Subgroups — C₂×#13116694 #16016193

Order 3328.

678 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×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718534	C₂	1	2	yes	maximal
3	#13116694	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718084	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₂	C₂²	1	2	yes	—
8	C₂×#88141	—	2	2	no	—
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#107	C₂²	1	2	yes	—
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	1	yes	characteristic
13	C₂×#88050	—	2	2	no	—
14	C₂×Dic₁₀₄	C₄	1	8	yes	complemented
15	C₂×Dic₁₀₄	C₂²	1	4	yes	—
16	#11718534	—	2	4	no	—
17	#11718534	C₂²	1	8	yes	—
18	C₁₃×#366	C₂²	1	4	yes	—
19	C₁₆×Dic₁₃	C₂²	1	4	yes	—
20	#11718084	C₂²	1	8	yes	—
21	C₂²×Dic₂₆	—	2	2	no	—
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×#6101	—	4	2	no	—
25	C₂×C₄×Dic₁₃	—	4	2	no	—
26	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
27	C₂²×#1055	D₄	1	1	yes	characteristic
28	C₂×Dic₅₂	—	2	2	no	—
29	C₂×Dic₅₂	—	2	4	no	—
30	C₂×Dic₅₂	C₂×C₄	1	8	yes	—
31	C₂×Dic₅₂	C₂³	1	4	yes	—
32	#88141	—	2	8	no	—
33	#88141	—	4	4	no	—
34	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
35	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
36	C₁₃×#107	—	2	2	no	—
37	C₁₃×#107	C₂³	1	4	yes	—
38	C₈×Dic₁₃	D₄	1	2	yes	—
39	C₈×Dic₁₃	C₂³	1	2	yes	—
40	C₂×#6090	—	2	2	no	—
41	C₂×#6090	C₂³	1	2	yes	—
42	#88050	—	2	8	no	—
43	Dic₁₀₄	C₂×C₄	1	16	yes	complemented
44	C₂×C₄×Q₃₂	—	13	1	no	maximal, Sylow
45	C₂²×C₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	4	2	no	—
48	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
49	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
50	C₂×Dic₂₆	—	2	8	no	—
51	C₂×Dic₂₆	—	2	4	no	—
52	C₂×Dic₂₆	—	4	2	no	—
53	C₂×Dic₂₆	—	4	4	no	—
54	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	#41	1	2	yes	—
57	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	—	2	4	no	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	#6101	—	4	8	no	—
61	C₄×Dic₁₃	—	4	8	no	—
62	C₄×Dic₁₃	D₈	1	2	yes	—
63	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	C₂×D₄	1	2	yes	—
66	Dic₅₂	—	2	8	no	—
67	Dic₅₂	—	2	4	no	—
68	Dic₅₂	C₂²×C₄	1	8	yes	—
69	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
70	#6090	—	2	2	no	—
71	C₂²×Q₃₂	—	13	1	no	—
72	C₂×C₄×Q₁₆	—	13	2	no	—
73	C₄×Q₃₂	—	13	8	no	—
74	C₂×#366	D₁₃	1	1	yes	characteristic
75	C₂×#358	—	13	2	no	—
76	C₂×C₄×C₁₆	—	13	1	no	—
77	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	—	8	4	no	—
80	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#135	1	2	yes	—
83	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	—	4	4	no	—
86	C₂×Dic₁₃	—	4	8	no	—
87	C₂×Dic₁₃	Q₃₂	1	4	yes	complemented
88	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
89	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
90	Dic₂₆	—	2	8	no	—
91	Dic₂₆	—	4	4	no	—
92	Dic₂₆	—	4	8	no	—
93	C₈×C₁₃	C₄×D₄	1	2	yes	—
94	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
96	#1055	—	2	2	no	—
97	C₂²×Q₁₆	—	13	2	no	—
98	C₂×C₄×Q₈	—	26	2	no	—
99	C₂×Q₃₂	—	13	8	no	—
100	C₂×Q₃₂	—	13	4	no	—
101	C₂²×C₁₆	—	13	1	no	—
102	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
103	C₄×Q₁₆	—	13	8	no	—
104	C₄×Q₁₆	—	26	4	no	—
105	C₂×#107	C₂×D₁₃	1	2	yes	—
106	C₂×#103	—	26	2	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	#366	C₂×D₁₃	1	4	yes	—
109	#358	—	13	8	no	—
110	C₄×C₁₆	—	13	4	no	—
111	C₂²×C₁₃	C₄×D₈	1	2	yes	—
112	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
113	C₂²×C₁₃	#508	1	2	yes	—
114	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₁₃	—	8	4	no	—
116	C₄×C₁₃	C₄×D₈	1	2	yes	—
