Subgroups — C₂×#13113732 #15832660

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718492	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13113732	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718076	C₂	1	2	yes	maximal
5	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₂	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₂₆	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88058	C₄	1	1	yes	characteristic
12	C₂×#88056	C₄	1	2	yes	—
13	C₂×#88055	—	2	2	no	—
14	C₂×#88054	—	2	2	no	—
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718492	—	2	4	no	—
18	#11718492	C₂²	1	8	yes	—
19	C₄×Dic₅₂	C₄	1	4	yes	—
20	C₄×Dic₅₂	C₂²	1	4	yes	—
21	C₁₃×#335	C₂²	1	4	yes	—
22	#11718076	C₂²	1	8	yes	—
23	C₈×#1055	C₂²	1	4	yes	—
24	C₂²×Dic₂₆	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
30	C₂×#6101	—	2	2	no	—
31	C₂×C₄×Dic₁₃	—	2	2	no	—
32	C₂²×#1055	—	2	1	no	—
33	C₂²×#1055	D₄	1	1	yes	characteristic
34	C₂²×#1055	D₄	1	1	yes	characteristic
35	C₂×Dic₅₂	—	2	4	no	—
36	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
37	C₄×Dic₂₆	—	2	2	no	—
38	C₄×Dic₂₆	—	2	2	no	—
39	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
40	C₄×Dic₂₆	C₂³	1	4	yes	—
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
43	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂³	1	2	yes	—
45	#88058	C₂×C₄	1	4	yes	—
46	#88056	C₂×C₄	1	8	yes	—
47	#88055	—	2	8	no	—
48	#88054	—	2	8	no	—
49	C₄×#1055	—	2	1	no	—
50	C₄×#1055	D₄	1	2	yes	—
51	C₄×#1055	C₂³	1	2	yes	—
52	C₄×#1055	C₂³	1	2	yes	—
53	C₂×#1591	—	13	1	no	maximal, Sylow
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	2	2	no	—
58	C₂²×Dic₁₃	—	2	2	no	—
59	C₂×Dic₂₆	—	2	4	no	—
60	C₂×Dic₂₆	—	2	4	no	—
61	C₂×Dic₂₆	—	2	2	no	—
62	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	4	4	no	—
66	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	#41	1	1	yes	characteristic
72	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₄⋊Dic₁₃	—	2	2	no	—
74	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
75	#6101	—	2	4	no	—
76	#6101	—	2	4	no	—
77	C₄×Dic₁₃	—	2	4	no	—
78	C₄×Dic₁₃	—	2	4	no	—
79	C₂×#1055	—	2	4	no	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	—	2	2	no	—
82	C₂×#1055	Q₁₆	1	4	yes	complemented
83	C₂×#1055	C₂×D₄	1	2	yes	—
84	C₂×#1055	C₂×D₄	1	2	yes	—
85	Dic₅₂	—	2	8	no	—
86	C₂×#446	—	13	2	no	—
87	C₂×C₄×Q₁₆	—	13	1	no	—
88	#1591	—	13	8	no	—
89	C₂×#335	D₁₃	1	1	yes	characteristic
90	C₂×Q₈⋊C₈	—	13	2	no	—
91	C₂×C₈²	—	13	1	no	—
92	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₁₃	#131	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₁₃	#135	1	2	yes	—
101	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
104	C₂×Dic₁₃	—	2	4	no	—
105	C₂×Dic₁₃	—	2	4	no	—
106	C₂×Dic₁₃	—	4	4	no	—
107	C₂×Dic₁₃	—	4	4	no	—
108	Dic₂₆	—	2	4	no	—
109	Dic₂₆	—	2	4	no	—
110	Dic₂₆	—	4	4	no	—
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	—	4	4	no	—
113	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
114	#1055	—	2	4	no	—
115	#1055	—	2	2	no	—
116	#1055	C₂×Q₁₆	1	4	yes	complemented
117	C₂²×Q₁₆	—	13	1	no	—
118	C₂×C₄×Q₈	—	13	2	no	—
119	#446	—	13	8	no	—
120	#446	—	26	4	no	—
121	C₄×Q₁₆	—	13	4	no	—
122	C₄×Q₁₆	—	13	4	no	—
123	C₂×#107	—	13	1	no	—
124	C₂×C₄⋊C₈	—	26	2	no	—
125	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
126	C₂×#103	—	13	2	no	—
127	C₂×#97	—	26	2	no	—
128	C₂×C₄×C₈	—	13	1	no	—
129	C₂×C₄×C₈	—	13	1	no	—
130	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
131	#335	C₂×D₁₃	1	4	yes	—
132	Q₈⋊C₈	—	13	8	no	—
133	C₈²	—	13	4	no	—
134	C₂²×C₁₃	#436	1	2	yes	—
135	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
136	C₂²×C₁₃	#443	1	2	yes	—
137	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	#436	1	2	yes	—
141	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
142	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
143	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
144	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
