Subgroups — C₂×#13116996 #16017537

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718539	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718533	C₂	1	1	yes	maximal, characteristic
3	#13116996	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718082	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718101	C₂	1	1	yes	maximal, characteristic
7	C₂×#11718083	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
10	C₂×#88147	—	2	1	no	—
11	C₂×#88144	—	2	1	no	—
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×#88058	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	C₂×#88049	—	2	1	no	—
19	C₂×#88048	—	2	1	no	—
20	#11718539	—	2	2	no	—
21	#11718539	C₂²	1	4	yes	complemented
22	#11718533	—	2	2	no	—
23	#11718533	C₂²	1	4	yes	—
24	C₁₃×#366	C₂²	1	4	yes	complemented
25	#11718101	C₂²	1	4	yes	—
26	#11718093	C₂²	1	4	yes	complemented
27	#11718083	C₂²	1	4	yes	—
28	#11718082	C₂²	1	4	yes	complemented
29	C₂²×D₅₂	—	2	1	no	—
30	C₂²×C₄×D₁₃	—	4	1	no	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×#6103	—	4	1	no	—
35	C₂×C₄⋊Dic₁₃	—	2	1	no	—
36	C₂×C₄⋊Dic₁₃	—	4	1	no	—
37	C₂×#6101	—	4	1	no	—
38	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
39	C₂²×#1055	D₄	1	1	yes	characteristic
40	C₂×D₁₀₄	—	2	1	no	—
41	C₂×D₁₀₄	—	2	2	no	—
42	C₂×D₁₀₄	Q₈	1	4	yes	—
43	C₂×D₁₀₄	C₂³	1	2	yes	—
44	#88147	—	2	4	no	—
45	#88147	—	4	2	no	—
46	#88144	—	2	4	no	—
47	#88144	—	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	—	2	1	no	—
52	C₁₃×#107	C₂³	1	2	yes	—
53	C₁₃×#107	C₂³	1	2	yes	—
54	#88058	—	2	1	no	—
55	#88058	C₂³	1	2	yes	—
56	C₈×Dic₁₃	D₄	1	2	yes	—
57	C₈×Dic₁₃	C₂³	1	2	yes	—
58	C₂×#6090	—	2	2	no	—
59	C₂×#6090	C₂³	1	2	yes	—
60	#88049	—	2	4	no	—
61	#88048	—	2	4	no	—
62	C₂×#2228	—	13	1	no	maximal, Sylow
63	C₂³×D₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	—	4	1	no	—
68	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
69	C₂²×Dic₁₃	D₈	1	1	yes	characteristic
70	C₂×D₅₂	—	2	4	no	—
71	C₂×D₅₂	—	2	2	no	—
72	C₂×D₅₂	—	4	1	no	—
73	C₂×D₅₂	—	4	2	no	—
74	C₂×C₄×D₁₃	—	4	4	no	—
75	C₂×C₄×D₁₃	—	8	4	no	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₁₃	#41	1	2	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	4	1	no	—
83	C₁₃×C₄⋊C₄	—	4	1	no	—
84	#6103	—	4	4	no	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	4	1	no	—
87	C₄⋊Dic₁₃	—	4	4	no	—
88	#6101	—	4	4	no	—
89	C₄×Dic₁₃	D₈	1	2	yes	—
90	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	C₂×D₄	1	2	yes	—
93	D₁₀₄	—	2	4	no	—
94	D₁₀₄	—	2	2	no	—
95	D₁₀₄	C₂×Q₈	1	4	yes	—
96	C₁₃×C₁₆	—	2	2	no	—
97	#6090	—	2	2	no	—
98	C₂×#499	—	13	1	no	—
99	C₂×C₄×D₈	—	13	1	no	—
100	#2228	—	13	8	no	—
101	C₂×#367	—	13	1	no	—
102	C₂×#366	D₁₃	1	1	yes	characteristic, complemented
103	C₂×C₄⋊C₁₆	—	13	1	no	—
104	C₂×#357	—	13	1	no	—
105	C₂×#357	—	13	1	no	—
106	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₁₃	—	4	2	no	—
108	C₂²×D₁₃	—	4	4	no	—
109	C₂²×D₁₃	—	8	2	no	—
110	C₂²×D₁₃	—	8	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	8	2	no	—
114	C₂×C₄×C₁₃	—	8	2	no	—
115	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#137	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₁₃	—	2	2	no	—
120	C₂×Dic₁₃	—	2	2	no	—
121	C₂×Dic₁₃	—	4	2	no	—
122	C₂×Dic₁₃	—	8	2	no	—
123	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
124	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
125	D₅₂	—	2	4	no	—
126	D₅₂	—	4	2	no	—
127	D₅₂	—	4	4	no	—
128	C₄×D₁₃	—	8	8	no	—
129	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
130	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
131	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
132	#1055	—	2	2	no	—
133	C₂²×D₈	—	13	1	no	—
134	C₂×#125	—	26	1	no	—
135	C₂×C₄×D₄	—	26	1	no	—
136	C₂²×C₁₆	—	13	1	no	—
137	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
138	#499	—	13	4	no	—
139	#499	—	26	2	no	—
140	C₄×D₈	—	13	4	no	—
141	C₄×D₈	—	26	2	no	—
142	C₂×#107	—	13	1	no	—
143	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
144	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
145	C₂×#106	—	26	1	no	—
146	C₂×#102	—	26	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	#367	—	13	4	no	—
149	#366	C₂×D₁₃	1	4	yes	complemented
150	C₄⋊C₁₆	—	13	4	no	—
151	#357	—	13	4	no	—
152	#357	—	13	4	no	—
