Subgroups — C₂×#13116971 #16017520

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718535	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13116971	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718096	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718081	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718154	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×D₄×Dic₁₃	—	2	1	no	—
11	C₂×#88142	—	2	1	no	—
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×#88072	—	2	1	no	—
15	D₈×Dic₁₃	—	2	2	no	—
16	D₈×Dic₁₃	C₂²	1	4	yes	—
17	C₂×#88058	C₂²	1	1	yes	characteristic
18	C₂×#88058	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂²×#6090	C₂²	1	1	yes	characteristic
21	C₂×#88047	—	2	1	no	—
22	#11718535	—	2	2	no	—
23	#11718535	C₂²	1	4	yes	complemented
24	C₁₃×#357	C₂²	1	4	yes	—
25	#11718154	C₂²	1	4	yes	—
26	#11718096	C₂²	1	4	yes	complemented
27	#11718093	C₂²	1	4	yes	complemented
28	#11718081	C₂²	1	4	yes	complemented
29	C₂²×D₄×C₁₃	—	2	1	no	—
30	C₂³×Dic₁₃	—	4	1	no	—
31	C₂²×Dic₂₆	—	4	1	no	—
32	C₂×C₁₃×D₈	—	2	1	no	—
33	C₂×C₁₃×D₈	—	2	2	no	—
34	C₂×C₁₃×D₈	Q₈	1	4	yes	complemented
35	C₂×C₁₃×D₈	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
39	D₄×Dic₁₃	—	2	4	no	—
40	D₄×Dic₁₃	—	4	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	1	no	—
42	C₂×C₄⋊Dic₁₃	—	2	1	no	—
43	C₂×#6101	—	4	1	no	—
44	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
45	C₂²×#1055	D₄	1	1	yes	characteristic
46	#88142	—	2	4	no	—
47	#88142	—	4	2	no	—
48	C₂×C₁₃×C₁₆	—	2	2	no	—
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₁₃×#107	—	2	1	no	—
51	C₁₃×#107	C₂³	1	2	yes	—
52	#88072	—	2	4	no	—
53	#88058	—	2	1	no	—
54	#88058	—	2	1	no	—
55	#88058	C₂³	1	2	yes	—
56	#88058	C₂³	1	2	yes	—
57	C₈×Dic₁₃	D₄	1	2	yes	—
58	C₈×Dic₁₃	C₂³	1	2	yes	—
59	C₂×#6090	—	2	2	no	—
60	C₂×#6090	C₂³	1	2	yes	—
61	#88047	—	2	4	no	—
62	C₂×#2224	—	13	1	no	maximal, Sylow
63	C₂⁴×C₁₃	—	4	1	no	—
64	C₂×D₄×C₁₃	—	2	4	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	4	1	no	—
67	C₂×D₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	—	4	1	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	4	1	no	—
71	C₂²×Dic₁₃	—	4	1	no	—
72	C₂²×Dic₁₃	—	4	4	no	—
73	C₂²×Dic₁₃	—	8	4	no	—
74	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
75	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
76	C₂×Dic₂₆	—	4	4	no	—
77	C₂×Dic₂₆	—	8	4	no	—
78	C₁₃×D₈	—	2	4	no	—
79	C₁₃×D₈	—	2	2	no	—
80	C₁₃×D₈	C₂×Q₈	1	4	yes	complemented
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₁₃	#41	1	2	yes	—
84	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	4	1	no	—
87	C₂²⋊Dic₁₃	—	4	4	no	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	2	2	no	—
90	C₄⋊Dic₁₃	—	4	1	no	—
91	C₄⋊Dic₁₃	—	4	1	no	—
92	#6101	—	4	4	no	—
93	C₄×Dic₁₃	D₈	1	2	yes	—
94	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	C₂×D₄	1	2	yes	—
97	C₁₃×C₁₆	—	2	2	no	—
98	#6090	—	2	2	no	—
99	C₂×#500	—	13	1	no	—
100	C₂×C₄×D₈	—	13	1	no	—
101	#2224	—	13	8	no	—
102	C₂×#366	—	13	1	no	—
103	C₂×#366	—	13	1	no	—
104	C₂×C₄⋊C₁₆	—	13	1	no	—
105	C₂×#357	—	13	1	no	—
106	C₂×#357	D₁₃	1	1	yes	characteristic
