Subgroups — C₂×#13123938 #16727883

Order 3328.

1794 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123938	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11718685	C₂	1	4	yes	maximal, complemented
3	C₂×Q₈×#1055	C₂	1	4	yes	maximal, complemented
4	C₂×#11718661	C₂	1	4	yes	maximal
5	C₂×#11718586	C₂	1	2	yes	maximal
6	C₂×C₁₃×#533	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#533	C₄	1	8	yes	—
8	C₁₃×#533	C₂²	1	8	yes	—
9	C₂×C₁₃×C₄⋊Q₈	C₄	1	2	yes	—
10	C₂×C₁₃×#125	C₄	1	2	yes	—
11	C₂×C₁₃×C₂²⋊Q₈	C₄	1	4	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
13	C₂×C₁₃×#117	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
16	#11718685	C₂²	1	32	yes	complemented
17	Q₈×#1055	C₂²	1	32	yes	complemented
18	#11718661	C₂²	1	32	yes	—
19	#11718586	C₂²	1	16	yes	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	C₂×C₄⋊#1055	C₂²	1	8	yes	—
23	C₂×#88043	C₂²	1	4	yes	—
24	C₂×C₄×#1055	C₂²	1	4	yes	—
25	C₂×C₄×#1055	C₂²	1	4	yes	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×C₄⋊Q₈	C₂×C₄	1	16	yes	—
30	C₁₃×#125	C₂×C₄	1	16	yes	—
31	C₁₃×C₂²⋊Q₈	C₂×C₄	1	32	yes	—
32	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	—
33	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
34	C₁₃×#117	C₂×C₄	1	8	yes	—
35	C₁₃×#117	C₂³	1	8	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
46	C₂²×#1055	C₂³	1	4	yes	—
47	C₂²×#1055	C₂³	1	4	yes	—
48	C₂²⋊#1055	C₂³	1	16	yes	—
49	C₄⋊#1055	C₂³	1	32	yes	—
50	C₄⋊#1055	C₂³	1	8	yes	—
51	C₄⋊#1055	C₂³	1	8	yes	—
52	#88043	C₂³	1	16	yes	—
53	C₄×#1055	C₂³	1	16	yes	—
54	C₄×#1055	C₂³	1	16	yes	—
55	C₂×#2986	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×Q₈×C₁₃	—	2	16	no	—
58	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
59	C₂²×C₄×C₁₃	—	2	8	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	C₂²×C₄	1	16	yes	—
70	C₁₃×C₄⋊C₄	C₂²×C₄	1	32	yes	—
71	C₁₃×C₄⋊C₄	C₂²×C₄	1	16	yes	—
72	C₁₃×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
73	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
74	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
77	C₄²×C₁₃	C₂⁴	1	4	yes	—
78	C₄²×C₁₃	C₂⁴	1	8	yes	—
79	C₂×#1055	—	2	8	no	—
80	C₂×#1055	—	2	8	no	—
81	C₂×#1055	C₂⁴	1	8	yes	—
82	C₂×#1055	C₂⁴	1	8	yes	—
83	C₂×#533	D₁₃	1	1	yes	characteristic
84	#2986	—	13	16	no	—
85	C₂×#446	—	13	4	no	—
86	C₂×C₈×Q₈	—	13	4	no	—
87	C₂×#433	—	13	4	no	—
88	C₂×#431	—	13	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	2	4	no	—
91	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
92	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₁₃	—	2	32	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	16	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	2	16	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	#131	1	8	yes	—
101	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₂³×C₄	1	16	yes	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
107	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
108	#1055	—	2	8	no	—
109	#1055	—	2	8	no	—
110	#533	Dic₁₃	1	8	yes	—
111	#533	C₂×D₁₃	1	8	yes	—
112	C₂×C₄⋊Q₈	Dic₁₃	1	2	yes	—
113	C₂×#125	Dic₁₃	1	2	yes	—
114	C₂×C₂²⋊Q₈	Dic₁₃	1	4	yes	—
115	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
116	C₂×#117	C₂×D₁₃	1	2	yes	—
117	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
118	#446	—	13	32	no	—
119	C₈×Q₈	—	13	32	no	—
120	#433	—	13	32	no	—
121	#431	—	13	16	no	—
122	C₂×C₄⋊C₈	—	13	2	no	—
123	C₂×C₄⋊C₈	—	13	2	no	—
124	C₂×C₄⋊C₈	—	13	8	no	—
125	C₂×C₂²⋊C₈	—	13	4	no	—
126	C₂×#97	—	13	4	no	—
127	C₂×C₄×C₈	—	13	4	no	—
128	C₂×C₄×C₈	—	13	4	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	4	4	no	—
132	C₂²×C₁₃	#519	1	2	yes	—
133	C₂²×C₁₃	C₂×#131	1	4	yes	—
134	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	—	2	16	no	—
138	C₄×C₁₃	#519	1	2	yes	—
139	C₄×C₁₃	C₂×#131	1	8	yes	—
