Subgroups — C₂×#13116600 #16015917

Order 3328.

698 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×D₁₃×M₃₂	C₂	1	2	yes	maximal
2	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
3	#13116600	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718103	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#6094	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88126	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88055	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	2	yes	—
15	C₂×#88043	C₄	1	1	yes	characteristic
16	D₁₃×M₃₂	—	2	4	no	—
17	D₁₃×M₃₂	C₂²	1	8	yes	—
18	C₂×#88038	—	2	2	no	—
19	C₂×C₁₆×D₁₃	—	2	2	no	—
20	#11718350	C₄	1	4	yes	—
21	#11718350	C₂²	1	4	yes	—
22	C₁₃×#364	C₂²	1	4	yes	complemented
23	#11718103	C₂²	1	8	yes	complemented
24	#11718055	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#6103	—	2	1	no	—
30	C₂×#6101	—	2	1	no	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂×#6094	C₈	1	4	yes	—
35	C₂×#6094	C₈	1	4	yes	—
36	C₂×#6094	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	D₄	1	2	yes	—
39	C₂×C₈×D₁₃	D₄	1	2	yes	—
40	C₂×C₈×D₁₃	Q₈	1	2	yes	—
41	C₂×C₈×D₁₃	Q₈	1	2	yes	—
42	C₂×C₈×D₁₃	C₂³	1	2	yes	—
43	#88126	—	2	2	no	—
44	#88126	C₂×C₄	1	4	yes	—
45	C₁₃×M₃₂	—	2	2	no	—
46	C₁₃×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₃×C₁₆	—	2	2	no	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	2	yes	—
50	#88055	C₂×C₄	1	4	yes	—
51	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
52	C₈×Dic₁₃	C₂³	1	2	yes	—
53	#88052	—	2	2	no	—
54	#88052	C₂³	1	4	yes	—
55	C₂×#6090	—	2	2	no	—
56	#88043	C₂×C₄	1	4	yes	—
57	#88038	—	2	8	no	—
58	C₁₆×D₁₃	—	2	8	no	—
59	C₂×#2153	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
65	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
70	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
82	#6103	—	2	4	no	—
83	#6101	—	2	4	no	—
84	C₄×Dic₁₃	—	2	1	no	—
85	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
86	C₂×#1055	C₄⋊C₄	1	2	yes	—
87	C₂×#1055	C₄⋊C₄	1	2	yes	—
88	C₂×#1055	C₂×C₈	1	2	yes	—
89	C₂×#1055	C₂×C₈	1	2	yes	—
90	C₂×#1055	C₂²×C₄	1	2	yes	—
91	C₂×#1055	C₂²×C₄	1	2	yes	—
92	#6094	C₂×C₈	1	16	yes	—
93	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
94	C₈×D₁₃	C₄⋊C₄	1	4	yes	—
95	C₈×D₁₃	C₂×D₄	1	4	yes	—
96	C₈×D₁₃	C₂×Q₈	1	4	yes	—
97	C₁₃×C₁₆	—	2	4	no	—
98	#6090	—	2	4	no	—
99	C₂²×M₃₂	—	13	2	no	—
100	C₂×#405	—	13	1	no	—
101	#2153	—	13	8	no	—
102	C₂×#364	—	13	2	no	—
103	C₂×#364	—	13	1	no	—
104	C₂×#364	D₁₃	1	1	yes	characteristic, complemented
105	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂²×D₁₃	—	4	2	no	—
107	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
108	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
109	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×Dic₁₃	—	2	2	no	—
119	C₂×Dic₁₃	—	4	2	no	—
120	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
121	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
122	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
123	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
124	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
125	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₄×D₁₃	C₂²×C₈	1	4	yes	—
127	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₈×C₁₃	C₂²×C₈	1	4	yes	—
130	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	#1055	C₂×C₄⋊C₄	1	4	yes	—
133	#1055	C₂²×C₈	1	4	yes	—
134	C₂²×M₁₆	—	13	1	no	—
135	C₂³×C₈	—	13	1	no	—
136	C₂×#117	—	13	1	no	—
137	C₂×M₃₂	—	13	2	no	—
138	C₂×M₃₂	—	13	8	no	—
139	C₂×M₃₂	—	26	2	no	—
140	C₂×M₃₂	—	26	4	no	—
141	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
142	C₂²×C₁₆	—	26	2	no	—
143	#405	—	13	4	no	—
144	#405	—	13	4	no	—
145	C₂×#97	—	13	1	no	—
146	C₂×#97	—	13	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
149	#364	—	13	8	no	—
150	#364	—	13	4	no	—
151	#364	C₂×D₁₃	1	4	yes	complemented
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
154	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
155	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
156	C₂×D₁₃	—	2	4	no	—
157	C₂×D₁₃	—	4	2	no	—
158	C₂×D₁₃	—	4	2	no	—
159	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
160	C₄×C₁₃	—	4	2	no	—
161	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
162	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
