Subgroups — C₂×#13123771 #16727285

Order 3328.

1982 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	#13123771	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11718685	C₂	1	4	yes	maximal, complemented
3	C₂×Q₈×#1055	C₂	1	4	yes	maximal, complemented
4	C₂×#11718660	C₂	1	2	yes	maximal
5	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718658	C₂	1	2	yes	maximal
7	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#533	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₄	1	8	yes	—
10	C₁₃×#533	C₂²	1	8	yes	—
11	C₂×C₁₃×C₄⋊Q₈	C₄	1	2	yes	—
12	C₂×C₁₃×#125	C₄	1	2	yes	—
13	C₂×C₁₃×C₂²⋊Q₈	C₄	1	4	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
15	C₂×C₁₃×#117	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂²×#6099	C₂²	1	2	yes	—
18	C₂³×#1055	C₂²	1	1	yes	characteristic
19	C₂³×#1055	C₂²	1	1	yes	characteristic
20	#11718685	C₂²	1	32	yes	complemented
21	Q₈×#1055	C₂²	1	32	yes	complemented
22	#11718660	C₂²	1	16	yes	—
23	#11718658	C₂²	1	16	yes	—
24	#11718585	C₂²	1	8	yes	—
25	C₂×C₄⋊#1055	C₂²	1	4	yes	—
26	C₂×C₄⋊#1055	C₂²	1	8	yes	—
27	C₂×#88043	C₂²	1	4	yes	—
28	C₂×C₄×#1055	C₂²	1	2	yes	—
29	C₂×C₄×#1055	C₂²	1	2	yes	—
30	C₂×C₄×#1055	C₂²	1	8	yes	—
31	C₂×C₄×#1055	C₂²	1	4	yes	—
32	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	16	yes	—
36	C₁₃×#125	C₂×C₄	1	16	yes	—
37	C₁₃×C₂²⋊Q₈	C₂×C₄	1	32	yes	—
38	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	—
39	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
40	C₁₃×#117	C₂×C₄	1	8	yes	—
41	C₁₃×#117	C₂³	1	8	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
45	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
52	C₂×#6099	—	2	8	no	—
53	C₂×#6099	C₂³	1	8	yes	—
54	C₂²×#1055	—	2	4	no	—
55	C₂²×#1055	Q₈	1	8	yes	complemented
56	C₂²×#1055	C₂³	1	4	yes	—
57	C₂²×#1055	C₂³	1	2	yes	—
58	C₂²×#1055	C₂³	1	4	yes	—
59	C₂²×#1055	C₂³	1	1	yes	characteristic
60	C₂²×#1055	C₂³	1	1	yes	characteristic
61	C₂²×#1055	C₂³	1	4	yes	—
62	C₄⋊#1055	C₂³	1	32	yes	—
63	C₄⋊#1055	C₂³	1	16	yes	—
64	#88043	C₂³	1	16	yes	—
65	C₄×#1055	C₂³	1	16	yes	—
66	C₄×#1055	C₂³	1	8	yes	—
67	C₄×#1055	C₂³	1	8	yes	—
68	C₂×#2960	—	13	1	no	maximal, Sylow
69	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×Q₈×C₁₃	—	2	16	no	—
71	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₁₃	#41	1	4	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	C₂²×C₄	1	16	yes	—
84	C₁₃×C₄⋊C₄	C₂²×C₄	1	32	yes	—
85	C₁₃×C₄⋊C₄	C₂²×C₄	1	16	yes	—
86	C₁₃×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
90	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
91	C₄²×C₁₃	C₂⁴	1	4	yes	—
92	C₄²×C₁₃	C₂⁴	1	8	yes	—
93	#6099	—	2	16	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	2	8	no	—
96	C₂×#1055	—	2	8	no	—
97	C₂×#1055	C₂×Q₈	1	4	yes	—
98	C₂×#1055	C₂×Q₈	1	4	yes	—
99	C₂×#1055	C₂×Q₈	1	16	yes	complemented
100	C₂×#1055	C₂⁴	1	4	yes	—
101	C₂×#1055	C₂⁴	1	8	yes	—
102	C₂×#1055	C₂⁴	1	2	yes	—
103	C₂×#1055	C₂⁴	1	2	yes	—
104	C₂×#533	D₁₃	1	1	yes	characteristic
105	#2960	—	13	16	no	—
106	C₂×#446	—	13	4	no	—
107	C₂×C₈×Q₈	—	13	4	no	—
108	C₂×#424	—	13	2	no	—
109	C₂×#423	—	13	1	no	—
110	C₂×#405	—	13	2	no	—
111	C₂²×C₄×C₈	—	13	1	no	—
112	C₂³×C₁₃	—	2	4	no	—
113	C₂³×C₁₃	C₄×Q₈	1	4	yes	—
114	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
