Subgroups — C₂×#13114553 #15860319

Order 3328.

2398 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88180	C₂	1	2	yes	maximal
2	#13114553	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718143	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	—	2	8	no	—
7	C₂³×C₈×C₁₃	C₄	1	8	yes	complemented
8	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	4	yes	—
10	C₂²×C₂²⋊Dic₁₃	—	2	2	no	—
11	C₂×#88180	—	2	4	no	—
12	C₂×#88180	—	2	8	no	—
13	C₂×#88180	C₄	1	16	yes	complemented
14	C₂×#88180	C₂²	1	8	yes	—
15	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
16	C₂²×C₄×Dic₁₃	—	2	2	no	—
17	#11718143	C₂²	1	16	yes	complemented
18	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₃	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	8	yes	—
24	C₂³×C₄×C₁₃	Q₈	1	4	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂³×Dic₁₃	—	2	4	no	—
28	C₂²×C₈×C₁₃	—	2	32	no	—
29	C₂²×C₈×C₁₃	—	2	8	no	—
30	C₂²×C₈×C₁₃	—	2	16	no	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	16	yes	complemented
33	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
34	C₂×C₂²⋊Dic₁₃	—	2	16	no	—
35	C₂×C₂²⋊Dic₁₃	—	2	8	no	—
36	#88180	—	2	32	no	—
37	#88180	—	2	16	no	—
38	#88180	C₂×C₄	1	32	yes	complemented
39	C₂×C₄⋊Dic₁₃	—	2	4	no	—
40	C₂×C₄⋊Dic₁₃	—	2	8	no	—
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	16	yes	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
43	C₂×C₄×Dic₁₃	—	2	16	no	—
44	C₂×C₄×Dic₁₃	—	2	8	no	—
45	C₂×#1735	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	—	2	8	no	—
47	C₂⁴×C₁₃	C₂²⋊C₄	1	8	yes	—
48	C₂⁴×C₁₃	C₄⋊C₄	1	4	yes	—
49	C₂⁴×C₁₃	C₂×D₄	1	2	yes	—
50	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₄²	1	16	yes	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	8	yes	—
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	16	yes	—
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	16	yes	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	32	yes	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	16	no	—
70	C₂²×Dic₁₃	—	2	8	no	—
71	C₂²×Dic₁₃	—	4	16	no	—
72	C₂×C₈×C₁₃	—	2	16	no	—
73	C₂×C₈×C₁₃	—	2	32	no	—
74	C₂×C₈×C₁₃	—	2	8	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
76	C₂²⋊Dic₁₃	—	2	32	no	—
77	C₄⋊Dic₁₃	—	2	16	no	—
78	C₄⋊Dic₁₃	—	2	8	no	—
79	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
80	C₄×Dic₁₃	—	2	32	no	—
81	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
82	C₂²×#117	—	13	2	no	—
83	#1735	—	13	8	no	—
84	C₂×#340	—	13	4	no	—
85	C₂³×C₁₃	—	2	8	no	—
86	C₂³×C₁₃	—	2	32	no	—
87	C₂³×C₁₃	—	2	16	no	—
88	C₂³×C₁₃	#95	1	16	yes	—
89	C₂³×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
90	C₂³×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
91	C₂³×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₁₃	#135	1	8	yes	—
94	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#95	1	32	yes	—
96	C₂×C₄×C₁₃	#95	1	16	yes	—
97	C₂×C₄×C₁₃	C₂×C₄²	1	8	yes	—
98	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×Dic₁₃	—	2	16	no	—
112	C₂×Dic₁₃	—	4	16	no	—
113	C₂×Dic₁₃	—	4	32	no	—
114	C₈×C₁₃	—	2	8	no	—
115	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
116	C₂³×C₈	—	2	8	no	—
117	C₂³×C₈	Dic₁₃	1	8	yes	complemented
118	C₂³×C₈	C₂×D₁₃	1	2	yes	—
119	C₂³×C₈	C₂×D₁₃	1	4	yes	—
120	C₂×#117	—	13	16	no	—
121	C₂×#117	—	13	8	no	—
122	C₂×#117	—	26	4	no	—
123	C₂×#117	—	26	8	no	—
124	C₂²×C₄⋊C₄	—	13	4	no	—
125	C₂²×C₂²⋊C₄	—	26	2	no	—
126	C₂²×C₄²	—	26	2	no	—
127	#340	—	13	16	no	—
128	C₂²×C₁₃	—	2	16	no	—
129	C₂²×C₁₃	—	2	8	no	—
130	C₂²×C₁₃	—	2	32	no	—
131	C₂²×C₁₃	C₂×#95	1	8	yes	—
132	C₂²×C₁₃	C₂×#95	1	4	yes	—
133	C₂²×C₁₃	#416	1	16	yes	—
134	C₂²×C₁₃	#427	1	8	yes	—
135	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	2	yes	—
136	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×#135	1	4	yes	—
138	C₄×C₁₃	C₂×#95	1	8	yes	—
139	C₄×C₁₃	C₂×#95	1	4	yes	—
140	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
141	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
143	Dic₁₃	—	4	16	no	—
144	C₂⁵	D₅₂	1	1	yes	characteristic
145	C₂³×C₄	—	26	4	no	—
146	C₂³×C₄	Dic₂₆	1	4	yes	—
147	C₂³×C₄	C₄×D₁₃	1	8	yes	—
148	C₂³×C₄	D₅₂	1	2	yes	—
149	C₂³×C₄	D₅₂	1	1	yes	characteristic
150	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
151	C₂³×C₄	C₂²⋊D₁₃	1	8	yes	—
152	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