107	C₂³×C₁₃	—	4	2	no	—
108	C₂³×C₁₃	—	4	4	no	—
109	C₂³×C₁₃	—	8	2	no	—
110	C₂³×C₁₃	—	8	2	no	—
111	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
112	D₄×C₁₃	—	2	4	no	—
113	D₄×C₁₃	—	4	2	no	—
114	D₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	8	2	no	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#137	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	4	2	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	8	2	no	—
125	C₂×Dic₁₃	—	8	2	no	—
126	C₂×Dic₁₃	—	8	8	no	—
127	C₂×Dic₁₃	D₁₆	1	4	yes	—
128	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
129	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
130	Dic₂₆	—	8	8	no	—
131	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
132	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
133	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
134	#1055	—	2	2	no	—
135	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂×C₄⋊Q₈	—	26	1	no	—
137	C₂×C₄×D₄	—	26	1	no	—
138	C₂²×C₁₆	—	13	1	no	—
139	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
140	#500	—	13	4	no	—
141	#500	—	26	2	no	—
142	C₄×D₈	—	13	4	no	—
143	C₄×D₈	—	26	2	no	—
144	C₂×#107	—	13	1	no	—
145	C₂×#107	—	13	1	no	—
146	C₂×#107	—	26	1	no	—
147	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
148	C₂×#102	—	26	1	no	—
149	C₂×C₄×C₈	—	13	1	no	—
150	#366	—	13	4	no	—
151	#366	—	13	4	no	—
152	C₄⋊C₁₆	—	13	4	no	—
153	#357	—	13	4	no	—
154	#357	C₂×D₁₃	1	4	yes	—
155	C₂²×C₁₃	—	4	4	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	—	8	8	no	—
159	C₂²×C₁₃	#474	1	2	yes	—
160	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
161	C₂²×C₁₃	#510	1	2	yes	—
162	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₁₃	—	8	2	no	—
164	C₄×C₁₃	#474	1	2	yes	—
165	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
167	Dic₁₃	—	2	2	no	—
168	Dic₁₃	—	8	2	no	—
169	Dic₁₃	—	8	2	no	—
170	Dic₁₃	C₂×D₁₆	1	4	yes	—
171	C₂²×Q₈	—	52	1	no	—
172	C₂²×D₄	—	2	1	no	—
173	C₂³×C₄	—	52	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	2	no	—
176	C₂×D₈	Dic₂₆	1	4	yes	complemented
177	C₂×D₈	C₂²×D₁₃	1	2	yes	—
178	C₂²×C₈	—	13	1	no	—
179	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	26	4	no	—
181	C₄⋊Q₈	—	52	2	no	—
182	C₄×D₄	—	26	4	no	—
183	C₄×D₄	—	52	2	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	26	1	no	—
186	C₂×C₄⋊C₄	—	26	1	no	—
187	C₂×C₄⋊C₄	—	52	1	no	—
188	C₂×C₂²⋊C₄	—	52	1	no	—
189	C₂×C₄²	—	13	1	no	—
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	13	2	no	—
192	C₂×C₁₆	—	26	2	no	—
193	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
194	#107	—	2	1	no	—
195	#107	—	13	2	no	—
196	#107	—	13	2	no	—
197	#107	—	26	4	no	—
198	#107	—	26	1	no	—
199	#107	—	26	1	no	—
200	#107	C₂²×D₁₃	1	2	yes	—
201	#102	—	26	4	no	—
202	C₄×C₈	—	13	2	no	—
203	C₄×C₈	—	13	2	no	—
204	C₂×C₁₃	—	8	4	no	—
205	C₂×C₁₃	#2224	1	4	yes	complemented
206	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
207	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
208	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
209	C₂⁴	—	4	1	no	—
210	C₂×Q₈	—	52	4	no	—
211	C₂×Q₈	—	104	4	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	13	1	no	—
