Subgroups — C₂×#13116447 #16015588

Order 3328.

1406 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	2	yes	maximal, complemented
2	#13116447	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂²⋊#6090	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	—	2	8	no	—
7	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₄	1	2	yes	—
9	C₂³×C₈×C₁₃	C₄	1	4	yes	—
10	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂³×C₈×C₁₃	C₂²	1	4	yes	—
12	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88052	—	2	8	no	—
14	C₂×#88052	C₂²	1	8	yes	complemented
15	C₂²×#6090	C₂²	1	4	yes	complemented
16	C₂²⋊#6090	C₂²	1	16	yes	complemented
17	C₂⁵×C₁₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	—	2	4	no	—
19	C₂³×C₄×C₁₃	—	2	4	no	—
20	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₈	1	4	yes	—
22	C₂³×C₄×C₁₃	C₈	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₃	—	2	16	no	—
28	C₂²×C₈×C₁₃	—	2	8	no	—
29	C₂²×C₈×C₁₃	—	2	8	no	—
30	C₂²×C₈×C₁₃	—	2	16	no	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₃	D₄	1	8	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	#88052	—	2	16	no	—
42	C₂×#6090	—	2	8	no	—
43	C₂×#6090	C₂³	1	8	yes	complemented
44	C₂×#2112	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	—	2	4	no	—
46	C₂⁴×C₁₃	—	2	8	no	—
47	C₂⁴×C₁₃	C₂×C₈	1	4	yes	—
48	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂⁴×C₁₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₃	—	2	4	no	—
51	C₂²×C₄×C₁₃	—	2	16	no	—
52	C₂²×C₄×C₁₃	—	2	8	no	—
53	C₂²×C₄×C₁₃	—	2	8	no	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	2	8	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
62	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	—	2	8	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	8	no	—
71	C₂×C₈×C₁₃	—	2	16	no	—
72	C₂×C₈×C₁₃	—	2	8	no	—
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	8	yes	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
81	#6090	—	2	8	no	—
82	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
83	C₂²×M₃₂	—	13	2	no	—
84	#2112	—	13	8	no	—
85	C₂×C₂²⋊C₁₆	—	13	4	no	—
86	C₂³×C₁₃	—	2	32	no	—
87	C₂³×C₁₃	—	2	8	no	—
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	—	2	4	no	—
90	C₂³×C₁₃	—	2	4	no	—
91	C₂³×C₁₃	—	2	16	no	—
92	C₂³×C₁₃	C₂²⋊C₈	1	4	yes	—
93	C₂³×C₁₃	M₃₂	1	8	yes	complemented
94	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
95	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
96	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	—	2	8	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	8	no	—
102	C₂×C₄×C₁₃	—	2	16	no	—
103	C₂×C₄×C₁₃	C₂²⋊C₈	1	8	yes	—
104	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₁₃	—	2	4	no	—
114	C₈×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
115	C₈×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
116	C₈×C₁₃	C₂²×D₄	1	2	yes	—
117	C₂⁴×C₄	Dic₁₃	1	1	yes	characteristic
118	C₂³×C₈	—	2	8	no	—
119	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
120	C₂³×C₈	Dic₁₃	1	4	yes	—
121	C₂³×C₈	Dic₁₃	1	2	yes	—
122	C₂³×C₈	C₂×D₁₃	1	4	yes	—
123	C₂³×C₈	C₂×D₁₃	1	2	yes	—
124	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
125	C₂×M₃₂	—	13	8	no	—
126	C₂×M₃₂	—	26	8	no	—
127	C₂²×C₁₆	—	13	4	no	—
128	C₂²⋊C₁₆	—	13	16	no	—
129	C₂²×C₁₃	—	2	32	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	2	8	no	—
132	C₂²×C₁₃	—	2	16	no	—
133	C₂²×C₁₃	—	2	4	no	—
134	C₂²×C₁₃	—	2	4	no	—
135	C₂²×C₁₃	C₂×C₂²⋊C₈	1	4	yes	—
136	C₂²×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
137	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
138	C₂²×C₁₃	C₂×M₃₂	1	8	yes	complemented
139	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	C₂×C₂²⋊C₈	1	2	yes	—
142	C₄×C₁₃	C₂×C₂²⋊C₈	1	4	yes	—
143	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
145	C₂⁵	#1055	1	1	yes	characteristic
146	C₂³×C₄	—	2	4	no	—
147	C₂³×C₄	—	2	4	no	—
148	C₂³×C₄	#1055	1	1	yes	characteristic
149	C₂³×C₄	#1055	1	4	yes	—
150	C₂³×C₄	#1055	1	2	yes	—
151	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
152	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
153	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	—	2	16	no	—
