Subgroups — C₂×#13131057 #16806752

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 238 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	2	yes	maximal
2	#13131057	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718232	C₂	1	4	yes	maximal, complemented
4	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
6	C₂²×#6099	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂×#88106	C₄	1	4	yes	complemented
9	C₂×#88106	C₄	1	8	yes	complemented
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂×#88106	C₂²	1	8	yes	—
12	C₂²×#6092	C₄	1	4	yes	complemented
13	#11718658	C₄	1	4	yes	—
14	#11718658	C₂²	1	4	yes	—
15	C₂×#88043	C₂²	1	2	yes	—
16	C₂×C₄×#1055	C₄	1	2	yes	—
17	#11718232	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₁₃×#117	—	2	2	no	—
20	C₁₃×#117	C₂×C₄	1	4	yes	—
21	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
22	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
24	C₂×#6099	—	2	2	no	—
25	C₂×#6099	—	2	2	no	—
26	C₂×#6099	C₂×C₄	1	4	yes	—
27	C₂²×#1055	C₂×C₄	1	2	yes	—
28	C₂²×#1055	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	C₂×C₄	1	2	yes	—
32	C₂²×#1055	C₂×C₄	1	4	yes	—
33	C₂²×#1055	C₂³	1	2	yes	—
34	C₂²×#1055	C₂³	1	1	yes	characteristic
35	#88106	C₂×C₄	1	8	yes	—
36	#88106	C₂×C₄	1	16	yes	complemented
37	#88106	C₂³	1	8	yes	—
38	C₂×#6092	C₂×C₄	1	8	yes	—
39	C₂×#6092	C₂×C₄	1	16	yes	complemented
40	#88043	C₂×C₄	1	4	yes	—
41	#88043	C₂³	1	4	yes	—
42	C₄×#1055	C₂×C₄	1	8	yes	—
43	C₂×#2110	—	13	1	no	maximal, Sylow
44	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
47	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
48	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
49	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
50	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
54	C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₁₃×C₂²⋊C₄	—	2	4	no	—
56	C₄²×C₁₃	—	2	2	no	—
57	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
58	#6099	—	2	8	no	—
59	C₂×#1055	—	2	2	no	—
60	C₂×#1055	—	2	2	no	—
61	C₂×#1055	C₄²	1	4	yes	—
62	C₂×#1055	C₄²	1	4	yes	—
63	C₂×#1055	C₄²	1	2	yes	—
64	C₂×#1055	C₄²	1	1	yes	characteristic
65	C₂×#1055	C₄²	1	1	yes	characteristic
66	C₂×#1055	C₂²×C₄	1	4	yes	—
67	C₂×#1055	C₂²×C₄	1	2	yes	—
68	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
69	C₂×#1055	C₂²×C₄	1	2	yes	—
70	C₂×#1055	C₂²×C₄	1	4	yes	—
71	C₂×#1055	C₂²×C₄	1	2	yes	—
72	C₂×#1055	C₂²×C₄	1	4	yes	—
73	C₂×#1055	C₂⁴	1	1	yes	characteristic
74	#6092	C₂²×C₄	1	16	yes	—
75	C₂²×M₃₂	—	13	2	no	—
76	C₂×#405	—	13	1	no	—
77	#2110	—	13	8	no	—
78	C₂×#347	—	13	4	no	—
79	C₂³×C₁₃	—	4	2	no	—
80	C₂³×C₁₃	#97	1	2	yes	—
81	C₂³×C₁₃	#97	1	2	yes	—
82	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	4	4	no	—
86	C₂×C₄×C₁₃	#97	1	2	yes	—
87	C₂×C₄×C₁₃	#97	1	4	yes	—
88	C₂×C₄×C₁₃	#97	1	4	yes	—
89	C₂×C₄×C₁₃	#97	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
91	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
96	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	#1055	—	2	4	no	—
100	#1055	C₂×C₄²	1	2	yes	—
101	#1055	C₂×C₄²	1	2	yes	—
102	#1055	C₂×C₄²	1	2	yes	—
103	#1055	C₂³×C₄	1	2	yes	—
104	C₂²×M₁₆	—	13	1	no	—
105	C₂³×C₈	—	13	1	no	—
106	C₂×#117	#261	1	1	yes	characteristic
107	C₂×M₃₂	—	13	4	no	—
108	C₂×M₃₂	—	13	8	no	—
109	C₂×M₃₂	—	13	4	no	—
110	C₂×M₃₂	—	13	8	no	—
111	C₂²×C₁₆	—	13	4	no	—
112	#405	—	13	4	no	—
113	#405	—	13	4	no	—
114	C₂×#97	—	13	2	no	—
115	C₂×C₄×C₈	—	13	2	no	—
116	#347	—	13	16	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	—	2	4	no	—
119	C₂²×C₁₃	—	4	2	no	—
120	C₂²×C₁₃	—	4	2	no	—
121	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
122	C₂²×C₁₃	C₂×#97	1	4	yes	—
123	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
124	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
125	C₄×C₁₃	—	4	4	no	—
126	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
127	C₄×C₁₃	C₂×#97	1	4	yes	—
