Subgroups — C₂×#13131805 #16809621

Order 3328.

2074 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131805	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11719096	C₂	1	8	yes	maximal, complemented
3	C₂×D₁₃⋊M₃₂	C₂	1	4	yes	maximal, complemented
4	C₂²×#88106	C₂	1	2	yes	maximal, complemented
5	C₂²×#88161	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6127	C₄	1	1	yes	characteristic
7	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₂²	1	1	yes	characteristic
9	C₂×#88161	C₄	1	4	yes	—
10	C₂×#88161	C₄	1	8	yes	—
11	C₂×#88161	C₂²	1	4	yes	—
12	C₂×#88161	C₂²	1	8	yes	—
13	C₂²×#6094	C₄	1	2	yes	—
14	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
15	C₂×#88106	C₂²	1	4	yes	—
16	C₂×#88106	C₂²	1	4	yes	complemented
17	C₂×#88106	C₂²	1	16	yes	complemented
18	C₂²×#6092	C₂²	1	4	yes	complemented
19	#11719096	C₂²	1	64	yes	complemented
20	D₁₃⋊M₃₂	C₂²	1	32	yes	complemented
21	C₂×#88098	C₂²	1	8	yes	complemented
22	C₂×D₁₃⋊C₁₆	C₂²	1	8	yes	complemented
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₂²⋊D₁₃	C₈	1	2	yes	—
25	C₂×#6127	C₈	1	4	yes	—
26	C₂×#6127	C₈	1	8	yes	—
27	C₂×#6127	C₂×C₄	1	4	yes	—
28	C₂×#6127	C₂×C₄	1	8	yes	—
29	C₂²×D₅₂	C₈	1	1	yes	characteristic
30	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
31	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
32	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	4	yes	—
37	C₂²×#1055	C₂×C₄	1	2	yes	—
38	C₂²×#1055	C₂³	1	1	yes	characteristic
39	C₂²×#1055	C₂³	1	2	yes	—
40	C₂²×#1055	C₂³	1	4	yes	—
41	#88161	C₂×C₄	1	16	yes	—
42	#88161	C₂×C₄	1	32	yes	—
43	#88161	C₂³	1	16	yes	—
44	C₂×#6094	C₂×C₄	1	8	yes	—
45	C₂×#6094	C₂×C₄	1	16	yes	—
46	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
47	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
48	C₂×C₈×D₁₃	C₂³	1	4	yes	—
49	C₂×C₈×D₁₃	C₂³	1	8	yes	—
50	#88106	C₂³	1	16	yes	—
51	#88106	C₂³	1	16	yes	complemented
52	C₂×#6092	C₂³	1	8	yes	—
53	C₂×#6092	C₂³	1	16	yes	complemented
54	#88098	C₂³	1	32	yes	complemented
55	D₁₃⋊C₁₆	C₂³	1	32	yes	complemented
56	C₂×#3407	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂³×D₁₃	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂×C₂²⋊D₁₃	C₂×C₈	1	8	yes	—
66	C₂×C₂²⋊D₁₃	C₂×C₈	1	16	yes	—
67	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
68	#6127	C₂×C₈	1	16	yes	—
69	#6127	C₂×C₈	1	32	yes	—
70	#6127	C₂²×C₄	1	16	yes	—
71	C₂×D₅₂	C₂×C₈	1	4	yes	—
72	C₂×D₅₂	C₂×C₈	1	8	yes	—
73	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
74	C₂×C₄×D₁₃	C₂×C₈	1	8	yes	—
75	C₂×C₄×D₁₃	C₂²×C₄	1	8	yes	—
76	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
77	C₂×Dic₂₆	C₂×C₈	1	8	yes	—
78	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
79	C₁₃×M₁₆	C₂²×C₄	1	16	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂×#1055	C₂²×C₄	1	2	yes	—
83	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
84	C₂×#1055	C₂²×C₄	1	4	yes	—
85	C₂×#1055	C₂²×C₄	1	2	yes	—
86	C₂×#1055	C₂²×C₄	1	4	yes	—
87	C₂×#1055	C₂²×C₄	1	8	yes	—
88	C₂×#1055	C₂⁴	1	1	yes	characteristic
89	C₂×#1055	C₂⁴	1	4	yes	—
90	C₂×#1055	C₂⁴	1	2	yes	—
91	#6094	C₂²×C₄	1	32	yes	—
92	C₈×D₁₃	C₂²×C₄	1	8	yes	—
93	C₈×D₁₃	C₂²×C₄	1	16	yes	—
94	C₈×D₁₃	C₂⁴	1	8	yes	—
95	#6092	C₂⁴	1	16	yes	—
96	C₂²×#131	—	13	1	no	—
