Subgroups — C₂×#13117011 #16017654

Order 3328.

526 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718519	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718556	C₂	1	1	yes	maximal, characteristic
3	#13117011	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718086	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718103	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718110	C₂	1	1	yes	maximal, characteristic
7	C₂×#11718080	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88161	—	2	1	no	—
10	C₂×#88158	C₂²	1	1	yes	characteristic
11	C₂×#88144	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
15	C₂×#88065	—	2	1	no	—
16	C₂×#88057	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88052	C₂²	1	1	yes	characteristic
19	C₂×#88047	—	2	1	no	—
20	#11718556	—	2	2	no	—
21	#11718556	C₂²	1	4	yes	—
22	#11718519	—	2	2	no	—
23	#11718519	C₂²	1	4	yes	complemented
24	C₁₃×#365	C₂²	1	4	yes	complemented
25	#11718110	C₂²	1	4	yes	—
26	#11718103	C₂²	1	4	yes	complemented
27	#11718086	C₂²	1	4	yes	complemented
28	#11718080	C₂²	1	4	yes	—
29	C₂×#6127	—	2	1	no	—
30	C₂×C₁₃×M₁₆	—	2	1	no	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃	—	4	1	no	—
35	C₂×#6101	—	4	1	no	—
36	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
37	C₂²×#1055	D₄	1	1	yes	characteristic
38	#88161	—	2	4	no	—
39	#88161	—	4	2	no	—
40	C₂×Dic₅₂	Q₈	1	1	yes	characteristic
41	#88158	—	2	1	no	—
42	#88158	—	2	1	no	—
43	#88158	Q₈	1	2	yes	—
44	#88158	C₂³	1	2	yes	—
45	C₂×D₁₀₄	Q₈	1	1	yes	characteristic
46	C₂×#6095	—	2	1	no	—
47	C₂×#6094	—	4	1	no	—
48	C₂×C₈×D₁₃	—	4	1	no	—
49	#88144	—	2	4	no	—
50	#88144	—	4	2	no	—
51	C₁₃×M₃₂	—	2	1	no	—
52	C₁₃×M₃₂	C₂³	1	2	yes	—
53	C₂×C₁₃×C₁₆	—	2	1	no	—
54	C₁₃×#108	—	2	1	no	—
55	C₁₃×#108	C₂³	1	2	yes	—
56	C₁₃×#106	—	2	1	no	—
57	C₁₃×#106	C₂³	1	2	yes	—
58	#88065	—	2	4	no	—
59	#88057	—	2	1	no	—
60	#88057	C₂³	1	2	yes	—
61	C₈×Dic₁₃	D₄	1	2	yes	—
62	C₈×Dic₁₃	C₂³	1	2	yes	—
63	#88052	—	2	1	no	—
64	#88052	C₂³	1	2	yes	—
65	C₂×#6090	—	2	1	no	—
66	#88047	—	2	4	no	—
67	C₂×#2231	—	13	1	no	maximal, Sylow
68	C₂²×C₄×C₁₃	—	4	1	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₃	—	4	1	no	—
71	C₂²×Dic₁₃	—	2	1	no	—
72	C₂²×Dic₁₃	—	4	1	no	—
73	#6127	—	2	2	no	—
74	#6127	—	2	2	no	—
75	#6127	—	4	1	no	—
76	#6127	—	4	1	no	—
77	C₂×D₅₂	—	2	1	no	—
78	C₂×C₄×D₁₃	—	4	1	no	—
79	C₂×Dic₂₆	—	2	1	no	—
80	C₁₃×M₁₆	—	2	2	no	—
81	C₁₃×M₁₆	—	4	1	no	—
82	C₂×C₈×C₁₃	—	4	1	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
87	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	4	1	no	—
91	C₄⋊Dic₁₃	—	2	2	no	—
92	C₄⋊Dic₁₃	—	4	4	no	—
93	C₄⋊Dic₁₃	—	4	1	no	—
94	#6101	—	4	4	no	—
95	C₄×Dic₁₃	—	2	1	no	—
96	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
97	C₂×#1055	SD₁₆	1	1	yes	characteristic
98	C₂×#1055	SD₁₆	1	1	yes	characteristic
99	C₂×#1055	SD₁₆	1	2	yes	—
100	C₂×#1055	C₂×D₄	1	2	yes	—
101	Dic₅₂	—	2	1	no	—
102	Dic₅₂	C₂×Q₈	1	2	yes	—
103	D₁₀₄	—	2	1	no	—
104	D₁₀₄	C₂×Q₈	1	2	yes	—
105	#6095	—	2	2	no	—
106	#6095	—	2	2	no	—
107	#6094	—	4	4	no	—
108	C₈×D₁₃	—	4	4	no	—
109	C₁₃×C₁₆	—	2	2	no	—
110	#6090	—	2	2	no	—
111	C₂×#499	—	13	1	no	—
112	C₂×#443	—	13	1	no	—
113	#2231	—	13	8	no	—
114	C₂×#365	—	13	1	no	—
115	C₂×#365	D₁₃	1	1	yes	characteristic, complemented