153	C₂²×C₁₃	#474	1	2	yes	—
154	C₂²×C₁₃	#508	1	2	yes	—
155	C₂²×C₁₃	#509	1	2	yes	—
156	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₁₃	—	4	4	no	—
158	C₂×D₁₃	—	8	2	no	—
159	C₂×D₁₃	—	8	2	no	—
160	C₂×D₁₃	—	8	8	no	—
161	C₄×C₁₃	—	8	2	no	—
162	C₄×C₁₃	—	8	2	no	—
163	C₄×C₁₃	#474	1	2	yes	—
164	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
166	Dic₁₃	—	2	2	no	—
167	Dic₁₃	—	2	2	no	—
168	Dic₁₃	—	8	2	no	—
169	C₂²×D₄	—	26	1	no	—
170	C₂³×C₄	—	52	1	no	—
171	C₂×D₈	—	13	4	no	—
172	C₂×D₈	—	13	2	no	—
173	C₂×D₈	—	26	2	no	—
174	C₂×D₈	—	26	1	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	#125	—	26	4	no	—
178	#125	—	52	2	no	—
179	C₄×D₄	—	26	4	no	—
180	C₄×D₄	—	52	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	26	1	no	—
184	C₂×C₄⋊C₄	—	52	1	no	—
185	C₂×C₄⋊C₄	—	52	1	no	—
186	C₂×C₂²⋊C₄	—	52	1	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	13	2	no	—
190	C₂×C₁₆	—	26	2	no	—
191	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
192	#107	—	2	1	no	—
193	#107	—	2	1	no	—
194	#107	—	13	2	no	—
195	#107	—	26	1	no	—
196	#107	C₂²×D₁₃	1	2	yes	—
197	#107	C₂²×D₁₃	1	2	yes	—
198	#106	—	26	4	no	—
199	#102	—	26	4	no	—
200	C₄×C₈	—	13	2	no	—
201	C₄×C₈	—	13	2	no	—
202	C₂×C₁₃	#2228	1	4	yes	complemented
203	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
206	D₁₃	—	8	4	no	—
207	C₂⁴	—	52	1	no	—
208	C₂×D₄	—	26	4	no	—
209	C₂×D₄	—	26	2	no	—
210	C₂×D₄	—	52	1	no	—
211	C₂×D₄	—	52	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	52	4	no	—
217	C₂²×C₄	—	52	1	no	—
218	C₂²×C₄	—	104	4	no	—
219	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
220	D₈	—	13	4	no	—
221	D₈	—	26	4	no	—
222	D₈	—	26	2	no	—
223	C₂×C₈	—	13	2	no	—
224	C₂×C₈	—	26	2	no	—
225	C₂×C₈	#6127	1	2	yes	—
226	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	Q₈×D₁₃	1	2	yes	—
228	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	26	2	no	—
234	C₄⋊C₄	—	52	4	no	—
235	C₄⋊C₄	—	52	4	no	—
236	C₄⋊C₄	—	52	1	no	—
237	C₂²⋊C₄	—	52	4	no	—
238	C₄²	—	13	2	no	—
239	C₄²	—	13	2	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	26	2	no	—
242	C₁₃	C₂×#2228	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	52	2	no	—
244	C₂³	—	52	4	no	—
245	C₂³	—	104	2	no	—
246	C₂³	—	104	2	no	—
247	C₂³	D₈×D₁₃	1	1	yes	characteristic
248	D₄	—	26	4	no	—
249	D₄	—	52	2	no	—
250	D₄	—	52	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	52	2	no	—
260	C₂×C₄	—	104	8	no	—
261	C₂×C₄	—	104	2	no	—
262	C₂×C₄	#88150	1	2	yes	—
263	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
264	C₂×C₄	#88172	1	2	yes	—
265	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
266	C₈	—	26	2	no	—
267	C₈	#88150	1	2	yes	—
268	C₈	C₂×#6127	1	1	yes	characteristic
269	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
270	C₂²	—	52	4	no	—
271	C₂²	—	104	2	no	—
272	C₂²	—	104	2	no	—
273	C₂²	—	104	8	no	—
274	C₂²	#11718551	1	2	yes	—
275	C₂²	#11718572	1	2	yes	—
276	C₂²	#11718580	1	2	yes	—
277	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	—	26	2	no	—
281	C₄	—	26	2	no	—
282	C₄	—	104	2	no	—
283	C₄	#11718551	1	2	yes	—
284	C₄	C₂×#88150	1	1	yes	characteristic
285	C₄	C₂×#88172	1	1	yes	characteristic
286	C₂	—	104	4	no	—
287	C₂	#13116996	1	4	yes	direct product factor
288	C₂	C₂×#11718551	1	1	yes	characteristic
289	C₂	C₂×#11718580	1	1	yes	characteristic
290	C₂	C₂×#11718572	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×#366	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
hypercenter	C₂×#366	D₁₃
hypocenter	C₁₃	C₂×#2228
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
1. C₂×#11718539	C₂	semidirect
3. #13116996	C₂	direct (N × H)
4. C₂×#11718082	C₂	semidirect
5. C₂×#11718093	C₂	semidirect
8. C₂×C₁₃×#366	C₂	semidirect
21. #11718539	C₂²	semidirect
24. C₁₃×#366	C₂²	semidirect
26. #11718093	C₂²	semidirect
28. #11718082	C₂²	semidirect
102. C₂×#366	D₁₃	semidirect
149. #366	C₂×D₁₃	semidirect
202. C₂×C₁₃	#2228	semidirect
242. C₁₃	C₂×#2228	semidirect
287. C₂	#13116996	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×#11718539	2
C₂×#11718533	2
#13116996	2
C₂×#11718082	2
C₂×#11718093	2
C₂×#11718101	2
C₂×#11718083	2
C₂×C₁₃×#366	2
C₂×#2228	13

Subgroups — C2×#13116996 #16017537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116996 #16017537