1	8	yes	complemented
44	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
45	C₄×C₈×C₁₃	D₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂³	1	2	yes	—
47	C₂×#2248	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₃	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	2	4	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	4	no	—
52	C₂×Q₈×C₁₃	—	4	4	no	—
53	C₂×D₄×C₁₃	—	2	4	no	—
54	C₂×D₄×C₁₃	—	4	4	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	—	4	8	no	—
57	C₂×D₄×C₁₃	—	4	2	no	—
58	C₂×D₄×C₁₃	—	8	8	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	D₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₁₃×Q₁₆	—	2	8	no	—
63	C₁₃×Q₁₆	—	4	4	no	—
64	C₁₃×Q₁₆	—	4	8	no	—
65	C₁₃×D₈	—	2	8	no	—
66	C₁₃×D₈	—	4	4	no	—
67	C₁₃×D₈	—	4	8	no	—
68	C₂×C₈×C₁₃	D₈	1	4	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	—	4	8	no	—
74	C₄²×C₁₃	D₈	1	2	yes	—
75	C₄²×C₁₃	C₂×D₄	1	2	yes	—
76	C₁₃×C₁₆	C₂×D₄	1	8	yes	complemented
77	C₂²×SD₃₂	C₂×C₁₃	1	4	yes	complemented
78	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
79	C₂×#493	C₂×C₁₃	1	1	yes	characteristic
80	#2248	C₂×C₁₃	1	8	yes	direct product factor
81	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
82	C₂³×C₁₃	—	4	4	no	—
83	C₂³×C₁₃	—	8	4	no	—
84	C₂³×C₁₃	—	8	8	no	—
85	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
86	Q₈×C₁₃	—	4	4	no	—
87	Q₈×C₁₃	—	4	4	no	—
88	Q₈×C₁₃	—	8	4	no	—
89	D₄×C₁₃	—	4	4	no	—
90	D₄×C₁₃	—	4	4	no	—
91	D₄×C₁₃	—	8	4	no	—
92	D₄×C₁₃	—	8	16	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	—	8	4	no	—
95	C₂×C₄×C₁₃	SD₃₂	1	8	yes	complemented
96	C₂×C₄×C₁₃	#127	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
100	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₁₃	#127	1	2	yes	—
102	C₈×C₁₃	C₂×D₈	1	4	yes	—
103	C₈×C₁₃	C₂²×D₄	1	2	yes	—
104	C₂²×Q₁₆	—	2	1	no	—
105	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
106	C₂²×D₈	—	2	1	no	—
107	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
108	C₂×C₄⋊Q₈	—	2	1	no	—
109	C₂×#127	—	2	1	no	—
110	C₂×SD₃₂	—	2	16	no	—
111	C₂×SD₃₂	C₂²×C₁₃	1	16	yes	complemented
112	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
113	#494	—	2	2	no	—
114	#494	C₂²×C₁₃	1	4	yes	complemented
115	#493	—	2	2	no	—
116	#493	C₂²×C₁₃	1	4	yes	—
117	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
118	C₄×C₁₆	C₂²×C₁₃	1	4	yes	complemented
119	C₂²×C₁₃	—	8	4	no	—
120	C₂²×C₁₃	—	8	8	no	—
121	C₂²×C₁₃	—	16	8	no	—
122	C₂²×C₁₃	#493	1	2	yes	—
123	C₂²×C₁₃	C₂×SD₃₂	1	4	yes	—
124	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₁₃	—	8	4	no	—
126	C₄×C₁₃	#493	1	2	yes	—
127	C₄×C₁₃	C₂×SD₃₂	1	8	yes	complemented
128	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
129	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×D₄	—	2	2	no	—
132	C₂²×D₄	—	4	2	no	—
133	C₂×Q₁₆	—	2	8	no	—
134	C₂×Q₁₆	—	2	4	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
138	C₂×D₈	—	2	8	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂×D₈	—	4	4	no	—
142	C₂×D₈	C₂³×C₁₃	1	4	yes	—
143	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
145	C₄⋊Q₈	—	2	4	no	—
146	C₄⋊Q₈	—	4	2	no	—
147	#127	—	2	4	no	—
148	#127	—	4	2	no	—
149	C₂×C₄⋊C₄	—	4	2	no	—
150	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
151	SD₃₂	—	2	32	no	—
152	C₂×C₁₆	D₄×C₁₃	1	8	yes	complemented
153	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
154	C₄×C₈	D₄×C₁₃	1	2	yes	—
155	C₄×C₈	C₂³×C₁₃	1	2	yes	—