115	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	Q₈×C₁₃	—	2	32	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	16	no	—
119	C₂×C₄×C₁₃	—	2	16	no	—
120	C₂×C₄×C₁₃	C₄×Q₈	1	8	yes	—
121	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
122	C₂×C₄×C₁₃	#131	1	8	yes	—
123	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	16	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
128	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
130	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
132	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
133	#1055	—	2	4	no	—
134	#1055	—	2	8	no	—
135	#1055	C₂²×Q₈	1	4	yes	—
136	#1055	C₂²×Q₈	1	4	yes	—
137	C₂²×M₁₆	—	13	2	no	—
138	C₂³×C₈	—	13	1	no	—
139	C₂³×C₈	—	13	1	no	—
140	#533	Dic₁₃	1	8	yes	—
141	#533	C₂×D₁₃	1	8	yes	—
142	C₂×C₄⋊Q₈	Dic₁₃	1	2	yes	—
143	C₂×#125	Dic₁₃	1	2	yes	—
144	C₂×C₂²⋊Q₈	Dic₁₃	1	4	yes	—
145	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
146	C₂×#117	C₂×D₁₃	1	2	yes	—
147	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
148	#446	—	13	32	no	—
149	C₈×Q₈	—	13	32	no	—
150	#424	—	13	16	no	—
151	#423	—	13	8	no	—
152	C₂×C₄⋊C₈	—	13	4	no	—
153	C₂×C₄⋊C₈	—	13	8	no	—
154	#405	—	13	16	no	—
155	C₂×#97	—	13	4	no	—
156	C₂×C₄×C₈	—	13	2	no	—
157	C₂×C₄×C₈	—	13	2	no	—
158	C₂×C₄×C₈	—	13	8	no	—
159	C₂×C₄×C₈	—	13	4	no	—
160	C₂²×C₁₃	—	2	8	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	C₂×C₄×Q₈	1	4	yes	—
163	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
164	C₂²×C₁₃	C₂×#131	1	4	yes	—
165	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
166	C₄×C₁₃	—	2	4	no	—
167	C₄×C₁₃	—	2	16	no	—
168	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
169	C₄×C₁₃	C₂×C₄×Q₈	1	4	yes	—
170	C₄×C₁₃	C₂×#131	1	8	yes	—
171	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
172	C₄×C₁₃	C₂²×#41	1	1	yes	characteristic
173	C₂²×Q₈	C₂×Dic₁₃	1	4	yes	—
174	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
175	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
176	C₂×M₁₆	—	13	8	no	—
177	C₂×M₁₆	—	26	8	no	—
178	C₂²×C₈	—	13	8	no	—
179	C₂²×C₈	—	13	4	no	—
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	—	13	4	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	13	4	no	—
185	C₂²×C₈	—	26	4	no	—
186	C₄⋊Q₈	C₂×Dic₁₃	1	16	yes	—
187	#125	C₂×Dic₁₃	1	16	yes	—
188	C₂²⋊Q₈	C₂×Dic₁₃	1	32	yes	—
189	C₄×Q₈	C₂×Dic₁₃	1	16	yes	—
190	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
191	#117	C₂×Dic₁₃	1	8	yes	—
192	#117	C₂²×D₁₃	1	8	yes	—
193	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
194	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
195	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
196	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
197	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
200	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
201	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
202	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
203	C₄⋊C₈	—	13	16	no	—
204	C₄⋊C₈	—	13	32	no	—
205	#97	—	13	16	no	—
206	C₄×C₈	—	13	8	no	—
207	C₄×C₈	—	13	8	no	—
208	C₄×C₈	—	13	16	no	—
209	C₂×C₁₃	—	2	4	no	—
210	C₂×C₁₃	#2960	1	4	yes	complemented
211	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
212	C₂×C₁₃	C₂²×#131	1	2	yes	—
