Subgroups — C₂×#13130771 #16798267

Order 3328.

1358 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	C₂²×#88101	C₂	1	4	yes	maximal
2	C₂×#11719059	C₂	1	2	yes	maximal
3	#13130771	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#261	—	2	4	no	—
6	C₂²×C₄⋊#261	C₂²	1	4	yes	—
7	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88101	—	2	16	no	—
10	C₂×#88101	C₂²	1	16	yes	—
11	#11719059	—	2	4	no	—
12	#11719059	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×Dic₁₃	C₄	1	2	yes	complemented
15	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
16	C₄×C₈×D₁₃	C₄	1	4	yes	complemented
17	C₄×C₈×D₁₃	C₂²	1	4	yes	—
18	C₂³×#261	—	2	4	no	—
19	C₂²×C₄×D₁₃	D₄	1	2	yes	—
20	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊#261	—	2	16	no	—
22	C₂×C₄⋊#261	—	2	4	no	—
23	C₂×C₄⋊#261	—	2	16	no	—
24	C₂×C₄⋊#261	—	4	16	no	—
25	C₂×C₄⋊#261	C₂³	1	8	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
30	C₂²×#1055	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₁₃	D₄	1	8	yes	—
33	C₂×C₈×D₁₃	Q₈	1	8	yes	—
34	C₂×C₈×D₁₃	C₂³	1	4	yes	—
35	C₄²×D₁₃	C₂×C₄	1	2	yes	—
36	C₄²×D₁₃	D₄	1	2	yes	—
37	C₄²×D₁₃	Q₈	1	2	yes	—
38	C₄²×D₁₃	C₂³	1	2	yes	—
39	#88101	—	2	32	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
41	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
42	C₄×#1055	C₂×C₄	1	4	yes	complemented
43	C₂×#1844	—	13	1	no	maximal, Sylow
44	C₂³×D₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×#261	—	2	8	no	—
46	C₂²×#261	—	4	8	no	—
47	C₂²×#261	—	4	16	no	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×D₁₃	SD₁₆	1	16	yes	—
54	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₃	C₂×D₄	1	4	yes	—
56	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×D₁₃	C₂×D₄	1	4	yes	—
58	C₂×C₄×D₁₃	C₂×Q₈	1	4	yes	—
59	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
61	C₄⋊#261	—	2	16	no	—
62	C₄⋊#261	—	4	8	no	—
63	C₄⋊#261	—	4	32	no	—
64	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	complemented
65	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
68	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
69	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
70	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
71	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
72	C₂×#1055	C₄⋊C₄	1	8	yes	complemented
73	C₂×#1055	C₂²×C₄	1	4	yes	—
74	C₈×D₁₃	C₄⋊C₄	1	16	yes	complemented
75	C₈×D₁₃	C₂×D₄	1	8	yes	—
76	C₈×D₁₃	C₂×Q₈	1	8	yes	—
77	C₂²×#106	—	13	4	no	—
78	C₂²×C₄×C₈	—	13	1	no	—
79	C₂×#384	—	13	2	no	—
80	#1844	—	13	8	no	—
81	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×D₁₃	#127	1	2	yes	—
84	C₂²×D₁₃	C₄⋊Q₈	1	2	yes	—
85	C₂²×D₁₃	C₂×SD₁₆	1	8	yes	—
86	C₂²×D₁₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×#261	—	4	8	no	—
88	C₂×#261	—	4	16	no	—
89	C₂×#261	—	8	16	no	—
90	C₂×C₄×C₁₃	#106	1	8	yes	complemented
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×Dic₁₃	#106	1	8	yes	complemented