140	C₄×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
141	C₂²×Q₈	C₂×Dic₁₃	1	4	yes	—
142	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
143	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
144	C₂²×C₈	—	13	4	no	—
145	C₂²×C₈	—	13	4	no	—
146	C₄⋊Q₈	C₂×Dic₁₃	1	16	yes	—
147	#125	C₂×Dic₁₃	1	16	yes	—
148	C₂²⋊Q₈	C₂×Dic₁₃	1	32	yes	—
149	C₄×Q₈	C₂×Dic₁₃	1	16	yes	—
150	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
151	#117	C₂×Dic₁₃	1	8	yes	—
152	#117	C₂²×D₁₃	1	8	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
155	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
156	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
157	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
160	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
161	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
162	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
163	C₄⋊C₈	—	13	8	no	—
164	C₄⋊C₈	—	13	8	no	—
165	C₄⋊C₈	—	13	32	no	—
166	C₂²⋊C₈	—	13	16	no	—
167	#97	—	13	16	no	—
168	C₄×C₈	—	13	16	no	—
169	C₄×C₈	—	13	16	no	—
170	C₂×C₁₃	—	4	2	no	—
171	C₂×C₁₃	#2986	1	4	yes	complemented
172	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
173	C₂×C₁₃	C₂²×#131	1	2	yes	—
174	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
175	C₂×Q₈	—	2	16	no	—
176	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
183	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
184	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
185	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
186	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
187	C₂×C₈	—	13	8	no	—
188	C₂×C₈	—	13	8	no	—
189	C₂×C₈	—	26	8	no	—
190	C₂×C₈	—	26	8	no	—
191	C₄⋊C₄	C₂²×Dic₁₃	1	16	yes	—
192	C₄⋊C₄	C₂²×Dic₁₃	1	32	yes	—
193	C₄⋊C₄	C₂²×Dic₁₃	1	16	yes	—
194	C₂²⋊C₄	C₂²×Dic₁₃	1	16	yes	—
195	C₄²	C₂²×Dic₁₃	1	4	yes	—
196	C₄²	C₂²×Dic₁₃	1	4	yes	—
197	C₄²	C₂²×Dic₁₃	1	4	yes	—
198	C₄²	C₂²×Dic₁₃	1	8	yes	—
199	C₄²	C₂³×D₁₃	1	4	yes	—
200	C₄²	C₂³×D₁₃	1	8	yes	—
201	C₁₃	C₂×#2986	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	2	2	no	—
203	C₂³	—	2	4	no	—
204	C₂³	C₂³×Dic₁₃	1	2	yes	—
205	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
206	Q₈	—	2	32	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	#88202	1	8	yes	—
214	C₂×C₄	#88254	1	4	yes	—
215	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
216	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
217	C₂×C₄	C₂³×Dic₁₃	1	16	yes	—
218	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
220	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
221	C₈	—	26	8	no	—
222	C₈	—	26	8	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#11719517	1	2	yes	—
227	C₂²	C₂×#88202	1	4	yes	—
228	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	2	16	no	—
231	C₄	—	2	2	no	—
232	C₄	#11719517	1	2	yes	—
233	C₄	C₂×#88202	1	8	yes	—
234	C₄	C₂×#88254	1	2	yes	—
235	C₂	—	4	2	no	—
236	C₂	#13123938	1	4	yes	direct product factor
237	C₂	C₂×#11719517	1	1	yes	characteristic
238	C₂	C₂²×#88202	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#533	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×#533	D₁₃
hypocenter	C₁₃	C₂×#2986
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #13123938	C₂	direct (N × H)
2. C₂×#11718685	C₂	semidirect
3. C₂×Q₈×#1055	C₂	semidirect
16. #11718685	C₂²	semidirect
17. Q₈×#1055	C₂²	semidirect
171. C₂×C₁₃	#2986	semidirect
201. C₁₃	C₂×#2986	semidirect
236. C₂	#13123938	direct (N × H)

Maximal subgroups

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

subgroup	index
#13123938	2
C₂×#11718685	2
C₂×Q₈×#1055	2
C₂×#11718661	2
C₂×#11718586	2
C₂×C₁₃×#533	2
C₂×#2986	13

Subgroups — C2×#13123938 #16727883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123938 #16727883