Subgroups — C₂×#13131201 #16807050

Order 3328.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131201	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#6092	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#6092	C₂	1	2	yes	maximal
4	C₂×#11718231	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
8	C₂²×#6099	C₄	1	1	yes	characteristic
9	C₂³×#1055	C₂²	1	1	yes	characteristic
10	C₂²×#6092	C₂²	1	2	yes	—
11	C₂²×#6092	C₂²	1	2	yes	—
12	#11718658	C₄	1	4	yes	—
13	#11718658	C₂²	1	4	yes	—
14	C₂×#88043	C₄	1	1	yes	characteristic
15	C₂×#88043	C₄	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	C₂²⋊#6092	C₂²	1	8	yes	—
19	#11718231	C₂²	1	4	yes	complemented
20	C₄⋊#6092	C₂²	1	8	yes	complemented
21	C₄×#6092	C₂²	1	4	yes	complemented
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₁₃×#117	C₈	1	4	yes	—
24	C₁₃×#117	C₂×C₄	1	4	yes	—
25	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
26	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#6099	—	2	4	no	—
30	C₂×#6099	C₂×C₄	1	4	yes	—
31	C₂²×#1055	—	2	4	no	—
32	C₂²×#1055	C₂×C₄	1	2	yes	—
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂³	1	2	yes	—
37	C₂²×#1055	C₂³	1	1	yes	characteristic
38	C₂×#6092	—	2	4	no	—
39	C₂×#6092	—	2	4	no	—
40	C₂×#6092	C₂³	1	4	yes	—
41	C₂×#6092	C₂³	1	4	yes	—
42	#88043	C₂×C₄	1	4	yes	—
43	#88043	C₂×C₄	1	4	yes	—
44	C₄×#1055	C₂×C₄	1	2	yes	—
45	C₄×#1055	C₂×C₄	1	2	yes	—
46	C₄×#1055	C₂³	1	2	yes	—
47	C₄×#1055	C₂³	1	2	yes	—
48	C₂×#2164	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
59	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
60	C₄²×C₁₃	C₂×C₈	1	2	yes	—
61	C₄²×C₁₃	C₂×C₈	1	2	yes	—
62	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
63	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
64	#6099	—	2	8	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	2	no	—
67	C₂×#1055	—	2	4	no	—
68	C₂×#1055	—	2	4	no	—
69	C₂×#1055	M₁₆	1	4	yes	—
70	C₂×#1055	C₂²×C₄	1	2	yes	—
71	C₂×#1055	C₂²×C₄	1	2	yes	—
72	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
73	C₂×#1055	C₂²×C₄	1	2	yes	—
74	C₂×#1055	#41	1	4	yes	—
75	C₂×#1055	C₂⁴	1	1	yes	characteristic
76	#6092	—	2	4	no	—
77	#6092	—	2	4	no	—
78	C₂×#405	—	13	1	no	—
79	#2164	—	13	8	no	—
80	C₂×C₄⋊C₁₆	—	13	2	no	—
81	C₂×C₂²⋊C₁₆	—	13	2	no	—
82	C₂×#346	—	13	1	no	—
83	C₂×C₄×C₁₆	—	13	1	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	—	2	4	no	—
86	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
87	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	4	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	#117	1	2	yes	—
95	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
96	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	#1055	—	2	2	no	—
102	#1055	—	2	2	no	—
103	#1055	#117	1	2	yes	—
104	#1055	C₂×M₁₆	1	4	yes	—
105	#1055	C₂×#41	1	2	yes	—
106	C₂²×M₁₆	—	13	1	no	—
107	C₂³×C₈	—	13	1	no	—
108	C₂×#117	#261	1	1	yes	characteristic
109	C₂²×C₁₆	—	13	2	no	—
110	C₂²×C₁₆	—	13	2	no	—
111	#405	—	13	4	no	—
112	#405	—	13	4	no	—
113	C₂×#97	—	13	1	no	—
114	C₂×#97	—	13	1	no	—
115	C₂×C₄×C₈	—	13	1	no	—
116	C₂×C₄×C₈	—	13	1	no	—
117	C₄⋊C₁₆	—	13	8	no	—
118	C₂²⋊C₁₆	—	13	8	no	—
119	#346	—	13	4	no	—
120	C₄×C₁₆	—	13	4	no	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	4	4	no	—
124	C₂²×C₁₃	#431	1	2	yes	—
125	C₂²×C₁₃	#504	1	4	yes	—
126	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	—	2	4	no	—
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	#431	1	2	yes	—
131	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