97	#3407	—	13	16	no	—
98	C₂×#504	—	13	8	no	—
99	C₂²×M₃₂	—	13	2	no	—
100	C₂²×M₃₂	—	13	4	no	—
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
104	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
105	C₂²×D₁₃	—	4	4	no	—
106	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
107	C₂²×D₁₃	C₂²×C₈	1	8	yes	—
108	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
114	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂²⋊D₁₃	C₂²×C₈	1	32	yes	—
118	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
119	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
120	D₅₂	C₂²×C₈	1	16	yes	—
121	C₄×D₁₃	C₂²×C₈	1	16	yes	—
122	C₄×D₁₃	C₂²×C₈	1	8	yes	—
123	C₄×D₁₃	C₂³×C₄	1	8	yes	—
124	Dic₂₆	C₂²×C₈	1	16	yes	—
125	C₈×C₁₃	C₂³×C₄	1	8	yes	—
126	#1055	C₂³×C₄	1	1	yes	characteristic
127	#1055	C₂³×C₄	1	2	yes	—
128	#1055	C₂³×C₄	1	4	yes	—
129	#1055	C₂⁵	1	1	yes	characteristic
130	C₂²×#41	—	13	1	no	—
131	C₂×#131	—	13	8	no	—
132	C₂×#131	—	13	4	no	—
133	C₂×#131	—	13	4	no	—
134	C₂×#131	—	13	8	no	—
135	C₂²×M₁₆	—	13	2	no	—
136	C₂²×M₁₆	#261	1	1	yes	characteristic
137	C₂³×C₈	—	13	1	no	—
138	C₂³×C₈	—	13	2	no	—
139	#504	—	13	64	no	—
140	C₂×M₃₂	—	13	4	no	—
141	C₂×M₃₂	—	13	8	no	—
142	C₂×M₃₂	—	13	4	no	—
143	C₂×M₃₂	—	13	32	no	—
144	C₂×M₃₂	—	13	16	no	—
145	C₂²×C₁₆	—	13	8	no	—
146	C₂²×C₁₆	—	13	4	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	4	no	—
149	C₂²×C₁₃	—	4	4	no	—
150	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
151	C₂²×C₁₃	C₂³×C₈	1	4	yes	—
152	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
153	C₂×D₁₃	—	2	8	no	—
154	C₂×D₁₃	—	4	8	no	—
155	C₂×D₁₃	C₂³×C₈	1	8	yes	—
156	C₄×C₁₃	C₂³×C₈	1	4	yes	—
157	C₄×C₁₃	C₂³×C₈	1	2	yes	—
158	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
159	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
160	Dic₁₃	C₂³×C₈	1	8	yes	—
161	C₂×#41	—	13	4	no	—
162	C₂×#41	—	13	8	no	—
163	C₂×#41	—	13	8	no	—
164	C₂×#41	—	13	4	no	—
165	C₂²×Q₈	—	13	1	no	—
166	C₂²×D₄	—	13	2	no	—
167	C₂²×D₄	—	13	1	no	—
168	C₂³×C₄	—	13	2	no	—
169	C₂³×C₄	C₂×#261	1	1	yes	characteristic
170	#131	—	13	16	no	—
171	#131	—	13	32	no	—
172	#131	—	13	16	no	—
173	C₂×M₁₆	—	13	16	no	—
174	C₂×M₁₆	—	13	8	no	—
175	C₂×M₁₆	C₂×#261	1	8	yes	—
176	C₂×M₁₆	C₂×#261	1	4	yes	—
177	C₂²×C₈	—	13	1	no	—
178	C₂²×C₈	—	13	1	no	—
179	C₂²×C₈	—	13	4	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	13	2	no	—
182	C₂²×C₈	—	13	4	no	—
183	C₂²×C₈	—	13	8	no	—
184	C₂²×C₈	—	13	4	no	—
185	C₂²×C₈	—	13	4	no	—
186	C₂²×C₈	—	13	8	no	—
187	C₂²×C₈	C₂×#261	1	2	yes	—
188	M₃₂	—	13	32	no	—
189	M₃₂	—	13	16	no	—
190	M₃₂	—	13	16	no	—
191	C₂×C₁₆	—	13	32	no	—
192	C₂×C₁₆	—	13	8	no	—
193	C₂×C₁₆	—	13	16	no	—
194	C₂×C₁₃	—	2	2	no	—
195	C₂×C₁₃	—	2	4	no	—
196	C₂×C₁₃	#3407	1	2	yes	complemented
197	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
198	D₁₃	—	2	8	no	—
199	C₂⁴	—	13	2	no	—
200	C₂⁴	D₁₃⋊C₈	1	1	yes	characteristic
201	#41	—	13	32	no	—
202	#41	—	13	16	no	—
203	#41	—	13	16	no	—
204	C₂×Q₈	—	13	8	no	—
205	C₂×Q₈	—	13	4	no	—
206	C₂×D₄	—	13	8	no	—
207	C₂×D₄	—	13	16	no	—
208	C₂×D₄	—	13	4	no	—
209	C₂×D₄	—	13	8	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	4	no	—
212	C₂²×C₄	—	13	8	no	—
213	C₂²×C₄	—	13	8	no	—
214	C₂²×C₄	—	13	4	no	—
