Subgroups — C₂×#13112978 #15783126

Order 3328.

294 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	C₂×#11718091	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#88034	C₂	1	1	yes	maximal, characteristic, complemented
3	#13112978	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
6	C₁₃×#504	—	2	1	no	—
7	C₁₃×#504	—	2	1	no	—
8	C₁₃×#504	C₄	1	2	yes	—
9	C₁₃×#504	C₂²	1	2	yes	—
10	C₂×C₁₃×M₃₂	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	—	2	1	no	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	#11718091	—	2	1	no	—
15	#11718091	C₂²	1	2	yes	complemented
16	C₂×#88034	—	2	1	no	—
17	C₂×#88034	—	2	1	no	—
18	C₂×#88034	—	2	1	no	—
19	C₂×#88034	C₂²	1	2	yes	complemented
20	C₂×C₁₃×#41	C₈	1	1	yes	characteristic
21	C₁₃×#131	—	2	1	no	—
22	C₁₃×#131	—	2	1	no	—
23	C₁₃×#131	C₈	1	2	yes	—
24	C₁₃×#131	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	—	2	1	no	—
26	C₂×C₁₃×M₁₆	C₈	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	—	2	1	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₁₃×M₃₂	—	2	1	no	—
30	C₁₃×M₃₂	—	2	2	no	—
31	C₁₃×M₃₂	—	2	2	no	—
32	C₁₃×M₃₂	C₂×C₄	1	2	yes	—
33	C₂×C₁₃×C₁₆	—	2	1	no	—
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
38	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
39	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
42	#88034	—	2	2	no	—
43	#88034	—	2	2	no	—
44	C₂×#1401	—	13	1	no	maximal, Sylow
45	C₁₃×#41	—	2	1	no	—
46	C₁₃×#41	—	2	1	no	—
47	C₁₃×#41	C₁₆	1	2	yes	—
48	C₁₃×#41	C₂×C₈	1	2	yes	—
49	C₂×Q₈×C₁₃	C₁₆	1	1	yes	characteristic
50	C₂×D₄×C₁₃	—	2	1	no	—
51	C₂×D₄×C₁₃	C₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₁₃×M₁₆	—	2	1	no	—
55	C₁₃×M₁₆	—	2	2	no	—
56	C₁₃×M₁₆	—	2	2	no	—
57	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₁₃×C₁₆	—	2	2	no	—
68	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
69	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
70	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
71	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
72	C₂×#504	D₁₃	1	1	yes	characteristic
73	C₂×M₆₄	—	13	1	no	—
74	C₂²×C₃₂	—	13	1	no	—
75	#1401	—	13	4	no	—
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	M₃₂	1	1	yes	characteristic
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	C₂×C₁₆	1	2	yes	—
80	D₄×C₁₃	—	2	2	no	—
81	D₄×C₁₃	—	2	2	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	C₂×C₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₃	M₃₂	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₈×C₁₃	—	2	2	no	—
94	C₈×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
95	C₈×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
96	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×#131	Dic₁₃	1	1	yes	characteristic
99	#504	—	2	1	no	—
100	#504	—	2	1	no	—
101	#504	Dic₁₃	1	2	yes	—
102	#504	C₂×D₁₃	1	2	yes	—
103	C₂×M₃₂	—	2	1	no	—
104	C₂×M₃₂	Dic₁₃	1	1	yes	characteristic
105	C₂²×C₁₆	—	2	1	no	—
106	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
107	M₆₄	—	13	2	no	—
108	M₆₄	—	26	1	no	—
109	C₂×C₃₂	—	13	2	no	—
110	C₂×C₃₂	—	26	1	no	—
111	C₂×C₃₂	—	26	1	no	—
112	C₂×C₃₂	—	26	1	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	C₂²⋊C₁₆	1	1	yes	characteristic
117	C₂²×C₁₃	C₂²⋊C₁₆	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	C₂²⋊C₁₆	1	1	yes	characteristic
121	C₄×C₁₃	C₂²⋊C₁₆	1	1	yes	characteristic
122	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
124	C₂×#41	#1055	1	1	yes	characteristic
125	#131	—	2	1	no	—
126	#131	—	2	1	no	—
127	#131	#1055	1	2	yes	—