156	C₂×C₁₃	—	16	4	no	—
157	C₂×C₁₃	#2248	1	4	yes	direct product factor
158	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
159	C₂×C₁₃	C₂²×SD₃₂	1	2	yes	—
160	C₂⁴	—	4	2	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	4	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	8	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	8	8	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₁₃×D₈	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
174	Q₁₆	—	2	8	no	—
175	Q₁₆	—	4	4	no	—
176	Q₁₆	—	4	8	no	—
177	D₈	—	2	8	no	—
178	D₈	—	4	4	no	—
179	D₈	—	4	8	no	—
180	C₂×C₈	C₁₃×D₈	1	4	yes	—
181	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
183	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
184	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	4	8	no	—
186	C₄²	C₁₃×D₈	1	2	yes	—
187	C₄²	C₂×D₄×C₁₃	1	2	yes	—
188	C₁₆	C₂×D₄×C₁₃	1	8	yes	complemented
189	C₁₃	C₂×#2248	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	—	4	4	no	—
191	C₂³	—	8	4	no	—
192	C₂³	—	8	8	no	—
193	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
194	Q₈	—	4	4	no	—
195	Q₈	—	4	4	no	—
196	Q₈	—	8	4	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	4	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	16	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	4	no	—
203	C₂×C₄	C₁₃×SD₃₂	1	8	yes	complemented
204	C₂×C₄	C₁₃×#127	1	2	yes	—
205	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
206	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
207	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
208	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
209	C₈	C₁₃×#127	1	2	yes	—
210	C₈	C₂×C₁₃×D₈	1	4	yes	—
211	C₈	C₂²×D₄×C₁₃	1	2	yes	—
212	C₂²	—	8	4	no	—
213	C₂²	—	8	8	no	—
214	C₂²	—	16	8	no	—
215	C₂²	C₁₃×#493	1	2	yes	—
216	C₂²	C₂×C₁₃×SD₃₂	1	4	yes	—
217	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
218	C₄	—	8	4	no	—
219	C₄	C₁₃×#493	1	2	yes	—
220	C₄	C₂×C₁₃×SD₃₂	1	8	yes	complemented
221	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
222	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
223	C₂	—	16	4	no	—
224	C₂	C₁₃×#2248	1	4	yes	direct product factor
225	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
226	C₂	C₂²×C₁₃×SD₃₂	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×SD₃₂	C₂	semidirect
2. C₂×C₁₃×#494	C₂	semidirect
4. C₁₃×#2248	C₂	direct (N × H)
5. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
13. C₂×C₁₃×SD₃₂	C₂²	semidirect
16. C₁₃×#494	C₂²	semidirect
20. C₄×C₁₃×C₁₆	C₂²	semidirect
43. C₂×C₁₃×C₁₆	D₄	semidirect
47. C₂×#2248	C₁₃	direct (N × H)
76. C₁₃×C₁₆	C₂×D₄	semidirect
77. C₂²×SD₃₂	C₂×C₁₃	semidirect
78. C₂×#494	C₂×C₁₃	semidirect
80. #2248	C₂×C₁₃	direct (N × H)
81. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
95. C₂×C₄×C₁₃	SD₃₂	semidirect
111. C₂×SD₃₂	C₂²×C₁₃	semidirect
114. #494	C₂²×C₁₃	semidirect
118. C₄×C₁₆	C₂²×C₁₃	semidirect
127. C₄×C₁₃	C₂×SD₃₂	semidirect
152. C₂×C₁₆	D₄×C₁₃	semidirect
157. C₂×C₁₃	#2248	direct (N × H)
188. C₁₆	C₂×D₄×C₁₃	semidirect
189. C₁₃	C₂×#2248	direct (N × H)
203. C₂×C₄	C₁₃×SD₃₂	semidirect
220. C₄	C₂×C₁₃×SD₃₂	semidirect
224. C₂	C₁₃×#2248	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×SD₃₂	2
C₂×C₁₃×#494	2
C₂×C₁₃×#493	2
C₁₃×#2248	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2248	13

Subgroups — C2×C13×#2248 #15720050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2248 #15720050