Subgroups — C₂×#13131045 #16806723

Order 3328.

854 conjugacy classes of subgroups.

The subgroup lattice has 224 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	#13131045	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718231	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#6092	C₂	1	2	yes	maximal, 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	C₂×#88106	C₄	1	8	yes	complemented
11	C₂×#88106	C₂²	1	4	yes	—
12	C₂²×#6092	C₄	1	8	yes	complemented
13	C₂²×#6092	C₂²	1	1	yes	characteristic
14	C₂²×#6092	C₂²	1	1	yes	characteristic
15	C₂²×#6092	C₂²	1	2	yes	—
16	C₂²×#6092	C₂²	1	4	yes	—
17	#11718658	C₄	1	4	yes	—
18	#11718658	C₂²	1	4	yes	—
19	C₂×#88043	C₄	1	2	yes	—
20	C₂×C₄×#1055	C₂²	1	2	yes	—
21	#11718231	C₂²	1	8	yes	complemented
22	C₄×#6092	C₂²	1	8	yes	complemented
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₁₃×#117	C₈	1	4	yes	—
25	C₁₃×#117	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
27	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×#6099	C₈	1	8	yes	—
30	C₂×#6099	C₂×C₄	1	4	yes	—
31	C₂²×#1055	C₂×C₄	1	2	yes	—
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	4	yes	—
34	C₂²×#1055	C₂×C₄	1	4	yes	—
35	C₂²×#1055	C₂×C₄	1	2	yes	—
36	C₂²×#1055	C₂³	1	2	yes	—
37	C₂²×#1055	C₂³	1	1	yes	characteristic
38	#88106	C₂×C₄	1	16	yes	complemented
39	C₂×#6092	C₂×C₄	1	4	yes	—
40	C₂×#6092	C₂×C₄	1	4	yes	—
41	C₂×#6092	C₂×C₄	1	8	yes	—
42	C₂×#6092	C₂×C₄	1	16	yes	complemented
43	C₂×#6092	C₂³	1	2	yes	—
44	C₂×#6092	C₂³	1	4	yes	—
45	C₂×#6092	C₂³	1	2	yes	—
46	#88043	C₂×C₄	1	8	yes	—
47	C₄×#1055	C₂×C₄	1	4	yes	—
48	C₄×#1055	C₂³	1	4	yes	—
49	C₂×#2108	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	4	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	4	yes	—
61	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
62	#6099	C₂×C₈	1	16	yes	—
63	C₂×#1055	C₄²	1	8	yes	—
64	C₂×#1055	C₄²	1	4	yes	—
65	C₂×#1055	C₂×C₈	1	8	yes	—
66	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
67	C₂×#1055	C₂²×C₄	1	2	yes	—
68	C₂×#1055	C₂²×C₄	1	2	yes	—
69	C₂×#1055	C₂²×C₄	1	2	yes	—
70	C₂×#1055	C₂²×C₄	1	4	yes	—
71	C₂×#1055	C₂²×C₄	1	4	yes	—
72	C₂×#1055	C₂²×C₄	1	4	yes	—
73	C₂×#1055	C₂⁴	1	1	yes	characteristic
74	#6092	C₂²×C₄	1	4	yes	—
75	#6092	C₂²×C₄	1	4	yes	—
76	#6092	C₂²×C₄	1	8	yes	—
77	C₂²×M₃₂	—	13	1	no	—
78	C₂³×C₁₆	—	13	1	no	—
79	C₂×#405	—	13	1	no	—
80	#2108	—	13	8	no	—
81	C₂×#346	—	13	2	no	—
82	C₂×C₄×C₁₆	—	13	2	no	—
83	C₂³×C₁₃	—	2	4	no	—
84	C₂³×C₁₃	C₄×C₈	1	4	yes	—
85	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
86	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	—	2	8	no	—
88	C₂×C₄×C₁₃	C₄×C₈	1	8	yes	—
89	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
90	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
91	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
93	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
95	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	#1055	C₂×C₄²	1	2	yes	—
99	#1055	C₂×C₄²	1	4	yes	—
100	#1055	C₂²×C₈	1	8	yes	—
101	#1055	C₂³×C₄	1	1	yes	characteristic
102	#1055	C₂³×C₄	1	1	yes	characteristic
103	C₂²×M₁₆	—	13	1	no	—
104	C₂³×C₈	—	13	1	no	—
105	C₂×#117	#261	1	1	yes	characteristic
106	C₂×M₃₂	—	13	4	no	—
107	C₂×M₃₂	—	13	8	no	—
108	C₂²×C₁₆	—	13	1	no	—
109	C₂²×C₁₆	—	13	2	no	—
110	C₂²×C₁₆	—	13	1	no	—
111	C₂²×C₁₆	—	13	4	no	—
112	C₂²×C₁₆	—	13	8	no	—
113	#405	—	13	4	no	—
114	#405	—	13	4	no	—
115	C₂×#97	—	13	2	no	—
116	C₂×C₄×C₈	—	13	2	no	—
117	#346	—	13	8	no	—
118	C₄×C₁₆	—	13	8	no	—
119	C₂²×C₁₃	—	2	8	no	—
120	C₂²×C₁₃	—	2	4	no	—
121	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
122	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
123	C₂²×C₁₃	#504	1	4	yes	—
124	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
125	C₄×C₁₃	—	2	8	no	—