128	#131	C₂×Dic₁₃	1	2	yes	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	#1055	1	1	yes	characteristic
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
133	M₃₂	—	2	1	no	—
134	M₃₂	—	2	2	no	—
135	M₃₂	—	2	2	no	—
136	M₃₂	C₂×Dic₁₃	1	2	yes	—
137	C₂×C₁₆	—	2	1	no	—
138	C₂×C₁₆	—	2	2	no	—
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
141	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
142	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
143	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
144	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
145	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
146	C₃₂	—	26	2	no	—
147	C₃₂	—	26	2	no	—
148	C₂×C₁₃	—	2	1	no	—
149	C₂×C₁₃	—	2	1	no	—
150	C₂×C₁₃	—	4	2	no	—
151	C₂×C₁₃	#1401	1	2	yes	complemented
152	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
153	#41	—	2	1	no	—
154	#41	—	2	1	no	—
155	#41	#6090	1	2	yes	—
156	#41	C₂×#1055	1	2	yes	—
157	C₂×Q₈	#6090	1	1	yes	characteristic
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	#6090	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
162	M₁₆	—	2	1	no	—
163	M₁₆	—	2	2	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	C₂×#1055	1	2	yes	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	C₂×#1055	1	1	yes	characteristic
170	C₂×C₈	#6099	1	1	yes	characteristic
171	C₂×C₈	#6099	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
175	C₁₆	—	2	2	no	—
176	C₁₆	C₂²⋊Dic₁₃	1	1	yes	characteristic
177	C₁₆	C₂²⋊Dic₁₃	1	1	yes	characteristic
178	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
179	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
180	C₁₃	C₂×#1401	1	1	yes	characteristic, complemented, Sylow
181	C₂³	—	2	1	no	—
182	C₂³	#88052	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	C₂×#6090	1	2	yes	—
185	D₄	—	2	2	no	—
186	D₄	—	2	2	no	—
187	D₄	—	2	1	no	—
188	D₄	C₂×#6090	1	2	yes	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	C₂×#6090	1	1	yes	characteristic
193	C₂×C₄	C₂×#6090	1	1	yes	characteristic
194	C₂×C₄	#88052	1	1	yes	characteristic
195	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
197	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₂²⋊#1055	1	1	yes	characteristic
200	C₈	C₂²⋊#1055	1	1	yes	characteristic
201	C₈	C₂×#6099	1	1	yes	characteristic
202	C₈	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²⋊#6090	1	1	yes	characteristic
207	C₂²	C₂²⋊#6090	1	1	yes	characteristic
208	C₂²	C₂×#88052	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	C₂²⋊#6090	1	1	yes	characteristic
211	C₄	C₂²⋊#6090	1	1	yes	characteristic
212	C₄	C₂²×#6090	1	1	yes	characteristic
213	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	4	2	no	—
217	C₂	#13112978	1	2	yes	direct product factor
218	C₂	C₂×C₂²⋊#6090	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₂×C₁₃×#504	C₂
socle	C₂²×C₁₃	C₂²⋊C₁₆
prime radical	C₂×C₁₃×#41	C₈
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	C₂×#504	D₁₃
hypocenter	C₁₃	C₂×#1401
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#11718091	C₂	semidirect
2. C₂²×#88034	C₂	semidirect
3. #13112978	C₂	direct (N × H)
15. #11718091	C₂²	semidirect
19. C₂×#88034	C₂²	semidirect
151. C₂×C₁₃	#1401	semidirect
180. C₁₃	C₂×#1401	semidirect
217. C₂	#13112978	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718091	2
C₂²×#88034	2
#13112978	2
C₂×C₁₃×#504	2
C₂×#1401	13

Subgroups — C2×#13112978 #15783126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13112978 #15783126