Subgroups — C₂×#13117041 #16017694

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
2	#13117041	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718082	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718081	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718156	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718156	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#467	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
12	C₂×#88180	—	2	1	no	—
13	C₁₃×#467	—	2	2	no	—
14	C₁₃×#467	C₂²	1	4	yes	—
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#103	—	2	1	no	—
17	#11718767	—	2	2	no	—
18	#11718767	C₂²	1	4	yes	—
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88058	C₂²	1	1	yes	characteristic
21	C₂×#88057	—	2	1	no	—
22	C₂²×#6090	C₂²	1	1	yes	characteristic
23	C₂²×#6090	C₂²	1	1	yes	characteristic
24	#11718156	C₂²	1	4	yes	complemented
25	#11718156	C₂²	1	4	yes	complemented
26	C₂²⋊#6090	C₂²	1	4	yes	—
27	#11718082	C₂²	1	4	yes	complemented
28	#11718081	C₂²	1	4	yes	complemented
29	C₂²×Q₈×C₁₃	—	2	1	no	—
30	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	2	no	—
33	C₂×C₁₃×Q₁₆	—	2	2	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	D₄	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₂²⋊Q₈	—	2	4	no	—
42	C₁₃×C₂²⋊Q₈	—	4	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
45	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
46	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
47	#88180	—	2	4	no	—
48	#88180	—	4	2	no	—
49	C₂×C₄⋊Dic₁₃	—	2	1	no	—
50	C₂×C₄⋊Dic₁₃	—	2	1	no	—
51	C₂×C₄×Dic₁₃	—	4	1	no	—
52	C₁₃×#107	—	2	1	no	—
53	C₁₃×#107	C₂³	1	2	yes	—
54	C₁₃×#103	—	2	4	no	—
55	#88058	—	2	1	no	—
56	#88058	—	2	1	no	—
57	#88058	C₂³	1	2	yes	—
58	#88058	C₂³	1	2	yes	—
59	#88057	—	2	4	no	—
60	C₂×#6090	—	2	2	no	—
61	C₂×#6090	—	2	2	no	—
62	C₂×#6090	C₂³	1	2	yes	—
63	C₂×#6090	C₂³	1	2	yes	—
64	C₂×#2237	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
66	C₂×Q₈×C₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	4	2	no	—
69	C₂×Q₈×C₁₃	—	4	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	D₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	—	4	1	no	—
79	C₂²×Dic₁₃	—	4	1	no	—
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₁₃×Q₁₆	—	2	2	no	—
82	C₁₃×Q₁₆	—	4	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	1	no	—
94	C₁₃×C₄⋊C₄	—	4	4	no	—
95	C₁₃×C₂²⋊C₄	—	4	4	no	—
96	C₂²⋊Dic₁₃	—	4	4	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	4	1	no	—
100	C₄⋊Dic₁₃	—	4	1	no	—
101	C₄×Dic₁₃	—	4	4	no	—
102	#6090	—	2	2	no	—
103	#6090	—	2	2	no	—
104	C₂×#467	D₁₃	1	1	yes	characteristic
105	C₂×#427	—	13	1	no	—
106	#2237	—	13	8	no	—
107	C₂×#367	—	13	1	no	—
108	C₂×#366	—	13	1	no	—
109	C₂×#358	—	13	1	no	—
110	C₂×#358	—	13	1	no	—
111	C₂×C₂²⋊C₁₆	—	13	1	no	—
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	C₂×D₈	1	2	yes	—
116	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
117	Q₈×C₁₃	—	4	2	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	8	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	8	2	no	—
127	C₂×C₄×C₁₃	—	8	2	no	—
128	C₂×C₄×C₁₃	#123	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#137	1	2	yes	—
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₁₃	—	4	2	no	—
134	C₂×Dic₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	8	2	no	—
136	C₂×Dic₁₃	—	8	2	no	—
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	#123	1	2	yes	—
139	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
142	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
143	C₂×C₂²⋊Q₈	—	2	1	no	—
144	C₂×#117	—	26	1	no	—
145	C₂²×C₁₆	—	13	1	no	—
146	C₂²×C₁₆	—	13	1	no	—
147	#467	—	2	2	no	—
148	#467	C₂×D₁₃	1	4	yes	—
149	#427	—	13	4	no	—
150	#427	—	26	2	no	—
151	C₂×#107	—	13	1	no	—
152	C₂×#107	—	13	1	no	—
153	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
154	C₂×#106	—	26	1	no	—
155	C₂×#103	—	2	1	no	—