96	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
99	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
100	C₄×D₁₃	#106	1	16	yes	complemented
101	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₄×D₁₃	#127	1	2	yes	—
103	C₄×D₁₃	C₄⋊Q₈	1	4	yes	—
104	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
105	C₄×D₁₃	C₂×SD₁₆	1	16	yes	—
106	C₄×D₁₃	C₂²×D₄	1	2	yes	—
107	C₄×D₁₃	C₂²×Q₈	1	2	yes	—
108	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	complemented
109	#1055	C₂×C₄⋊C₄	1	8	yes	complemented
110	C₂³×C₈	—	13	2	no	—
111	C₂²×C₄⋊C₄	—	13	4	no	—
112	C₂²×C₄⋊C₄	—	26	4	no	—
113	C₂²×C₄²	—	13	1	no	—
114	C₂×#106	—	13	16	no	—
115	C₂×#106	—	26	16	no	—
116	C₂×C₄×C₈	—	13	2	no	—
117	C₂×C₄×C₈	—	13	1	no	—
118	C₂×C₄×C₈	—	13	4	no	—
119	C₂×C₄×C₈	—	13	4	no	—
120	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
121	#384	—	13	8	no	—
122	#384	—	26	4	no	—
123	C₂²×C₁₃	#384	1	2	yes	—
124	C₂²×C₁₃	C₂×#106	1	4	yes	—
125	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂×D₁₃	#384	1	2	yes	—
127	C₂×D₁₃	C₂×#106	1	4	yes	—
128	C₂×D₁₃	#492	1	8	yes	—
129	C₂×D₁₃	#498	1	8	yes	—
130	C₂×D₁₃	C₂×#127	1	1	yes	characteristic
131	C₂×D₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×D₁₃	C₂²×SD₁₆	1	4	yes	—
133	#261	—	8	8	no	—
134	C₄×C₁₃	#384	1	2	yes	—
135	C₄×C₁₃	C₂×#106	1	8	yes	complemented
136	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
137	Dic₁₃	#384	1	2	yes	—
138	Dic₁₃	C₂×#106	1	8	yes	complemented
139	Dic₁₃	C₂×C₄⋊Q₈	1	2	yes	—
140	C₂³×C₄	—	13	2	no	—
141	C₂³×C₄	—	13	1	no	—
142	C₂³×C₄	—	26	4	no	—
143	C₂²×C₈	—	13	2	no	—
144	C₂²×C₈	—	13	8	no	—
145	C₂²×C₈	—	13	8	no	—
146	C₂²×C₈	—	13	4	no	—
147	C₂²×C₈	—	13	4	no	—
148	C₂²×C₈	C₂×#261	1	2	yes	—
149	C₂×C₄⋊C₄	—	13	8	no	—
150	C₂×C₄⋊C₄	—	26	16	no	—
151	C₂×C₄⋊C₄	—	26	4	no	—
152	C₂×C₄⋊C₄	—	26	16	no	—
153	C₂×C₄⋊C₄	—	52	16	no	—
154	C₂×C₄²	—	13	2	no	—
155	C₂×C₄²	—	13	1	no	—
156	C₂×C₄²	—	13	2	no	—
157	C₂×C₄²	—	13	2	no	—
158	C₂×C₄²	—	13	2	no	—
159	C₂×C₄²	—	13	2	no	—
160	C₂×C₄²	C₂×#261	1	1	yes	characteristic
161	#106	—	26	32	no	—
162	C₄×C₈	—	13	8	no	—
163	C₄×C₈	—	13	4	no	—
164	C₄×C₈	C₂×#261	1	4	yes	complemented
165	C₂×C₁₃	#1844	1	4	yes	complemented
166	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
167	C₂×C₁₃	C₂²×#106	1	2	yes	—
168	D₁₃	#1844	1	4	yes	complemented
169	D₁₃	C₂×#492	1	2	yes	—
170	D₁₃	C₂×#498	1	2	yes	—
171	C₂⁴	—	13	1	no	—
172	C₂²×C₄	—	13	2	no	—
173	C₂²×C₄	—	13	1	no	—
174	C₂²×C₄	—	13	16	no	—
175	C₂²×C₄	—	13	1	no	—
176	C₂²×C₄	—	13	4	no	—
177	C₂²×C₄	—	13	4	no	—
178	C₂²×C₄	—	13	1	no	—
179	C₂²×C₄	—	13	4	no	—
180	C₂²×C₄	—	13	4	no	—
181	C₂²×C₄	—	13	1	no	—
182	C₂²×C₄	—	13	1	no	—
183	C₂²×C₄	—	26	8	no	—
184	C₂²×C₄	—	52	8	no	—
185	C₂²×C₄	—	52	16	no	—
186	C₂²×C₄	C₄⋊#261	1	2	yes	—
187	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
188	C₂×C₈	—	13	8	no	—
189	C₂×C₈	—	13	4	no	—
190	C₂×C₈	—	13	8	no	—
191	C₂×C₈	—	13	16	no	—
192	C₂×C₈	—	13	8	no	—
193	C₂×C₈	C₄⋊#261	1	8	yes	complemented