126	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
127	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
128	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
129	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
130	C₂³×C₄	C₂×#261	1	1	yes	characteristic
131	C₂×M₁₆	—	13	8	no	—
132	C₂×M₁₆	—	13	4	no	—
133	C₂²×C₈	—	13	4	no	—
134	C₂²×C₈	—	13	4	no	—
135	C₂²×C₈	—	13	2	no	—
136	C₂²×C₈	—	13	2	no	—
137	C₂²×C₈	—	13	1	no	—
138	C₂²×C₈	—	13	1	no	—
139	C₂²×C₈	—	13	2	no	—
140	#117	#1057	1	4	yes	—
141	#117	C₂×#261	1	4	yes	—
142	C₂×C₄⋊C₄	#1057	1	2	yes	—
143	C₂×C₂²⋊C₄	#1057	1	2	yes	—
144	C₂×C₄²	C₂×#261	1	2	yes	—
145	M₃₂	—	13	16	no	—
146	C₂×C₁₆	—	13	4	no	—
147	C₂×C₁₆	—	13	4	no	—
148	C₂×C₁₆	—	13	8	no	—
149	C₂×C₁₆	—	13	16	no	—
150	C₂×C₁₆	—	13	2	no	—
151	C₂×C₁₆	—	13	4	no	—
152	C₂×C₁₆	—	13	2	no	—
153	#97	—	13	8	no	—
154	C₄×C₈	—	13	4	no	—
155	C₄×C₈	—	13	4	no	—
156	C₂×C₁₃	—	2	4	no	—
157	C₂×C₁₃	#2108	1	4	yes	complemented
158	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
159	C₂×C₁₃	C₂×#504	1	2	yes	—
160	C₂⁴	C₂×#1057	1	1	yes	characteristic
161	C₂²×C₄	D₁₃⋊C₈	1	4	yes	—
162	C₂²×C₄	C₄×#261	1	4	yes	—
163	C₂²×C₄	C₂×#1057	1	4	yes	—
164	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
165	C₂²×C₄	C₂×#1057	1	2	yes	—
166	C₂²×C₄	C₂²×#261	1	2	yes	—
167	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
168	M₁₆	—	13	16	no	—
169	C₂×C₈	—	13	8	no	—
170	C₂×C₈	—	13	4	no	—
171	C₂×C₈	—	13	2	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	13	4	no	—
174	C₂×C₈	—	13	8	no	—
175	C₂×C₈	—	13	4	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	1	no	—
178	C₂×C₈	—	13	4	no	—
179	C₂×C₈	—	13	2	no	—
180	C₄⋊C₄	C₂×#1057	1	8	yes	—
181	C₂²⋊C₄	C₂×#1057	1	8	yes	—
182	C₄²	C₂×#1057	1	4	yes	—
183	C₄²	C₂²×#261	1	4	yes	—
184	C₁₆	—	13	4	no	—
185	C₁₆	—	13	4	no	—
186	C₁₆	—	13	8	no	—
187	C₁₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	2	4	no	—
189	C₂³	C₄×#1057	1	4	yes	—
190	C₂³	C₂²×#1057	1	2	yes	—
191	C₂³	C₂²×#1057	1	1	yes	characteristic
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	C₄×#1057	1	8	yes	—
194	C₂×C₄	C₄×#1057	1	4	yes	—
195	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
196	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
197	C₂×C₄	C₂×C₄×#261	1	4	yes	—
198	C₂×C₄	C₂×C₄×#261	1	2	yes	—
199	C₂×C₄	C₂²×#1057	1	8	yes	—
200	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
201	C₂×C₄	C₂²×#1057	1	2	yes	—
202	C₂×C₄	C₂³×#261	1	1	yes	characteristic
203	C₈	—	13	8	no	—
204	C₈	—	13	1	no	—
205	C₈	—	13	1	no	—
206	C₈	—	13	2	no	—
207	C₈	—	13	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	8	no	—
210	C₂²	C₂×C₄×#1057	1	4	yes	—
211	C₂²	C₂×C₄×#1057	1	2	yes	—
212	C₂²	#11719162	1	4	yes	—
213	C₂²	C₂³×#1057	1	1	yes	characteristic
214	C₄	—	2	8	no	—
215	C₄	C₂×C₄×#1057	1	2	yes	—
216	C₄	C₂×C₄×#1057	1	4	yes	—
217	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
218	C₄	C₂²×C₄×#261	1	1	yes	characteristic
219	C₂	—	2	4	no	—
220	C₂	#13131045	1	4	yes	direct product factor
221	C₂	C₂²×C₄×#1057	1	1	yes	characteristic
222	C₂	C₂×#11719162	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂²×C₄×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×#2108
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #13131045	C₂	direct (N × H)
4. C₂×#11718231	C₂	semidirect
5. C₂×C₄×#6092	C₂	semidirect
10. C₂×#88106	C₄	semidirect
12. C₂²×#6092	C₄	semidirect
21. #11718231	C₂²	semidirect
22. C₄×#6092	C₂²	semidirect
38. #88106	C₂×C₄	semidirect
42. C₂×#6092	C₂×C₄	semidirect
157. C₂×C₁₃	#2108	semidirect
187. C₁₃	C₂×#2108	semidirect
220. C₂	#13131045	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
#13131045	2
C₂×#11718231	2
C₂×C₄×#6092	2
C₂×#11718658	2
C₂×#2108	13

Subgroups — C2×#13131045 #16806723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131045 #16806723