215	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
216	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
217	C₂²×C₄	D₁₃⋊C₈	1	4	yes	—
218	C₂²×C₄	C₂²×#261	1	4	yes	—
219	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
220	C₂²×C₄	C₂²×#261	1	2	yes	—
221	M₁₆	—	13	32	no	—
222	M₁₆	C₂²×#261	1	16	yes	—
223	C₂×C₈	—	13	4	no	—
224	C₂×C₈	—	13	2	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	4	no	—
227	C₂×C₈	—	13	4	no	—
228	C₂×C₈	—	13	8	no	—
229	C₂×C₈	—	13	1	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	2	no	—
232	C₂×C₈	—	13	8	no	—
233	C₂×C₈	—	13	8	no	—
234	C₂×C₈	—	13	16	no	—
235	C₂×C₈	C₂²×#261	1	8	yes	—
236	C₂×C₈	C₂²×#261	1	4	yes	—
237	C₁₆	—	13	16	no	—
238	C₁₃	C₂×#3407	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	4	2	no	—
240	C₂³	—	13	4	no	—
241	C₂³	—	13	8	no	—
242	C₂³	—	52	4	no	—
243	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
244	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
245	C₂³	C₂×D₁₃⋊C₈	1	4	yes	—
246	Q₈	—	13	16	no	—
247	D₄	—	13	32	no	—
248	D₄	—	13	16	no	—
249	C₂×C₄	—	13	8	no	—
250	C₂×C₄	—	13	4	no	—
251	C₂×C₄	—	13	16	no	—
252	C₂×C₄	—	13	8	no	—
253	C₂×C₄	—	13	8	no	—
254	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
255	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
256	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
257	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
258	C₂×C₄	C₂×D₁₃⋊C₈	1	8	yes	—
259	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
260	C₂×C₄	C₂³×#261	1	2	yes	—
261	C₂×C₄	C₂³×#261	1	4	yes	—
262	C₂×C₄	C₂³×#261	1	1	yes	characteristic
263	C₈	—	13	1	no	—
264	C₈	—	13	2	no	—
265	C₈	—	13	1	no	—
266	C₈	—	13	4	no	—
267	C₈	C₂³×#261	1	8	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	13	8	no	—
272	C₂²	—	26	8	no	—
273	C₂²	—	52	8	no	—
274	C₂²	C₂²×D₁₃⋊C₈	1	2	yes	—
275	C₂²	C₂²×D₁₃⋊C₈	1	4	yes	—
276	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
277	C₄	—	13	8	no	—
278	C₄	C₂²×D₁₃⋊C₈	1	2	yes	—
279	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
280	C₄	C₂²×D₁₃⋊C₈	1	4	yes	—
281	C₄	C₂⁴×#261	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	2	4	no	—
284	C₂	—	26	8	no	—
285	C₂	#13131805	1	2	yes	direct product factor
286	C₂	C₂³×D₁₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂²×#6127	C₄
cosocle	#1055	C₂⁵
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	C₂×#3407
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. #13131805	C₂	direct (N × H)
2. C₂×#11719096	C₂	semidirect
3. C₂×D₁₃⋊M₃₂	C₂	semidirect
4. C₂²×#88106	C₂	semidirect
16. C₂×#88106	C₂²	semidirect
17. C₂×#88106	C₂²	semidirect
18. C₂²×#6092	C₂²	semidirect
19. #11719096	C₂²	semidirect
20. D₁₃⋊M₃₂	C₂²	semidirect
21. C₂×#88098	C₂²	semidirect
22. C₂×D₁₃⋊C₁₆	C₂²	semidirect
51. #88106	C₂³	semidirect
53. C₂×#6092	C₂³	semidirect
54. #88098	C₂³	semidirect
55. D₁₃⋊C₁₆	C₂³	semidirect
196. C₂×C₁₃	#3407	semidirect
238. C₁₃	C₂×#3407	semidirect
285. C₂	#13131805	direct (N × H)

Maximal subgroups

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

subgroup	index
#13131805	2
C₂×#11719096	2
C₂×D₁₃⋊M₃₂	2
C₂²×#88106	2
C₂²×#88161	2
C₂×#3407	13

Subgroups — C2×#13131805 #16809621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131805 #16809621