153	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
154	C₂²×C₈	—	2	32	no	—
155	C₂²×C₈	—	2	8	no	—
156	C₂²×C₈	—	2	16	no	—
157	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
158	C₂²×C₈	C₂×Dic₁₃	1	16	yes	complemented
159	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
160	#117	—	13	32	no	—
161	#117	—	26	32	no	—
162	#117	—	26	16	no	—
163	C₂×C₄⋊C₄	—	13	16	no	—
164	C₂×C₄⋊C₄	—	13	8	no	—
165	C₂×C₄⋊C₄	—	26	4	no	—
166	C₂×C₄⋊C₄	—	26	8	no	—
167	C₂×C₂²⋊C₄	—	26	16	no	—
168	C₂×C₂²⋊C₄	—	26	8	no	—
169	C₂×C₄²	—	26	16	no	—
170	C₂×C₄²	—	26	8	no	—
171	C₂×C₁₃	—	2	8	no	—
172	C₂×C₁₃	#1735	1	8	yes	complemented
173	C₂×C₁₃	C₂²×#95	1	1	yes	characteristic
174	C₂×C₁₃	C₂×#416	1	4	yes	—
175	C₂×C₁₃	C₂×#427	1	2	yes	—
176	C₂⁴	—	2	8	no	—
177	C₂⁴	C₄⋊Dic₁₃	1	4	yes	—
178	C₂⁴	#6103	1	8	yes	—
179	C₂⁴	C₂×D₅₂	1	2	yes	—
180	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
181	C₂²×C₄	—	26	16	no	—
182	C₂²×C₄	—	26	8	no	—
183	C₂²×C₄	—	52	16	no	—
184	C₂²×C₄	C₄×Dic₁₃	1	16	yes	—
185	C₂²×C₄	#6101	1	32	yes	—
186	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
187	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
188	C₂²×C₄	#6103	1	8	yes	—
189	C₂²×C₄	#6103	1	16	yes	—
190	C₂²×C₄	C₂²⋊Dic₁₃	1	16	yes	—
191	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
192	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
193	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
194	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
195	C₂²×C₄	C₂×D₅₂	1	2	yes	—
196	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₁₃	1	8	yes	—
201	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
202	C₂×C₈	—	2	16	no	—
203	C₂×C₈	—	2	32	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
206	C₄⋊C₄	—	13	16	no	—
207	C₄⋊C₄	—	26	16	no	—
208	C₄⋊C₄	—	26	8	no	—
209	C₂²⋊C₄	—	26	32	no	—
210	C₄²	—	26	32	no	—
211	C₁₃	C₂×#1735	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	8	no	—
213	C₂³	—	2	16	no	—
214	C₂³	—	2	32	no	—
215	C₂³	#88071	1	16	yes	—
216	C₂³	#88158	1	8	yes	—
217	C₂³	C₂×C₄⋊Dic₁₃	1	4	yes	—
218	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
219	C₂³	C₂×#6103	1	8	yes	—
220	C₂³	C₂×#6103	1	4	yes	—
221	C₂³	C₂²×D₅₂	1	1	yes	characteristic
222	C₂×C₄	—	26	16	no	—
223	C₂×C₄	—	52	16	no	—
224	C₂×C₄	—	52	32	no	—
225	C₂×C₄	#88071	1	32	yes	—
226	C₂×C₄	#88071	1	16	yes	—
227	C₂×C₄	C₂×C₄×Dic₁₃	1	8	yes	—
228	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
229	C₂×C₄	C₂×#6101	1	16	yes	—
230	C₂×C₄	C₂×#6101	1	8	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
233	C₂×C₄	C₂×#6103	1	8	yes	—
234	C₂×C₄	C₂×#6103	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	8	yes	—
236	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	4	yes	—
237	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
239	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₂²⋊D₁₃	1	2	yes	—
241	C₈	—	2	8	no	—
242	C₂²	—	2	16	no	—
243	C₂²	—	2	8	no	—
244	C₂²	—	2	32	no	—
245	C₂²	#11718767	1	8	yes	—
246	C₂²	#11718774	1	16	yes	—
247	C₂²	C₂×#88071	1	8	yes	—
248	C₂²	C₂×#88071	1	4	yes	—
249	C₂²	C₂×#88158	1	4	yes	—
250	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
251	C₂²	C₂²×#6103	1	2	yes	—
252	C₄	—	52	16	no	—
253	C₄	C₂×#88071	1	8	yes	—
254	C₄	C₂×#88071	1	4	yes	—
255	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
256	C₄	C₂²×#6101	1	2	yes	—
257	C₄	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
258	C₂	—	2	8	no	—
259	C₂	#13114553	1	8	yes	direct product factor
260	C₂	C₂²×#88071	1	1	yes	characteristic
261	C₂	C₂×#11718774	1	4	yes	—
262	C₂	C₂×#11718767	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂×D₄
prime radical	C₂⁵×C₁₃	D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	C₂×#1735
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. #13114553	C₂	direct (N × H)
3. C₂×#11718143	C₂	semidirect
7. C₂³×C₈×C₁₃	C₄	semidirect
13. C₂×#88180	C₄	semidirect
17. #11718143	C₂²	semidirect
32. C₂²×C₈×C₁₃	C₂×C₄	semidirect
38. #88180	C₂×C₄	semidirect
117. C₂³×C₈	Dic₁₃	semidirect
158. C₂²×C₈	C₂×Dic₁₃	semidirect
172. C₂×C₁₃	#1735	semidirect
211. C₁₃	C₂×#1735	semidirect
259. C₂	#13114553	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×#88180	2
#13114553	2
C₂×#11718143	2
C₂⁴×C₈×C₁₃	2
C₂×#1735	13

Subgroups — C2×#13114553 #15860319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114553 #15860319