128	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
129	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
130	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
131	C₂³×C₄	C₂×#261	1	1	yes	characteristic
132	C₂×M₁₆	—	13	4	no	—
133	C₂×M₁₆	—	26	2	no	—
134	C₂×M₁₆	—	26	2	no	—
135	C₂²×C₈	—	13	2	no	—
136	C₂²×C₈	—	13	4	no	—
137	C₂²×C₈	—	13	2	no	—
138	C₂²×C₈	—	13	1	no	—
139	C₂²×C₈	—	13	2	no	—
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	2	no	—
142	C₂²×C₈	—	13	2	no	—
143	#117	—	2	2	no	—
144	#117	C₂×#261	1	4	yes	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₂²⋊C₄	—	2	1	no	—
147	C₂×C₄²	C₂×#261	1	2	yes	—
148	M₃₂	—	13	8	no	—
149	M₃₂	—	13	16	no	—
150	M₃₂	—	13	8	no	—
151	C₂×C₁₆	—	13	8	no	—
152	C₂×C₁₆	—	13	16	no	—
153	#97	—	13	4	no	—
154	#97	—	13	4	no	—
155	C₄×C₈	—	13	8	no	—
156	C₂×C₁₃	—	2	1	no	—
157	C₂×C₁₃	—	2	1	no	—
158	C₂×C₁₃	—	2	4	no	—
159	C₂×C₁₃	#2110	1	2	yes	complemented
160	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
161	C₂⁴	#6122	1	1	yes	characteristic
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	#6118	1	2	yes	—
164	C₂²×C₄	#6118	1	2	yes	—
165	C₂²×C₄	C₄×#261	1	2	yes	—
166	C₂²×C₄	C₄×#261	1	2	yes	—
167	C₂²×C₄	#6122	1	1	yes	characteristic
168	C₂²×C₄	#6122	1	2	yes	—
169	C₂²×C₄	C₂²×#261	1	2	yes	—
170	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
171	M₁₆	—	26	8	no	—
172	C₂×C₈	—	13	4	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	13	4	no	—
175	C₂×C₈	—	13	4	no	—
176	C₂×C₈	—	13	2	no	—
177	C₂×C₈	—	13	4	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	13	1	no	—
180	C₂×C₈	—	13	1	no	—
181	C₂×C₈	—	13	2	no	—
182	C₂×C₈	—	13	1	no	—
183	C₂×C₈	—	13	1	no	—
184	C₂×C₈	—	13	4	no	—
185	C₂×C₈	—	26	2	no	—
186	C₂×C₈	—	26	2	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₂²⋊C₄	—	2	4	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂²×#261	1	4	yes	—
191	C₁₆	—	13	16	no	—
192	C₁₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	4	2	no	—
194	C₂³	#88110	1	2	yes	—
195	C₂³	#88110	1	2	yes	—
196	C₂³	C₂×#6122	1	1	yes	characteristic
197	C₂³	C₂×#6122	1	2	yes	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	#88110	1	2	yes	—
201	C₂×C₄	#88110	1	4	yes	—
202	C₂×C₄	#88110	1	2	yes	—
203	C₂×C₄	#88110	1	4	yes	—
204	C₂×C₄	C₂×#6118	1	1	yes	characteristic
205	C₂×C₄	C₂×#6118	1	1	yes	characteristic
206	C₂×C₄	C₂×#6118	1	4	yes	—
207	C₂×C₄	C₂×C₄×#261	1	4	yes	—
208	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
210	C₂×C₄	C₂×#6122	1	1	yes	characteristic
211	C₂×C₄	C₂×#6122	1	2	yes	—
212	C₂×C₄	C₂³×#261	1	1	yes	characteristic
213	C₈	—	13	2	no	—
214	C₈	—	13	2	no	—
215	C₈	—	13	2	no	—
216	C₈	—	13	2	no	—
217	C₈	—	26	4	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×#88110	1	1	yes	characteristic
223	C₂²	C₂×#88110	1	4	yes	—
224	C₂²	C₂×#88110	1	1	yes	characteristic
225	C₂²	C₂²×#6122	1	1	yes	characteristic
226	C₄	—	4	4	no	—
227	C₄	C₂×#88110	1	1	yes	characteristic
228	C₄	C₂×#88110	1	4	yes	—
229	C₄	C₂×#88110	1	1	yes	characteristic
230	C₄	C₂²×#6118	1	1	yes	characteristic
231	C₄	C₂²×C₄×#261	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	2	4	no	—
235	C₂	#13131057	1	2	yes	direct product factor
236	C₂	C₂²×#88110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#117	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#117	#261
hypocenter	C₁₃	C₂×#2110
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #13131057	C₂	direct (N × H)
3. C₂×#11718232	C₂	semidirect
8. C₂×#88106	C₄	semidirect
9. C₂×#88106	C₄	semidirect
12. C₂²×#6092	C₄	semidirect
17. #11718232	C₂²	semidirect
36. #88106	C₂×C₄	semidirect
39. C₂×#6092	C₂×C₄	semidirect
159. C₂×C₁₃	#2110	semidirect
192. C₁₃	C₂×#2110	semidirect
235. C₂	#13131057	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
#13131057	2
C₂×#11718232	2
C₂×#11718658	2
C₂×#2110	13

Subgroups — C2×#13131057 #16806752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131057 #16806752