194	C₂×C₈	C₂²×#261	1	4	yes	—
195	C₄⋊C₄	—	26	16	no	—
196	C₄⋊C₄	—	52	8	no	—
197	C₄⋊C₄	—	52	32	no	—
198	C₄²	—	13	2	no	—
199	C₄²	—	13	4	no	—
200	C₄²	—	13	4	no	—
201	C₄²	—	13	2	no	—
202	C₄²	C₄⋊#261	1	2	yes	—
203	C₄²	C₂²×#261	1	2	yes	—
204	C₁₃	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
205	C₂³	—	13	2	no	—
206	C₂³	—	13	2	no	—
207	C₂³	—	13	8	no	—
208	C₂³	—	13	1	no	—
209	C₂³	—	13	1	no	—
210	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
211	C₂×C₄	—	13	4	no	—
212	C₂×C₄	—	13	8	no	—
213	C₂×C₄	—	13	1	no	—
214	C₂×C₄	—	13	4	no	—
215	C₂×C₄	—	13	1	no	—
216	C₂×C₄	—	13	2	no	—
217	C₂×C₄	—	13	4	no	—
218	C₂×C₄	—	13	2	no	—
219	C₂×C₄	—	13	16	no	—
220	C₂×C₄	—	13	16	no	—
221	C₂×C₄	—	13	2	no	—
222	C₂×C₄	—	13	4	no	—
223	C₂×C₄	—	13	2	no	—
224	C₂×C₄	—	52	8	no	—
225	C₂×C₄	—	52	16	no	—
226	C₂×C₄	—	104	16	no	—
227	C₂×C₄	#88101	1	8	yes	complemented
228	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
230	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
231	C₂×C₄	C₂³×#261	1	1	yes	characteristic
232	C₈	—	13	8	no	—
233	C₈	C₂×C₄⋊#261	1	8	yes	complemented
234	C₂²	—	13	8	no	—
235	C₂²	—	13	8	no	—
236	C₂²	—	13	1	no	—
237	C₂²	—	13	1	no	—
238	C₂²	—	13	2	no	—
239	C₂²	—	13	4	no	—
240	C₂²	—	13	4	no	—
241	C₂²	#11719059	1	2	yes	—
242	C₂²	C₂×#88101	1	4	yes	—
243	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
244	C₄	—	13	2	no	—
245	C₄	—	13	2	no	—
246	C₄	—	13	8	no	—
247	C₄	—	104	8	no	—
248	C₄	#11719059	1	2	yes	—
249	C₄	C₂×#88101	1	8	yes	complemented
250	C₄	C₂²×C₄⋊#261	1	2	yes	—
251	C₂	—	13	2	no	—
252	C₂	—	13	4	no	—
253	C₂	—	13	2	no	—
254	C₂	#13130771	1	4	yes	direct product factor
255	C₂	C₂×#11719059	1	1	yes	characteristic
256	C₂	C₂²×#88101	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1844
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #13130771	C₂	direct (N × H)
13. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. C₂×C₈×Dic₁₃	C₄	semidirect
15. C₂×C₄×#1055	C₄	semidirect
16. C₄×C₈×D₁₃	C₄	semidirect
40. C₄×C₈×C₁₃	C₂×C₄	semidirect
41. C₈×Dic₁₃	C₂×C₄	semidirect
42. C₄×#1055	C₂×C₄	semidirect
64. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
72. C₂×#1055	C₄⋊C₄	semidirect
74. C₈×D₁₃	C₄⋊C₄	semidirect
90. C₂×C₄×C₁₃	#106	semidirect
95. C₂×Dic₁₃	#106	semidirect
100. C₄×D₁₃	#106	semidirect
108. C₈×C₁₃	C₂×C₄⋊C₄	semidirect
109. #1055	C₂×C₄⋊C₄	semidirect
120. C₂×C₄×C₈	#261	semidirect
135. C₄×C₁₃	C₂×#106	semidirect
138. Dic₁₃	C₂×#106	semidirect
164. C₄×C₈	C₂×#261	semidirect
165. C₂×C₁₃	#1844	semidirect
168. D₁₃	#1844	semidirect
193. C₂×C₈	C₄⋊#261	semidirect
204. C₁₃	C₂×#1844	semidirect
227. C₂×C₄	#88101	semidirect
233. C₈	C₂×C₄⋊#261	semidirect
249. C₄	C₂×#88101	semidirect
254. C₂	#13130771	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×#88101	2
C₂×#11719059	2
#13130771	2
C₂×C₄×C₈×D₁₃	2
C₂×#1844	13

Subgroups — C2×#13130771 #16798267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130771 #16798267