156	C₂²×C₈	—	2	8	no	—
157	C₂²×C₈	—	2	8	no	—
158	C₂²×C₈	—	2	16	no	—
159	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
160	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
161	C₂²×C₈	C₂×Dic₁₃	1	8	yes	—
162	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
163	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
165	C₂²×C₈	C₂²⋊D₁₃	1	8	yes	—
166	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
167	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
168	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	M₃₂	—	26	16	no	—
170	C₂×C₁₆	—	13	8	no	—
171	C₂×C₁₆	—	26	8	no	—
172	C₂×C₁₃	—	2	4	no	—
173	C₂×C₁₃	—	2	8	no	—
174	C₂×C₁₃	#2112	1	4	yes	complemented
175	C₂×C₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
176	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
177	C₂⁴	—	2	4	no	—
178	C₂⁴	—	2	8	no	—
179	C₂⁴	C₂×#1055	1	4	yes	—
180	C₂⁴	C₂×#1055	1	1	yes	characteristic
181	C₂⁴	C₂×#1055	1	2	yes	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	16	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	C₂×#1055	1	4	yes	—
190	C₂²×C₄	C₂×#1055	1	2	yes	—
191	C₂²×C₄	C₂×#1055	1	8	yes	—
192	C₂²×C₄	C₂×#1055	1	4	yes	—
193	C₂²×C₄	C₂×#1055	1	2	yes	—
194	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
195	C₂²×C₄	#6099	1	4	yes	—
196	C₂²×C₄	C₂²⋊Dic₁₃	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	16	no	—
203	C₂×C₈	—	2	8	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
206	C₂×C₈	C₂²⋊Dic₁₃	1	8	yes	—
207	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
208	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
209	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
211	C₂×C₈	C₂×C₂²⋊D₁₃	1	8	yes	—
212	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
213	C₁₆	—	26	8	no	—
214	C₁₃	C₂×#2112	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	32	no	—
216	C₂³	—	2	4	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	16	no	—
219	C₂³	—	2	8	no	—
220	C₂³	—	2	4	no	—
221	C₂³	#88052	1	8	yes	complemented
222	C₂³	C₂²⋊#1055	1	4	yes	—
223	C₂³	C₂²×#1055	1	4	yes	—
224	C₂³	C₂²×#1055	1	2	yes	—
225	C₂³	C₂²×#1055	1	1	yes	characteristic
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	C₂²⋊#1055	1	8	yes	—
233	C₂×C₄	C₂²⋊#1055	1	4	yes	—
234	C₂×C₄	C₂²×#1055	1	4	yes	—
235	C₂×C₄	C₂²×#1055	1	2	yes	—
236	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
237	C₂×C₄	C₂×#6099	1	4	yes	—
238	C₂×C₄	C₂×#6099	1	2	yes	—
239	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	4	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	2	yes	—
241	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	C₂×C₂²⋊Dic₁₃	1	4	yes	—
244	C₈	C₂×C₂²⋊Dic₁₃	1	2	yes	—
245	C₈	C₂²×C₂²⋊D₁₃	1	2	yes	—
246	C₂²	—	2	32	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	16	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	C₂×#88052	1	4	yes	—
253	C₂²	C₂×#88052	1	8	yes	complemented
254	C₂²	C₂×C₂²⋊#1055	1	4	yes	—
255	C₂²	C₂×C₂²⋊#1055	1	2	yes	—
256	C₂²	C₂³×#1055	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	C₂×C₂²⋊#1055	1	2	yes	—
259	C₄	C₂×C₂²⋊#1055	1	4	yes	—
260	C₄	C₂²×#6099	1	1	yes	characteristic
261	C₄	C₂²×C₂²⋊Dic₁₃	1	1	yes	characteristic
262	C₂	—	2	8	no	—
263	C₂	—	2	4	no	—
264	C₂	#13116447	1	4	yes	direct product factor
265	C₂	C₂²×C₂²⋊#1055	1	1	yes	characteristic
266	C₂	C₂²×#88052	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁵×C₁₃	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	C₂×#2112
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#88052	C₂	semidirect
2. #13116447	C₂	direct (N × H)
3. C₂×C₂²⋊#6090	C₂	semidirect
14. C₂×#88052	C₂²	semidirect
15. C₂²×#6090	C₂²	semidirect
16. C₂²⋊#6090	C₂²	semidirect
43. C₂×#6090	C₂³	semidirect
93. C₂³×C₁₃	M₃₂	semidirect
138. C₂²×C₁₃	C₂×M₃₂	semidirect
174. C₂×C₁₃	#2112	semidirect
214. C₁₃	C₂×#2112	semidirect
221. C₂³	#88052	semidirect
253. C₂²	C₂×#88052	semidirect
264. C₂	#13116447	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×#88052	2
#13116447	2
C₂×C₂²⋊#6090	2
C₂⁴×C₈×C₁₃	2
C₂×#2112	13

Subgroups — C2×#13116447 #16015588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116447 #16015588