Subgroups — C₂×#13131173 #16806998

Order 3328.

734 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×#88106	C₂	1	1	yes	maximal, characteristic
2	C₂³×#6092	C₂	1	1	yes	maximal, characteristic
3	#13131173	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718260	C₂	1	4	yes	maximal, complemented
5	C₂×#11718783	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂×#88180	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	1	yes	characteristic
9	C₂×#88106	—	2	4	no	—
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂²×#6092	—	2	4	no	—
12	C₂²×#6092	C₂²	1	2	yes	—
13	C₂²×#6092	C₂²	1	4	yes	—
14	C₂²×#6092	C₂²	1	2	yes	—
15	#11718783	C₄	1	4	yes	—
16	#11718783	C₂²	1	4	yes	—
17	C₂×#88055	C₄	1	2	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
19	#11718260	C₂²	1	16	yes	complemented
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	2	4	no	—
22	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₂²⋊Dic₁₃	C₈	1	2	yes	—
25	#88180	C₈	1	4	yes	—
26	#88180	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊Dic₁₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	2	yes	—
31	C₂²×#1055	D₄	1	4	yes	—
32	C₂²×#1055	Q₈	1	4	yes	—
33	C₂²×#1055	C₂³	1	1	yes	characteristic
34	C₂²×#1055	C₂³	1	2	yes	—
35	#88106	—	2	8	no	—
36	C₂×#6092	—	2	4	no	—
37	C₂×#6092	—	2	8	no	—
38	C₂×#6092	—	2	4	no	—
39	C₂×#6092	C₂³	1	4	yes	—
40	C₂×#6092	C₂³	1	4	yes	—
41	#88055	C₂×C₄	1	8	yes	—
42	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
43	C₈×Dic₁₃	C₂³	1	4	yes	—
44	C₂×#2151	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
53	C₁₃×M₁₆	—	2	8	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
56	C₂²⋊Dic₁₃	C₂×C₈	1	8	yes	—
57	C₄⋊Dic₁₃	C₂×C₈	1	8	yes	—
58	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
59	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
60	C₂×#1055	C₄⋊C₄	1	4	yes	—
61	C₂×#1055	C₄⋊C₄	1	8	yes	—
62	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
63	C₂×#1055	C₂²×C₄	1	2	yes	—
64	C₂×#1055	C₂×D₄	1	2	yes	—
65	C₂×#1055	C₂×D₄	1	4	yes	—
66	C₂×#1055	C₂×Q₈	1	2	yes	—
67	C₂×#1055	C₂×Q₈	1	4	yes	—
68	C₂×#1055	C₂⁴	1	1	yes	characteristic
69	#6092	—	2	4	no	—
70	#6092	—	2	4	no	—
71	C₂²×M₃₂	—	13	1	no	—
72	C₂³×C₁₆	—	13	1	no	—
73	C₂×#405	—	13	1	no	—
74	#2151	—	13	8	no	—
75	C₂×C₄⋊C₁₆	—	13	4	no	—
76	C₂³×C₁₃	—	2	4	no	—
77	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
78	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
79	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
81	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
88	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₁₃	—	2	8	no	—
90	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
91	C₈×C₁₃	—	2	4	no	—
92	#1055	C₂×C₄⋊C₄	1	2	yes	—
93	#1055	C₂×C₄⋊C₄	1	4	yes	—
94	#1055	C₂²×D₄	1	1	yes	characteristic
95	#1055	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×M₁₆	#261	1	1	yes	characteristic
97	C₂³×C₈	—	13	1	no	—
98	C₂×#117	—	13	1	no	—
99	C₂×M₃₂	—	13	4	no	—
100	C₂×M₃₂	—	26	4	no	—
101	C₂²×C₁₆	—	13	2	no	—
102	C₂²×C₁₆	—	13	4	no	—
103	C₂²×C₁₆	—	13	2	no	—
104	C₂²×C₁₆	—	26	4	no	—
105	#405	—	13	4	no	—
106	#405	—	13	4	no	—
107	C₂×#97	—	13	2	no	—
108	C₂×C₄×C₈	—	13	2	no	—
109	C₄⋊C₁₆	—	13	16	no	—
110	C₂²×C₁₃	—	2	4	no	—
111	C₂²×C₁₃	—	2	8	no	—
112	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
113	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
114	C₂²×C₁₃	#504	1	4	yes	—
115	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
