Subgroups — C₂×#13131058 #16806754

Order 3328.

738 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊M₃₂	C₂	1	2	yes	maximal
2	#13131058	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718262	C₂	1	2	yes	maximal, complemented
4	C₂×#11718232	C₂	1	2	yes	maximal, complemented
5	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88126	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	4	yes	—
10	D₁₃⋊M₃₂	C₄	1	8	yes	complemented
11	D₁₃⋊M₃₂	C₂²	1	8	yes	—
12	C₂×#88098	C₄	1	4	yes	complemented
13	C₂×D₁₃⋊C₁₆	C₄	1	4	yes	complemented
14	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
15	C₂×#88055	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
17	C₂×#88043	C₂²	1	1	yes	characteristic
18	#11718350	C₄	1	4	yes	—
19	#11718350	C₂²	1	4	yes	—
20	#11718262	C₂²	1	8	yes	complemented
21	#11718232	C₂²	1	8	yes	complemented
22	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#6103	—	2	1	no	—
27	C₂×#6101	—	2	1	no	—
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	C₂×#6094	—	2	2	no	—
32	C₂×#6094	—	2	2	no	—
33	C₂×#6094	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₁₃	C₂³	1	2	yes	—
39	#88126	—	2	2	no	—
40	#88126	C₂×C₄	1	4	yes	—
41	#88106	C₂×C₄	1	8	yes	—
42	#88106	C₂³	1	8	yes	—
43	C₂×#6092	C₂×C₄	1	8	yes	—
44	#88098	C₂×C₄	1	16	yes	complemented
45	D₁₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
46	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
47	#88055	C₂×C₄	1	2	yes	—
48	#88055	C₂³	1	2	yes	—
49	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
50	#88043	C₂×C₄	1	2	yes	—
51	#88043	C₂³	1	2	yes	—
52	C₂×#2110	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	1	no	—
57	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
58	C₂×C₄×D₁₃	C₄²	1	2	yes	—
59	C₂×C₄×D₁₃	C₄²	1	2	yes	—
60	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
61	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
62	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
63	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₄²	1	2	yes	—
68	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
73	#6103	—	2	4	no	—
74	#6101	—	2	4	no	—
75	C₄×Dic₁₃	—	2	1	no	—
76	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂×#1055	—	2	1	no	—
78	C₂×#1055	—	2	1	no	—
79	C₂×#1055	C₂²×C₄	1	2	yes	—
80	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
81	C₂×#1055	C₂²×C₄	1	2	yes	—
82	C₂×#1055	C₂²×C₄	1	2	yes	—
83	C₂×#1055	C₂⁴	1	1	yes	characteristic
84	#6094	—	2	8	no	—
85	C₈×D₁₃	C₄²	1	4	yes	—
86	C₈×D₁₃	C₄²	1	4	yes	—
87	C₈×D₁₃	C₂²×C₄	1	4	yes	—
88	C₈×D₁₃	C₂²×C₄	1	4	yes	—
89	#6092	C₂²×C₄	1	16	yes	—
90	C₂²×M₃₂	—	13	2	no	—
91	C₂×#405	—	13	1	no	—
92	#2110	—	13	8	no	—
93	C₂×#347	—	13	2	no	—
94	C₂×#347	—	13	2	no	—
95	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂²×D₁₃	—	4	2	no	—
97	C₂²×D₁₃	#97	1	2	yes	—
98	C₂²×D₁₃	#97	1	2	yes	—
99	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×Dic₁₃	—	2	2	no	—
109	C₂×Dic₁₃	—	4	2	no	—
110	C₂×Dic₁₃	#97	1	2	yes	—
111	C₂×Dic₁₃	#97	1	2	yes	—
112	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
113	C₄×D₁₃	#97	1	4	yes	—
114	C₄×D₁₃	#97	1	4	yes	—
115	C₄×D₁₃	C₂×C₄²	1	4	yes	—
116	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	C₂×C₄²	1	2	yes	—
119	C₈×C₁₃	C₂×C₄²	1	2	yes	—
120	#1055	—	2	2	no	—
121	#1055	C₂×C₄²	1	2	yes	—
122	#1055	C₂³×C₄	1	2	yes	—
123	C₂²×M₁₆	—	13	1	no	—
124	C₂³×C₈	—	13	1	no	—
125	C₂×#117	—	13	1	no	—
126	C₂×M₃₂	—	13	4	no	—
127	C₂×M₃₂	—	13	8	no	—
128	C₂×M₃₂	—	13	4	no	—
129	C₂×M₃₂	—	13	8	no	—
130	C₂²×C₁₆	—	13	4	no	—
131	#405	—	13	4	no	—
132	#405	—	13	4	no	—
133	C₂×#97	—	13	1	no	—
134	C₂×#97	—	13	1	no	—
135	C₂×C₄×C₈	—	13	1	no	—
136	C₂×C₄×C₈	#261	1	1	yes	characteristic
137	#347	—	13	8	no	—
138	#347	—	13	8	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
142	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
143	C₂×D₁₃	—	2	4	no	—
144	C₂×D₁₃	—	4	2	no	—
145	C₂×D₁₃	—	4	2	no	—
146	C₂×D₁₃	C₂×#97	1	4	yes	—
147	C₄×C₁₃	—	4	2	no	—
148	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
149	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
150	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