145	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
146	Dic₁₃	—	4	4	no	—
147	Dic₁₃	—	4	4	no	—
148	C₂²×Q₈	—	13	2	no	—
149	C₂×Q₁₆	—	13	4	no	—
150	C₂×Q₁₆	—	26	4	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	13	1	no	—
154	C₂²×C₈	—	26	1	no	—
155	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
156	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
157	C₄×Q₈	—	13	4	no	—
158	C₄×Q₈	—	13	4	no	—
159	C₄×Q₈	—	26	2	no	—
160	C₄×Q₈	—	26	2	no	—
161	C₂×C₄⋊C₄	—	13	2	no	—
162	C₂×C₄⋊C₄	—	26	2	no	—
163	C₂×C₄²	—	26	2	no	—
164	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
165	#107	—	13	4	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	26	8	no	—
168	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
169	#103	—	13	8	no	—
170	#97	—	26	8	no	—
171	C₄×C₈	—	13	2	no	—
172	C₄×C₈	—	13	2	no	—
173	C₄×C₈	—	13	2	no	—
174	C₄×C₈	—	26	1	no	—
175	C₄×C₈	C₄×D₁₃	1	2	yes	—
176	C₄×C₈	C₂²×D₁₃	1	2	yes	—
177	C₂×C₁₃	#1591	1	4	yes	complemented
178	C₂×C₁₃	C₂×#436	1	1	yes	characteristic
179	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
181	C₂×Q₈	—	13	4	no	—
182	C₂×Q₈	—	26	2	no	—
183	C₂×Q₈	—	26	4	no	—
184	C₂×Q₈	—	26	4	no	—
185	C₂²×C₄	—	26	2	no	—
186	C₂²×C₄	—	26	2	no	—
187	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
188	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
189	C₂²×C₄	#6131	1	1	yes	characteristic
190	Q₁₆	—	26	8	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	—	13	4	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	26	4	no	—
198	C₂×C₈	—	26	2	no	—
199	C₂×C₈	—	26	2	no	—
200	C₂×C₈	#6094	1	4	yes	—
201	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
202	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
203	C₄⋊C₄	—	13	4	no	—
204	C₄⋊C₄	—	26	2	no	—
205	C₄⋊C₄	—	26	4	no	—
206	C₄⋊C₄	—	26	4	no	—
207	C₄²	—	26	4	no	—
208	C₄²	—	26	4	no	—
209	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₄²	D₄×D₁₃	1	1	yes	characteristic
211	C₄²	#6131	1	1	yes	characteristic
212	C₄²	C₂³×D₁₃	1	1	yes	characteristic
213	C₁₃	C₂×#1591	1	1	yes	characteristic, complemented, Sylow
214	C₂³	#88147	1	1	yes	characteristic
215	Q₈	—	26	4	no	—
216	Q₈	—	26	4	no	—
217	Q₈	—	52	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	26	4	no	—
220	C₂×C₄	—	26	4	no	—
221	C₂×C₄	—	52	4	no	—
222	C₂×C₄	—	52	4	no	—
223	C₂×C₄	#88147	1	1	yes	characteristic
224	C₂×C₄	#88147	1	1	yes	characteristic
225	C₂×C₄	#88147	1	1	yes	characteristic
226	C₂×C₄	C₂×#6094	1	2	yes	—
227	C₂×C₄	#88161	1	2	yes	—
228	C₂×C₄	#88166	1	2	yes	—
229	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
230	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#6131	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	4	4	no	—
235	C₈	—	13	4	no	—
236	C₈	—	26	4	no	—
237	C₈	—	26	2	no	—
238	C₈	C₂×#6094	1	4	yes	—
239	C₂²	#11718499	1	2	yes	—
240	C₂²	#11718534	1	2	yes	—
241	C₂²	#11718556	1	2	yes	—
242	C₂²	C₂×#88147	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	52	4	no	—
246	C₄	—	52	4	no	—
247	C₄	#11718499	1	2	yes	—
248	C₄	#11718534	1	2	yes	—
249	C₄	C₂²×#6094	1	1	yes	characteristic
250	C₄	C₂×#88161	1	1	yes	characteristic
251	C₄	C₂×#88166	1	1	yes	characteristic
252	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
253	C₂	#13113732	1	4	yes	direct product factor
254	C₂	C₂×#11718499	1	1	yes	characteristic
255	C₂	C₂×#11718534	1	1	yes	characteristic
256	C₂	C₂×#11718556	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	C₂×#1591
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #13113732	C₂	direct (N × H)
82. C₂×#1055	Q₁₆	semidirect
116. #1055	C₂×Q₁₆	semidirect
177. C₂×C₁₃	#1591	semidirect
213. C₁₃	C₂×#1591	semidirect
253. C₂	#13113732	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718492	2
C₂×C₄×Dic₅₂	2
#13113732	2
C₂×#11718076	2
C₂×C₈×#1055	2
C₂×C₁₃×#335	2
C₂×#1591	13

Subgroups — C2×#13113732 #15832660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113732 #15832660