213	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
214	C₂×Q₈	—	2	16	no	—
215	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
218	C₂²×C₄	#6131	1	4	yes	—
219	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
221	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
222	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
225	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
226	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
227	M₁₆	—	26	16	no	—
228	C₂×C₈	—	13	4	no	—
229	C₂×C₈	—	13	4	no	—
230	C₂×C₈	—	13	16	no	—
231	C₂×C₈	—	13	2	no	—
232	C₂×C₈	—	13	2	no	—
233	C₂×C₈	—	13	4	no	—
234	C₂×C₈	—	13	8	no	—
235	C₂×C₈	—	26	4	no	—
236	C₂×C₈	—	26	8	no	—
237	C₂×C₈	—	26	8	no	—
238	C₄⋊C₄	C₂²×Dic₁₃	1	16	yes	—
239	C₄⋊C₄	C₂²×Dic₁₃	1	32	yes	—
240	C₄⋊C₄	C₂²×Dic₁₃	1	16	yes	—
241	C₂²⋊C₄	C₂²×Dic₁₃	1	16	yes	—
242	C₄²	C₂²×Dic₁₃	1	4	yes	—
243	C₄²	C₂²×Dic₁₃	1	4	yes	—
244	C₄²	C₂²×Dic₁₃	1	4	yes	—
245	C₄²	C₂²×Dic₁₃	1	8	yes	—
246	C₄²	C₂³×D₁₃	1	4	yes	—
247	C₄²	C₂³×D₁₃	1	8	yes	—
248	C₁₃	C₂×#2960	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	2	4	no	—
250	C₂³	Q₈×Dic₁₃	1	4	yes	—
251	C₂³	C₂³×Dic₁₃	1	2	yes	—
252	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
253	Q₈	—	2	32	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	Q₈×Dic₁₃	1	8	yes	—
258	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
259	C₂×C₄	#88202	1	8	yes	—
260	C₂×C₄	C₂×Q₈×D₁₃	1	4	yes	—
261	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
262	C₂×C₄	C₂×#6131	1	2	yes	—
263	C₂×C₄	C₂×#6131	1	4	yes	—
264	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
265	C₂×C₄	C₂³×Dic₁₃	1	16	yes	—
266	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
268	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
269	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
270	C₈	—	13	4	no	—
271	C₈	—	13	4	no	—
272	C₈	—	26	4	no	—
273	C₈	—	26	8	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	8	no	—
276	C₂²	C₂×Q₈×Dic₁₃	1	4	yes	—
277	C₂²	C₂×Q₈×Dic₁₃	1	2	yes	—
278	C₂²	C₂×#88202	1	4	yes	—
279	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	16	no	—
282	C₄	C₂×Q₈×Dic₁₃	1	4	yes	—
283	C₄	C₂×Q₈×Dic₁₃	1	2	yes	—
284	C₄	C₂×#88202	1	8	yes	—
285	C₄	C₂²×Q₈×D₁₃	1	1	yes	characteristic
286	C₄	C₂²×#6131	1	1	yes	characteristic
287	C₂	—	2	4	no	—
288	C₂	#13123771	1	4	yes	direct product factor
289	C₂	C₂²×Q₈×Dic₁₃	1	1	yes	characteristic
290	C₂	C₂²×#88202	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#533	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×#533	D₁₃
hypocenter	C₁₃	C₂×#2960
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #13123771	C₂	direct (N × H)
2. C₂×#11718685	C₂	semidirect
3. C₂×Q₈×#1055	C₂	semidirect
20. #11718685	C₂²	semidirect
21. Q₈×#1055	C₂²	semidirect
55. C₂²×#1055	Q₈	semidirect
99. C₂×#1055	C₂×Q₈	semidirect
210. C₂×C₁₃	#2960	semidirect
248. C₁₃	C₂×#2960	semidirect
288. C₂	#13123771	direct (N × H)

Maximal subgroups

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

subgroup	index
#13123771	2
C₂×#11718685	2
C₂×Q₈×#1055	2
C₂×#11718660	2
C₂×#11718585	2
C₂×#11718658	2
C₂²×C₄×#1055	2
C₂×C₁₃×#533	2
C₂×#2960	13

Subgroups — C2×#13123771 #16727285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123771 #16727285