151	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
152	Dic₁₃	—	4	2	no	—
153	Dic₁₃	C₂×#97	1	4	yes	—
154	C₂³×C₄	—	13	1	no	—
155	C₂×M₁₆	—	13	4	no	—
156	C₂×M₁₆	—	26	2	no	—
157	C₂×M₁₆	—	26	2	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	2	no	—
160	C₂²×C₈	—	13	4	no	—
161	C₂²×C₈	—	13	2	no	—
162	C₂²×C₈	—	13	2	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	1	no	—
165	C₂²×C₈	C₂×#261	1	1	yes	characteristic
166	C₂²×C₈	C₂×#261	1	1	yes	characteristic
167	#117	—	13	4	no	—
168	#117	—	26	2	no	—
169	C₂×C₄⋊C₄	—	26	1	no	—
170	C₂×C₂²⋊C₄	—	26	1	no	—
171	C₂×C₄²	—	13	1	no	—
172	C₂×C₄²	C₂×#261	1	1	yes	characteristic
173	M₃₂	—	13	16	no	—
174	M₃₂	—	13	8	no	—
175	M₃₂	—	13	8	no	—
176	C₂×C₁₆	—	13	16	no	—
177	C₂×C₁₆	—	13	8	no	—
178	#97	—	13	2	no	—
179	#97	—	13	2	no	—
180	#97	—	13	2	no	—
181	#97	—	13	2	no	—
182	C₄×C₈	—	13	4	no	—
183	C₄×C₈	C₂×#261	1	4	yes	—
184	C₂×C₁₃	—	2	1	no	—
185	C₂×C₁₃	—	2	1	no	—
186	C₂×C₁₃	#2110	1	2	yes	complemented
187	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
188	D₁₃	—	2	4	no	—
189	C₂⁴	—	13	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	—	13	2	no	—
195	C₂²×C₄	—	13	2	no	—
196	C₂²×C₄	—	13	2	no	—
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	—	26	1	no	—
199	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
200	M₁₆	—	26	8	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	13	2	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	1	no	—
206	C₂×C₈	—	13	1	no	—
207	C₂×C₈	—	13	4	no	—
208	C₂×C₈	—	13	4	no	—
209	C₂×C₈	—	13	4	no	—
210	C₂×C₈	—	13	4	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	26	1	no	—
213	C₂×C₈	—	26	1	no	—
214	C₂×C₈	C₄×#261	1	1	yes	characteristic
215	C₂×C₈	C₄×#261	1	2	yes	—
216	C₂×C₈	C₄×#261	1	1	yes	characteristic
217	C₂×C₈	C₂²×#261	1	2	yes	—
218	C₂×C₈	C₂²×#261	1	2	yes	—
219	C₄⋊C₄	—	26	4	no	—
220	C₂²⋊C₄	—	26	4	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	13	2	no	—
223	C₄²	—	26	1	no	—
224	C₄²	C₂²×#261	1	2	yes	—
225	C₁₆	—	13	16	no	—
226	C₁₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	13	2	no	—
228	C₂³	—	13	2	no	—
229	C₂³	—	13	2	no	—
230	C₂³	—	52	2	no	—
231	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	13	2	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	13	2	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	4	no	—
240	C₂×C₄	—	13	4	no	—
241	C₂×C₄	—	26	2	no	—
242	C₂×C₄	—	52	2	no	—
243	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
244	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
245	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
248	C₂×C₄	C₂³×#261	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	13	2	no	—
251	C₈	—	13	2	no	—
252	C₈	—	26	2	no	—
253	C₈	C₂×C₄×#261	1	2	yes	—
254	C₈	C₂×C₄×#261	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	13	4	no	—
257	C₂²	—	26	4	no	—
258	C₂²	—	52	2	no	—
259	C₂²	—	52	2	no	—
260	C₂²	#11719047	1	1	yes	characteristic
261	C₂²	#11719047	1	1	yes	characteristic
262	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
263	C₄	—	4	2	no	—
264	C₄	—	13	4	no	—
265	C₄	—	52	2	no	—
266	C₄	#11719047	1	1	yes	characteristic
267	C₄	#11719047	1	1	yes	characteristic
268	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
269	C₄	C₂²×C₄×#261	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	26	4	no	—
273	C₂	#13131058	1	2	yes	direct product factor
274	C₂	C₂×#11719047	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#2110
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13131058	C₂	direct (N × H)
3. C₂×#11718262	C₂	semidirect
4. C₂×#11718232	C₂	semidirect
10. D₁₃⋊M₃₂	C₄	semidirect
12. C₂×#88098	C₄	semidirect
13. C₂×D₁₃⋊C₁₆	C₄	semidirect
20. #11718262	C₂²	semidirect
21. #11718232	C₂²	semidirect
44. #88098	C₂×C₄	semidirect
45. D₁₃⋊C₁₆	C₂×C₄	semidirect
186. C₂×C₁₃	#2110	semidirect
226. C₁₃	C₂×#2110	semidirect
273. C₂	#13131058	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×D₁₃⋊M₃₂	2
#13131058	2
C₂×#11718262	2
C₂×#11718232	2
C₂×#11718350	2
C₂×#2110	13

Subgroups — C2×#13131058 #16806754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131058 #16806754