218	C₂²×C₄	—	13	1	no	—
219	C₂²×C₄	—	52	4	no	—
220	C₂²×C₄	—	52	1	no	—
221	C₂²×C₄	—	52	1	no	—
222	C₂²×C₄	—	104	4	no	—
223	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
224	D₈	—	2	4	no	—
225	D₈	—	2	2	no	—
226	D₈	C₂×Dic₂₆	1	4	yes	complemented
227	C₂×C₈	—	13	2	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
230	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
231	C₂×C₈	#6129	1	2	yes	—
232	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	26	2	no	—
236	C₄⋊C₄	—	26	2	no	—
237	C₄⋊C₄	—	52	4	no	—
238	C₄⋊C₄	—	52	1	no	—
239	C₄⋊C₄	—	52	1	no	—
240	C₂²⋊C₄	—	52	4	no	—
241	C₄²	—	13	2	no	—
242	C₄²	—	13	2	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	26	2	no	—
245	C₁₃	C₂×#2224	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	4	4	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	8	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	D₈×D₁₃	1	1	yes	characteristic
251	Q₈	—	104	8	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	13	2	no	—
258	C₂×C₄	—	13	4	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	52	2	no	—
262	C₂×C₄	—	52	2	no	—
263	C₂×C₄	—	104	8	no	—
264	C₂×C₄	—	104	2	no	—
265	C₂×C₄	—	104	2	no	—
266	C₂×C₄	#88132	1	2	yes	—
267	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
268	C₂×C₄	#88169	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
270	C₈	—	26	2	no	—
271	C₈	#88132	1	2	yes	—
272	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
273	C₈	C₂×#6129	1	1	yes	characteristic
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	8	8	no	—
278	C₂²	#11718420	1	2	yes	—
279	C₂²	D₁₃×D₁₆	1	2	yes	—
280	C₂²	#11718576	1	2	yes	—
281	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
282	C₄	—	8	2	no	—
283	C₄	—	13	4	no	—
284	C₄	—	26	2	no	—
285	C₄	—	104	2	no	—
286	C₄	—	104	2	no	—
287	C₄	#11718420	1	2	yes	—
288	C₄	C₂×#88132	1	1	yes	characteristic
289	C₄	C₂×#88169	1	1	yes	characteristic
290	C₂	—	8	4	no	—
291	C₂	#13116971	1	4	yes	direct product factor
292	C₂	C₂×#11718420	1	1	yes	characteristic
293	C₂	C₂×D₁₃×D₁₆	1	1	yes	characteristic
294	C₂	C₂×#11718576	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₁₃
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#357	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×C₁₃×D₈	C₂²
hypercenter	C₂×#357	D₁₃
hypocenter	C₁₃	C₂×#2224
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
1. C₂×#11718535	C₂	semidirect
3. #13116971	C₂	direct (N × H)
4. C₂×#11718096	C₂	semidirect
5. C₂×#11718081	C₂	semidirect
6. C₂×#11718093	C₂	semidirect
23. #11718535	C₂²	semidirect
26. #11718096	C₂²	semidirect
27. #11718093	C₂²	semidirect
28. #11718081	C₂²	semidirect
34. C₂×C₁₃×D₈	Q₈	semidirect
80. C₁₃×D₈	C₂×Q₈	semidirect
176. C₂×D₈	Dic₂₆	semidirect
205. C₂×C₁₃	#2224	semidirect
226. D₈	C₂×Dic₂₆	semidirect
245. C₁₃	C₂×#2224	semidirect
291. C₂	#13116971	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×#11718535	2
C₂×D₈×Dic₁₃	2
#13116971	2
C₂×#11718096	2
C₂×#11718081	2
C₂×#11718093	2
C₂×#11718154	2
C₂×C₁₃×#357	2
C₂×#2224	13

Subgroups — C2×#13116971 #16017520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116971 #16017520