116	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
117	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
118	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
120	Dic₁₃	—	2	8	no	—
121	C₂³×C₄	C₂×#261	1	1	yes	characteristic
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	C₂×#261	1	4	yes	—
124	C₂²×C₈	—	13	4	no	—
125	C₂²×C₈	—	13	4	no	—
126	C₂²×C₈	—	13	2	no	—
127	C₂²×C₈	—	13	2	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	C₂×#261	1	2	yes	—
131	#117	—	13	4	no	—
132	#117	—	13	4	no	—
133	C₂×C₄⋊C₄	—	13	2	no	—
134	C₂×C₂²⋊C₄	—	13	2	no	—
135	C₂×C₄²	—	13	2	no	—
136	M₃₂	—	26	8	no	—
137	C₂×C₁₆	—	13	4	no	—
138	C₂×C₁₆	—	13	4	no	—
139	C₂×C₁₆	—	26	4	no	—
140	C₂×C₁₆	—	26	8	no	—
141	C₂×C₁₆	—	26	4	no	—
142	#97	—	13	8	no	—
143	C₄×C₈	—	13	4	no	—
144	C₄×C₈	—	13	4	no	—
145	C₂×C₁₃	—	2	4	no	—
146	C₂×C₁₃	#2151	1	4	yes	complemented
147	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
148	C₂×C₁₃	C₂×#504	1	2	yes	—
149	C₂⁴	D₁₃⋊C₈	1	1	yes	characteristic
150	C₂²×C₄	—	13	4	no	—
151	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
152	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
153	C₂²×C₄	C₄⋊#261	1	4	yes	—
154	C₂²×C₄	#6122	1	4	yes	—
155	C₂²×C₄	C₂²×#261	1	2	yes	—
156	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
157	M₁₆	—	2	8	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	13	2	no	—
160	C₂×C₈	—	13	4	no	—
161	C₂×C₈	—	13	8	no	—
162	C₂×C₈	—	13	4	no	—
163	C₂×C₈	—	13	4	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	—	13	1	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	C₂²×#261	1	4	yes	—
169	C₄⋊C₄	—	13	8	no	—
170	C₂²⋊C₄	—	13	8	no	—
171	C₄²	—	13	4	no	—
172	C₄²	—	13	4	no	—
173	C₁₆	—	26	4	no	—
174	C₁₆	—	26	4	no	—
175	C₁₃	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
176	C₂³	—	2	4	no	—
177	C₂³	#88112	1	4	yes	—
178	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
179	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
180	C₂×C₄	—	13	8	no	—
181	C₂×C₄	—	26	8	no	—
182	C₂×C₄	#88112	1	8	yes	—
183	C₂×C₄	#88112	1	4	yes	—
184	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
186	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
187	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
188	C₂×C₄	C₂×#6122	1	2	yes	—
189	C₂×C₄	C₂×#6122	1	4	yes	—
190	C₂×C₄	C₂³×#261	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₈	—	13	1	no	—
193	C₈	—	13	2	no	—
194	C₈	—	13	4	no	—
195	C₈	—	13	1	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	8	no	—
198	C₂²	#11719096	1	4	yes	—
199	C₂²	C₂×#88112	1	4	yes	—
200	C₂²	C₂×#88112	1	2	yes	—
201	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
202	C₄	—	26	8	no	—
203	C₄	C₂×#88112	1	2	yes	—
204	C₄	C₂×#88112	1	4	yes	—
205	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
206	C₄	C₂²×#6122	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#13131173	1	4	yes	direct product factor
209	C₂	C₂²×#88112	1	1	yes	characteristic
210	C₂	C₂×#11719096	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	C₂×#2151
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13131173	C₂	direct (N × H)
4. C₂×#11718260	C₂	semidirect
19. #11718260	C₂²	semidirect
146. C₂×C₁₃	#2151	semidirect
175. C₁₃	C₂×#2151	semidirect
208. C₂	#13131173	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×#88106	2
C₂³×#6092	2
#13131173	2
C₂×#11718260	2
C₂×#11718783	2
C₂×#2151	13

Subgroups — C2×#13131173 #16806998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131173 #16806998