163	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
165	Dic₁₃	—	4	2	no	—
166	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
167	C₂³×C₄	—	13	1	no	—
168	C₂×M₁₆	—	13	4	no	—
169	C₂×M₁₆	—	13	4	no	—
170	C₂×M₁₆	—	13	4	no	—
171	C₂²×C₈	—	13	1	no	—
172	C₂²×C₈	—	13	1	no	—
173	C₂²×C₈	—	13	2	no	—
174	C₂²×C₈	—	13	2	no	—
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	—	13	2	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	#117	—	13	4	no	—
182	#117	—	26	2	no	—
183	C₂×C₄⋊C₄	—	26	1	no	—
184	C₂×C₂²⋊C₄	—	26	1	no	—
185	C₂×C₄²	—	13	1	no	—
186	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
187	M₃₂	—	2	2	no	—
188	M₃₂	—	13	4	no	—
189	M₃₂	—	26	8	no	—
190	M₃₂	—	26	2	no	—
191	M₃₂	C₂²×D₁₃	1	4	yes	—
192	C₂×C₁₆	—	2	2	no	—
193	C₂×C₁₆	—	26	8	no	—
194	C₂×C₁₆	—	26	2	no	—
195	#97	—	13	4	no	—
196	#97	—	13	4	no	—
197	C₄×C₈	—	13	2	no	—
198	C₄×C₈	—	13	2	no	—
199	C₄×C₈	C₄×D₁₃	1	2	yes	—
200	C₄×C₈	C₂²×D₁₃	1	2	yes	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	#2153	1	2	yes	complemented
204	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
205	D₁₃	—	2	4	no	—
206	C₂⁴	—	13	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	13	1	no	—
209	C₂²×C₄	—	13	2	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	13	2	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	2	no	—
215	C₂²×C₄	—	26	1	no	—
216	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
217	M₁₆	—	13	16	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	—	13	2	no	—
220	C₂×C₈	—	13	2	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	13	2	no	—
223	C₂×C₈	—	13	2	no	—
224	C₂×C₈	—	13	4	no	—
225	C₂×C₈	—	13	4	no	—
226	C₂×C₈	—	13	4	no	—
227	C₂×C₈	—	13	4	no	—
228	C₂×C₈	C₈×D₁₃	1	2	yes	—
229	C₂×C₈	C₈×D₁₃	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
232	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
233	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
237	C₄⋊C₄	—	26	4	no	—
238	C₂²⋊C₄	—	26	4	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	13	2	no	—
241	C₄²	—	26	1	no	—
242	C₄²	C₂×C₄×D₁₃	1	2	yes	—
243	C₁₆	—	2	4	no	—
244	C₁₆	—	26	4	no	—
245	C₁₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	13	2	no	—
247	C₂³	—	13	2	no	—
248	C₂³	—	13	2	no	—
249	C₂³	—	52	2	no	—
250	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	13	2	no	—
254	C₂×C₄	—	13	2	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	13	4	no	—
257	C₂×C₄	—	13	4	no	—
258	C₂×C₄	—	13	4	no	—
259	C₂×C₄	—	13	4	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	52	2	no	—
262	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
268	C₈	—	13	4	no	—
269	C₈	—	13	4	no	—
270	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
271	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
272	C₈	C₂×C₈×D₁₃	1	4	yes	—
273	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
274	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	13	4	no	—
277	C₂²	—	26	4	no	—
278	C₂²	—	52	2	no	—
279	C₂²	—	52	2	no	—
280	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
281	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
282	C₂²	C₂×M₁₆×D₁₃	1	1	yes	characteristic
283	C₄	—	4	2	no	—
284	C₄	—	13	4	no	—
285	C₄	—	52	2	no	—
286	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
287	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
288	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
289	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	26	4	no	—
293	C₂	#13116600	1	2	yes	direct product factor
294	C₂	C₂×D₁₃×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#364	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#364	D₁₃
hypocenter	C₁₃	C₂×#2153
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #13116600	C₂	direct (N × H)
4. C₂×#11718055	C₂	semidirect
5. C₂×#11718103	C₂	semidirect
6. C₂×C₁₃×#364	C₂	semidirect
22. C₁₃×#364	C₂²	semidirect
23. #11718103	C₂²	semidirect
24. #11718055	C₂²	semidirect
104. C₂×#364	D₁₃	semidirect
151. #364	C₂×D₁₃	semidirect
203. C₂×C₁₃	#2153	semidirect
245. C₁₃	C₂×#2153	semidirect
293. C₂	#13116600	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×D₁₃×M₃₂	2
C₂×#11718350	2
#13116600	2
C₂×#11718055	2
C₂×#11718103	2
C₂×C₁₃×#364	2
C₂×#2153	13

Subgroups — C2×#13116600 #16015917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116600 #16015917