156	#367	—	13	4	no	—
157	#366	—	13	4	no	—
158	#358	—	13	4	no	—
159	#358	—	13	4	no	—
160	C₂²⋊C₁₆	—	13	4	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	4	4	no	—
165	C₂²×C₁₃	#480	1	2	yes	—
166	C₂²×C₁₃	#508	1	2	yes	—
167	C₂²×C₁₃	#510	1	2	yes	—
168	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	8	2	no	—
171	C₄×C₁₃	—	8	2	no	—
172	C₄×C₁₃	#480	1	2	yes	—
173	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
175	Dic₁₃	—	8	2	no	—
176	Dic₁₃	—	8	2	no	—
177	C₂²×Q₈	—	2	1	no	—
178	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	1	no	—
182	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
187	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	C₂²⋊Q₈	—	2	4	no	—
190	C₂²⋊Q₈	—	4	2	no	—
191	#117	—	26	4	no	—
192	#117	—	52	2	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	4	1	no	—
195	C₂×C₄⋊C₄	—	26	1	no	—
196	C₂×C₄⋊C₄	—	26	1	no	—
197	C₂×C₂²⋊C₄	—	4	1	no	—
198	C₂×C₂²⋊C₄	—	52	1	no	—
199	C₂×C₄²	—	52	1	no	—
200	C₂×C₁₆	—	13	2	no	—
201	C₂×C₁₆	—	13	2	no	—
202	C₂×C₁₆	—	26	2	no	—
203	C₂×C₁₆	—	26	2	no	—
204	#107	—	2	1	no	—
205	#107	—	13	2	no	—
206	#107	—	13	2	no	—
207	#107	—	26	1	no	—
208	#107	—	26	1	no	—
209	#107	C₂²×D₁₃	1	2	yes	—
210	#106	—	26	4	no	—
211	#103	—	2	4	no	—
212	C₂×C₁₃	—	4	2	no	—
213	C₂×C₁₃	#2237	1	4	yes	complemented
214	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
215	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
217	C₂⁴	#6104	1	1	yes	characteristic
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	52	1	no	—
227	C₂²×C₄	—	52	1	no	—
228	C₂²×C₄	#6104	1	2	yes	—
229	C₂²×C₄	#6104	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	4	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	#6129	1	2	yes	—
240	C₂×C₈	#6129	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	26	2	no	—
248	C₄⋊C₄	—	26	2	no	—
249	C₄⋊C₄	—	52	1	no	—
250	C₄⋊C₄	—	52	1	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	52	4	no	—
253	C₄²	—	52	4	no	—
254	C₁₆	—	26	2	no	—
255	C₁₆	—	26	2	no	—
256	C₁₃	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×#6104	1	2	yes	—
261	C₂³	C₂×#6104	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	52	2	no	—
274	C₂×C₄	—	52	2	no	—
275	C₂×C₄	—	104	2	no	—
276	C₂×C₄	—	104	2	no	—
277	C₂×C₄	C₂×#6104	1	2	yes	—
278	C₂×C₄	C₂×#6104	1	1	yes	characteristic
279	C₂×C₄	#88189	1	2	yes	—
280	C₂×C₄	#88205	1	2	yes	—
281	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	#88189	1	2	yes	—
284	C₈	C₂×#6129	1	1	yes	characteristic
285	C₈	C₂×#6129	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	#11718694	1	2	yes	—
291	C₂²	#11718840	1	2	yes	—
292	C₂²	#11718855	1	2	yes	—
293	C₂²	C₂²×#6104	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	104	2	no	—
298	C₄	—	104	2	no	—
299	C₄	#11718694	1	2	yes	—
300	C₄	C₂×#88189	1	1	yes	characteristic
301	C₄	C₂×#88205	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	#13117041	1	4	yes	direct product factor
304	C₂	C₂×#11718694	1	1	yes	characteristic
305	C₂	C₂×#11718855	1	1	yes	characteristic
306	C₂	C₂×#11718840	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#467	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂⁴×C₁₃	D₈
hypercenter	C₂×#467	D₁₃
hypocenter	C₁₃	C₂×#2237
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. #13117041	C₂	direct (N × H)
3. C₂×#11718082	C₂	semidirect
4. C₂×#11718081	C₂	semidirect
5. C₂×#11718156	C₂	semidirect
6. C₂×#11718156	C₂	semidirect
24. #11718156	C₂²	semidirect
25. #11718156	C₂²	semidirect
27. #11718082	C₂²	semidirect
28. #11718081	C₂²	semidirect
213. C₂×C₁₃	#2237	semidirect
256. C₁₃	C₂×#2237	semidirect
303. C₂	#13117041	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×#11718767	2
#13117041	2
C₂×#11718082	2
C₂×#11718081	2
C₂×#11718156	2
C₂×C₂²⋊#6090	2
C₂×C₁₃×#467	2
C₂×#2237	13

Subgroups — C2×#13117041 #16017694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117041 #16017694