132	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
133	C₂³×C₄	C₂×#261	1	1	yes	characteristic
134	C₂×M₁₆	—	13	4	no	—
135	C₂×M₁₆	—	26	4	no	—
136	C₂²×C₈	—	13	1	no	—
137	C₂²×C₈	—	13	1	no	—
138	C₂²×C₈	—	13	2	no	—
139	C₂²×C₈	—	13	2	no	—
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	1	no	—
142	C₂²×C₈	—	26	4	no	—
143	#117	#1057	1	4	yes	—
144	#117	C₂×#261	1	4	yes	—
145	C₂×C₄⋊C₄	#1057	1	2	yes	—
146	C₂×C₂²⋊C₄	#1057	1	2	yes	—
147	C₂×C₄²	C₂×#261	1	1	yes	characteristic
148	C₂×C₄²	C₂×#261	1	1	yes	characteristic
149	C₂×C₁₆	—	13	4	no	—
150	C₂×C₁₆	—	13	4	no	—
151	C₂×C₁₆	—	26	4	no	—
152	C₂×C₁₆	—	26	4	no	—
153	#97	—	13	4	no	—
154	#97	—	13	4	no	—
155	C₄×C₈	—	13	2	no	—
156	C₄×C₈	—	13	2	no	—
157	C₄×C₈	—	13	2	no	—
158	C₄×C₈	—	13	2	no	—
159	C₂×C₁₃	—	4	2	no	—
160	C₂×C₁₃	#2164	1	4	yes	complemented
161	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
162	C₂×C₁₃	C₂×#504	1	2	yes	—
163	C₂⁴	C₂×#1057	1	1	yes	characteristic
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂×#1057	1	2	yes	—
167	C₂²×C₄	C₂×#1057	1	2	yes	—
168	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
169	C₂²×C₄	C₂×#1057	1	2	yes	—
170	C₂²×C₄	C₂²×#261	1	2	yes	—
171	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
172	M₁₆	—	26	8	no	—
173	C₂×C₈	—	13	4	no	—
174	C₂×C₈	—	13	4	no	—
175	C₂×C₈	—	13	1	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	2	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	26	4	no	—
181	C₂×C₈	—	26	4	no	—
182	C₂×C₈	—	26	2	no	—
183	C₂×C₈	—	26	2	no	—
184	C₄⋊C₄	C₂×#1057	1	8	yes	—
185	C₂²⋊C₄	C₂×#1057	1	8	yes	—
186	C₄²	C₂×#1057	1	2	yes	—
187	C₄²	C₂×#1057	1	2	yes	—
188	C₄²	C₂²×#261	1	2	yes	—
189	C₄²	C₂²×#261	1	2	yes	—
190	C₁₆	—	26	4	no	—
191	C₁₆	—	26	4	no	—
192	C₁₃	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	2	no	—
194	C₂³	—	2	4	no	—
195	C₂³	C₂²×#1057	1	2	yes	—
196	C₂³	C₂²×#1057	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
204	C₂×C₄	#88237	1	2	yes	—
205	C₂×C₄	C₂²×#1057	1	4	yes	—
206	C₂×C₄	C₂²×#1057	1	4	yes	—
207	C₂×C₄	C₂²×#1057	1	2	yes	—
208	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
209	C₂×C₄	C₂³×#261	1	1	yes	characteristic
210	C₈	—	13	4	no	—
211	C₈	—	13	2	no	—
212	C₈	—	13	2	no	—
213	C₈	—	26	2	no	—
214	C₈	—	26	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	#11719119	1	2	yes	—
219	C₂²	#11719162	1	4	yes	—
220	C₂²	C₂³×#1057	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	#11719119	1	2	yes	—
225	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
226	C₄	C₂×#88237	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	#13131201	1	4	yes	direct product factor
229	C₂	C₂×#11719119	1	1	yes	characteristic
230	C₂	C₂×#11719162	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#117	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#117	#261
hypocenter	C₁₃	C₂×#2164
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #13131201	C₂	direct (N × H)
2. C₂×C₄⋊#6092	C₂	semidirect
4. C₂×#11718231	C₂	semidirect
5. C₂×C₄×#6092	C₂	semidirect
19. #11718231	C₂²	semidirect
20. C₄⋊#6092	C₂²	semidirect
21. C₄×#6092	C₂²	semidirect
160. C₂×C₁₃	#2164	semidirect
192. C₁₃	C₂×#2164	semidirect
228. C₂	#13131201	direct (N × H)

Maximal subgroups

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

subgroup	index
#13131201	2
C₂×C₄⋊#6092	2
C₂×C₂²⋊#6092	2
C₂×#11718231	2
C₂×C₄×#6092	2
C₂×#11718658	2
C₂×#2164	13

Subgroups — C2×#13131201 #16807050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131201 #16807050