116	C₂×#364	—	13	1	no	—
117	C₂×#360	—	13	1	no	—
118	C₂×#360	—	13	1	no	—
119	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂²×D₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	8	2	no	—
123	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₁₃	—	4	2	no	—
130	C₂²⋊D₁₃	—	4	2	no	—
131	C₂×Dic₁₃	—	2	2	no	—
132	C₂×Dic₁₃	—	4	2	no	—
133	C₂×Dic₁₃	—	4	1	no	—
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	8	2	no	—
136	D₅₂	—	2	2	no	—
137	D₅₂	—	4	1	no	—
138	C₄×D₁₃	—	4	2	no	—
139	C₄×D₁₃	—	4	2	no	—
140	Dic₂₆	—	2	2	no	—
141	Dic₂₆	—	4	1	no	—
142	C₈×C₁₃	—	4	2	no	—
143	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
146	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
147	#1055	C₂×SD₁₆	1	2	yes	—
148	#1055	C₂×SD₁₆	1	2	yes	—
149	C₂×#135	—	13	1	no	—
150	C₂×#131	—	26	1	no	—
151	C₂×#125	—	26	1	no	—
152	C₂×M₃₂	—	13	1	no	—
153	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
154	#499	—	13	4	no	—
155	#499	—	26	2	no	—
156	#443	—	13	4	no	—
157	#443	—	26	2	no	—
158	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
159	C₂×#107	—	26	1	no	—
160	C₂×#106	—	13	1	no	—
161	C₂×#106	C₂×D₁₃	1	1	yes	characteristic
162	C₂×#104	—	26	1	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	#365	—	13	4	no	—
165	#365	C₂×D₁₃	1	4	yes	complemented
166	#364	—	13	4	no	—
167	#360	—	13	4	no	—
168	#360	—	13	4	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	#476	1	1	yes	characteristic
171	C₂²×C₁₃	#476	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×D₁₃	—	4	2	no	—
174	C₂×D₁₃	—	8	2	no	—
175	C₄×C₁₃	—	8	2	no	—
176	C₄×C₁₃	#476	1	1	yes	characteristic
177	C₄×C₁₃	#476	1	1	yes	characteristic
178	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
180	Dic₁₃	—	4	2	no	—
181	Dic₁₃	—	4	2	no	—
182	Dic₁₃	—	8	2	no	—
183	C₂×#41	—	26	1	no	—
184	#135	—	13	2	no	—
185	#135	—	13	2	no	—
186	#135	—	26	1	no	—
187	#135	—	26	1	no	—
188	C₂×Q₁₆	—	13	1	no	—
189	C₂×SD₁₆	—	26	1	no	—
190	C₂×D₈	—	13	1	no	—
191	#131	—	26	4	no	—
192	#131	—	52	2	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	52	1	no	—
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	—	52	1	no	—
197	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	#125	—	26	4	no	—
199	#125	—	52	2	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	26	1	no	—
202	C₂×C₄⋊C₄	—	52	1	no	—
203	C₂×C₄⋊C₄	—	52	1	no	—
204	C₂×C₄²	—	13	1	no	—
205	M₃₂	—	2	1	no	—
206	M₃₂	—	13	2	no	—
207	M₃₂	—	26	1	no	—
208	M₃₂	C₂²×D₁₃	1	2	yes	—
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	26	1	no	—
211	#108	—	2	1	no	—
212	#108	C₂²×D₁₃	1	2	yes	—
213	#107	—	26	4	no	—
214	#106	—	2	1	no	—
215	#106	—	13	2	no	—
216	#106	—	26	1	no	—
217	#106	C₂²×D₁₃	1	2	yes	—
218	#104	—	26	4	no	—
219	C₄×C₈	—	13	2	no	—
220	C₄×C₈	—	13	2	no	—
221	C₂×C₁₃	—	2	1	no	—
222	C₂×C₁₃	—	2	1	no	—
223	C₂×C₁₃	#2231	1	2	yes	complemented
224	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
225	D₁₃	—	8	2	no	—
226	#41	—	26	2	no	—
227	#41	—	26	2	no	—
228	#41	—	52	1	no	—
229	#41	—	52	1	no	—
230	C₂×Q₈	—	26	1	no	—
231	C₂×D₄	—	26	1	no	—
232	C₂×D₄	—	52	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	26	1	no	—
235	C₂²×C₄	—	52	1	no	—
236	C₂²×C₄	—	52	1	no	—
237	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