117	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
119	Dic₁₃	—	2	2	no	—
120	Dic₁₃	—	4	8	no	—
121	Dic₁₃	C₂×Q₃₂	1	4	yes	complemented
122	C₂²×Q₈	—	26	2	no	—
123	C₂×Q₁₆	—	13	8	no	—
124	C₂×Q₁₆	—	13	4	no	—
125	C₂×Q₁₆	—	26	2	no	—
126	C₂×Q₁₆	—	26	4	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
129	C₄×Q₈	—	26	8	no	—
130	C₄×Q₈	—	52	4	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	52	2	no	—
133	C₂×C₄²	—	13	1	no	—
134	C₂×C₄²	—	52	2	no	—
135	Q₃₂	—	13	16	no	—
136	C₂×C₁₆	—	13	2	no	—
137	C₂×C₁₆	—	26	2	no	—
138	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
139	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
140	#107	—	2	2	no	—
141	#107	C₂²×D₁₃	1	4	yes	—
142	#103	—	26	8	no	—
143	C₄×C₈	—	13	2	no	—
144	C₄×C₈	—	13	2	no	—
145	C₂×C₁₃	C₄×Q₃₂	1	4	yes	complemented
146	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
147	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
148	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
149	C₂×Q₈	—	26	8	no	—
150	C₂×Q₈	—	26	4	no	—
151	C₂×Q₈	—	52	2	no	—
152	C₂×Q₈	—	52	4	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	13	1	no	—
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	52	2	no	—
157	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
158	Q₁₆	—	13	8	no	—
159	Q₁₆	—	26	8	no	—
160	Q₁₆	—	26	4	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	26	2	no	—
163	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
164	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
165	C₂×C₈	#6131	1	2	yes	—
166	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	52	8	no	—
170	C₄²	—	13	2	no	—
171	C₄²	—	13	2	no	—
172	C₄²	—	52	8	no	—
173	C₁₆	—	26	2	no	—
174	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
175	C₁₃	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
176	C₂³	D₈×D₁₃	1	1	yes	characteristic
177	Q₈	—	26	8	no	—
178	Q₈	—	52	4	no	—
179	Q₈	—	52	8	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	—	13	2	no	—
183	C₂×C₄	—	13	4	no	—
184	C₂×C₄	—	13	2	no	—
185	C₂×C₄	—	26	2	no	—
186	C₂×C₄	—	52	4	no	—
187	C₂×C₄	—	52	8	no	—
188	C₂×C₄	#88147	1	2	yes	—
189	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
190	C₂×C₄	#88173	1	2	yes	—
191	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
192	C₈	—	26	2	no	—
193	C₈	#88147	1	2	yes	—
194	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
195	C₈	C₂×#6131	1	1	yes	characteristic
196	C₂²	#11718533	1	2	yes	—
197	C₂²	#11718573	1	2	yes	—
198	C₂²	D₁₃×Q₃₂	1	2	yes	—
199	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	13	4	no	—
202	C₄	—	26	2	no	—
203	C₄	—	52	8	no	—
204	C₄	#11718533	1	2	yes	—
205	C₄	C₂×#88147	1	1	yes	characteristic
206	C₄	C₂×#88173	1	1	yes	characteristic
207	C₂	#13116694	1	4	yes	direct product factor
208	C₂	C₂×#11718533	1	1	yes	characteristic
209	C₂	C₂×D₁₃×Q₃₂	1	1	yes	characteristic
210	C₂	C₂×#11718573	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#366	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#366	D₁₃
hypocenter	C₁₃	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
3. #13116694	C₂	direct (N × H)
14. C₂×Dic₁₀₄	C₄	semidirect
43. Dic₁₀₄	C₂×C₄	semidirect
87. C₂×Dic₁₃	Q₃₂	semidirect
121. Dic₁₃	C₂×Q₃₂	semidirect
145. C₂×C₁₃	C₄×Q₃₂	semidirect
175. C₁₃	C₂×C₄×Q₃₂	semidirect
207. C₂	#13116694	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×Dic₁₀₄	2
C₂×#11718534	2
#13116694	2
C₂×#11718084	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×#366	2
C₂×C₄×Q₃₂	13

Subgroups — C2×#13116694 #16016193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116694 #16016193