Subgroups — C₂×#13116592 #16015898

Order 3328.

798 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	2	yes	maximal
2	#13116592	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#6090	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
9	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
12	C₂×#88052	—	2	8	no	—
13	C₂×#88052	C₂²	1	8	yes	—
14	C₂²×#6090	C₂²	1	4	yes	—
15	C₄⋊#6090	C₂²	1	16	yes	complemented
16	C₂³×C₄×C₁₃	C₈	1	2	yes	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	—	2	4	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	D₄	1	4	yes	—
24	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₈	1	4	yes	—
29	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₁₃	C₂³	1	4	yes	—
35	#88052	—	2	16	no	—
36	C₂×#6090	—	2	8	no	—
37	C₂×#6090	C₂³	1	8	yes	—
38	C₂×#2150	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	—	2	8	no	—
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	—	2	8	no	—
51	C₂×C₈×C₁₃	—	2	4	no	—
52	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
53	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
54	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
59	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
62	C₄²×C₁₃	C₂×C₈	1	4	yes	—
63	C₄²×C₁₃	C₂×C₈	1	4	yes	—
64	C₄²×C₁₃	C₂×C₈	1	4	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
66	#6090	—	2	8	no	—
67	C₂²×M₃₂	—	13	2	no	—
68	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
69	#2150	—	13	8	no	—
70	C₂×C₄⋊C₁₆	—	13	4	no	—
71	C₂³×C₁₃	—	2	4	no	—
72	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
73	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
74	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	2	16	no	—
77	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
78	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
79	C₂×C₄×C₁₃	M₃₂	1	8	yes	complemented
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₁₃	—	2	4	no	—
90	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
94	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
95	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
96	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
97	C₂×M₃₂	—	13	8	no	—
98	C₂×M₃₂	—	26	8	no	—
99	C₂²×C₁₆	—	13	4	no	—
100	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
101	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
102	C₂×C₄×C₈	C₂×D₁₃	1	4	yes	—
103	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
104	C₄⋊C₁₆	—	13	16	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	—	2	8	no	—
107	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
108	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
109	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
110	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
113	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
114	C₄×C₁₃	C₂×M₃₂	1	8	yes	complemented
115	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
117	C₂³×C₄	#1055	1	2	yes	—
118	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	Dic₂₆	1	4	yes	—
121	C₂²×C₈	D₅₂	1	4	yes	—
122	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
123	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
124	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
126	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
127	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
128	C₂×C₄²	#1055	1	1	yes	characteristic
129	C₂×C₄²	#1055	1	4	yes	—
130	C₂×C₄²	#1055	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
132	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
133	M₃₂	—	26	16	no	—
134	C₂×C₁₆	—	13	8	no	—
135	C₂×C₁₆	—	26	8	no	—
136	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
137	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
138	C₄×C₈	C₂²×D₁₃	1	4	yes	—
139	C₂×C₁₃	—	2	4	no	—
140	C₂×C₁₃	#2150	1	4	yes	complemented
141	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
142	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
143	C₂⁴	C₂×#1055	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	C₂×#1055	1	2	yes	—
146	C₂²×C₄	C₂×#1055	1	8	yes	—
147	C₂²×C₄	C₂×#1055	1	2	yes	—
148	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
149	C₂²×C₄	C₂×#1055	1	2	yes	—
150	C₂²×C₄	#6099	1	4	yes	—
151	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
152	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
153	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
157	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
158	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
159	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
160	C₂×C₈	C₂×D₅₂	1	2	yes	—
161	C₂×C₈	C₂×D₅₂	1	4	yes	—
162	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
163	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
164	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
165	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
166	C₄²	C₂×#1055	1	4	yes	—
167	C₄²	C₂×#1055	1	4	yes	—
168	C₄²	C₂×#1055	1	4	yes	—
169	C₄²	C₂²×Dic₁₃	1	4	yes	—
170	C₁₆	—	26	8	no	—
171	C₁₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
172	C₂³	—	2	4	no	—
173	C₂³	C₄⋊#1055	1	4	yes	—
174	C₂³	C₂²×#1055	1	2	yes	—
175	C₂³	C₂²×#1055	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	C₄⋊#1055	1	8	yes	—
179	C₂×C₄	C₄⋊#1055	1	4	yes	—
180	C₂×C₄	#88052	1	8	yes	complemented
181	C₂×C₄	C₂²×#1055	1	4	yes	—
182	C₂×C₄	C₂²×#1055	1	4	yes	—
183	C₂×C₄	C₂²×#1055	1	2	yes	—
184	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
185	C₂×C₄	C₂×#6099	1	4	yes	—
186	C₂×C₄	C₂×#6099	1	2	yes	—
187	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
188	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
189	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
192	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
193	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
194	C₈	C₂²×D₅₂	1	1	yes	characteristic
195	C₂²	—	2	8	no	—
196	C₂²	—	2	4	no	—
197	C₂²	C₂×C₄⋊#1055	1	4	yes	—
198	C₂²	C₂×C₄⋊#1055	1	2	yes	—
199	C₂²	C₂×#88052	1	4	yes	—
200	C₂²	C₂³×#1055	1	1	yes	characteristic
201	C₄	—	2	4	no	—
202	C₄	C₂×C₄⋊#1055	1	4	yes	—
203	C₄	C₂×C₄⋊#1055	1	2	yes	—
204	C₄	C₂×#88052	1	8	yes	complemented
205	C₄	C₂²×#6099	1	1	yes	characteristic
206	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#13116592	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic
210	C₂	C₂²×#88052	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#2150
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13116592	C₂	direct (N × H)
3. C₂×C₄⋊#6090	C₂	semidirect
15. C₄⋊#6090	C₂²	semidirect
79. C₂×C₄×C₁₃	M₃₂	semidirect
114. C₄×C₁₃	C₂×M₃₂	semidirect
140. C₂×C₁₃	#2150	semidirect
171. C₁₃	C₂×#2150	semidirect
180. C₂×C₄	#88052	semidirect
204. C₄	C₂×#88052	semidirect
208. C₂	#13116592	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×#88052	2
#13116592	2
C₂×C₄⋊#6090	2
C₂²×C₄×C₈×C₁₃	2
C₂×#2150	13

Subgroups — C2×#13116592 #16015898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116592 #16015898