238	Q₁₆	—	13	2	no	—
239	Q₁₆	—	26	1	no	—
240	SD₁₆	—	26	2	no	—
241	SD₁₆	—	26	2	no	—
242	D₈	—	13	2	no	—
243	D₈	—	26	1	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	52	4	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	13	1	no	—
249	C₂×C₈	—	13	1	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	52	4	no	—
253	C₂×C₈	#6127	1	1	yes	characteristic
254	C₂×C₈	#6127	1	1	yes	characteristic
255	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
256	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
257	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	26	2	no	—
262	C₄⋊C₄	—	52	4	no	—
263	C₄⋊C₄	—	52	4	no	—
264	C₄⋊C₄	—	52	1	no	—
265	C₄²	—	13	2	no	—
266	C₄²	—	26	1	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	26	2	no	—
269	C₁₃	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	52	1	no	—
271	C₂³	#88168	1	1	yes	characteristic
272	Q₈	—	26	2	no	—
273	Q₈	—	52	1	no	—
274	D₄	—	26	2	no	—
275	D₄	—	52	2	no	—
276	D₄	—	52	2	no	—
277	D₄	—	52	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	26	2	no	—
281	C₂×C₄	—	52	2	no	—
282	C₂×C₄	—	52	1	no	—
283	C₂×C₄	—	52	2	no	—
284	C₂×C₄	—	52	2	no	—
285	C₂×C₄	—	52	2	no	—
286	C₂×C₄	—	104	2	no	—
287	C₂×C₄	#88150	1	1	yes	characteristic
288	C₂×C₄	#88150	1	1	yes	characteristic
289	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
290	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
291	C₂×C₄	#88168	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
293	C₈	—	4	2	no	—
294	C₈	—	13	2	no	—
295	C₈	—	13	2	no	—
296	C₈	#88150	1	1	yes	characteristic
297	C₈	#88150	1	1	yes	characteristic
298	C₈	C₂×#6127	1	1	yes	characteristic
299	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
300	C₂²	—	2	2	no	—
301	C₂²	—	52	2	no	—
302	C₂²	—	104	2	no	—
303	C₂²	#11718531	1	1	yes	characteristic
304	C₂²	#11718531	1	1	yes	characteristic
305	C₂²	C₂×#88168	1	1	yes	characteristic
306	C₄	—	8	2	no	—
307	C₄	—	52	2	no	—
308	C₄	—	52	2	no	—
309	C₄	—	104	2	no	—
310	C₄	#11718531	1	1	yes	characteristic
311	C₄	#11718531	1	1	yes	characteristic
312	C₄	C₂×#88150	1	1	yes	characteristic
313	C₄	C₂×SD₁₆×D₁₃	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	104	2	no	—
317	C₂	#13117011	1	2	yes	direct product factor
318	C₂	C₂×#11718531	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718531
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#365	C₂
socle	C₂²×C₁₃	#476
prime radical	C₂×#88158	C₂²
hypercenter	C₂×#365	D₁₃
hypocenter	C₁₃	C₂×#2231
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
1. C₂×#11718519	C₂	semidirect
3. #13117011	C₂	direct (N × H)
4. C₂×#11718086	C₂	semidirect
5. C₂×#11718103	C₂	semidirect
8. C₂×C₁₃×#365	C₂	semidirect
23. #11718519	C₂²	semidirect
24. C₁₃×#365	C₂²	semidirect
26. #11718103	C₂²	semidirect
27. #11718086	C₂²	semidirect
115. C₂×#365	D₁₃	semidirect
165. #365	C₂×D₁₃	semidirect
223. C₂×C₁₃	#2231	semidirect
269. C₁₃	C₂×#2231	semidirect
317. C₂	#13117011	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×#11718519	2
C₂×#11718556	2
#13117011	2
C₂×#11718086	2
C₂×#11718103	2
C₂×#11718110	2
C₂×#11718080	2
C₂×C₁₃×#365	2
C₂×#2231	13

Subgroups — C2×#13117011 #16017654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117011 #16017654