Subgroups — C₂×#13113612 #15831981

Order 3328.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	#13113612	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718071	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	—	2	1	no	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×#1055	—	2	1	no	—
9	C₁₃×#423	—	2	2	no	—
10	C₁₃×#423	C₂²	1	4	yes	—
11	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂²⋊#1055	C₄	1	2	yes	—
14	#11718586	C₄	1	4	yes	—
15	#11718586	C₂²	1	4	yes	—
16	C₂×C₄⋊#1055	—	2	1	no	—
17	C₂×C₄⋊#1055	—	2	2	no	—
18	C₂×C₄⋊#1055	C₄	1	2	yes	—
19	C₂×C₄⋊#1055	C₂²	1	4	yes	—
20	C₂×C₄⋊#1055	C₂²	1	2	yes	—
21	C₂×C₄×#1055	C₂²	1	2	yes	—
22	#11718071	C₂²	1	16	yes	complemented
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	4	no	—
27	C₂×C₁₃×M₁₆	—	4	4	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
37	C₂²×#1055	—	2	4	no	—
38	C₂²×#1055	—	2	2	no	—
39	C₂²×#1055	—	4	4	no	—
40	C₂²×#1055	C₂×C₄	1	4	yes	—
41	C₁₃×C₄⋊C₈	—	2	4	no	—
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
44	C₂²⋊#1055	C₂×C₄	1	8	yes	—
45	C₄⋊#1055	—	2	4	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄⋊#1055	C₂×C₄	1	8	yes	—
48	C₄⋊#1055	C₂³	1	4	yes	—
49	C₄×#1055	C₂×C₄	1	4	yes	—
50	C₄×#1055	C₂³	1	4	yes	—
51	C₂×#1571	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	D₈	1	4	yes	—
62	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₁₃×M₁₆	—	4	8	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	4	4	no	—
72	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₃	C₂×D₄	1	2	yes	—
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
81	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×#1055	—	2	8	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	4	4	no	—
85	C₂×#1055	—	4	8	no	—
86	C₂×#1055	C₂²×C₄	1	8	yes	—
87	C₂×#431	—	13	1	no	—
88	C₂×#423	D₁₃	1	1	yes	characteristic
89	C₂²×C₄⋊C₈	—	13	1	no	—
90	#1571	—	13	8	no	—
91	C₂×#335	—	13	4	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	#107	1	4	yes	—
95	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	#107	1	8	yes	—
105	C₂×C₄×C₁₃	#107	1	4	yes	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#131	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
116	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₁₃	—	4	4	no	—
121	#1055	—	2	8	no	—
122	#1055	—	4	4	no	—
123	C₂²×M₁₆	—	2	1	no	—
124	C₂³×C₈	—	26	1	no	—
125	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
126	#431	—	13	4	no	—
127	#431	—	13	4	no	—
128	#423	—	2	2	no	—
129	#423	C₂×D₁₃	1	4	yes	—
130	C₂×C₄⋊C₈	—	2	1	no	—
131	C₂×C₄⋊C₈	—	13	2	no	—
132	C₂×C₄⋊C₈	—	13	2	no	—
133	C₂×C₄⋊C₈	—	13	4	no	—
134	C₂×C₄⋊C₈	—	26	2	no	—
135	C₂×C₄⋊C₈	—	26	1	no	—
136	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
137	C₂×C₂²⋊C₈	—	13	2	no	—
138	C₂×C₄×C₈	—	13	2	no	—
139	#335	—	13	16	no	—
140	C₂²×C₁₃	—	2	8	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	#424	1	2	yes	—
144	C₂²×C₁₃	C₂×#107	1	4	yes	—
145	C₂²×C₁₃	C₂×#107	1	2	yes	—
146	C₂²×C₁₃	#430	1	2	yes	—
147	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	#424	1	2	yes	—
153	C₄×C₁₃	C₂×#107	1	2	yes	—
154	C₄×C₁₃	C₂×#107	1	4	yes	—
155	C₄×C₁₃	C₂×#131	1	2	yes	—
156	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
158	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
159	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
160	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	4	4	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	13	4	no	—
165	C₂²×C₈	—	26	4	no	—
166	C₂²×C₈	—	26	2	no	—
167	C₂²×C₈	—	52	4	no	—
168	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
169	C₂×C₄²	Dic₂₆	1	2	yes	—
170	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₄×D₁₃	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
173	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
175	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	13	8	no	—
179	C₄⋊C₈	—	13	4	no	—
180	C₄⋊C₈	—	26	2	no	—
181	C₄⋊C₈	—	26	4	no	—
182	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
183	C₂²⋊C₈	—	13	8	no	—
184	C₄×C₈	—	13	4	no	—
185	C₄×C₈	—	13	4	no	—
186	C₂×C₁₃	—	2	4	no	—
187	C₂×C₁₃	#1571	1	4	yes	complemented
188	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
189	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
191	C₂⁴	#6101	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	#6101	1	1	yes	characteristic
195	C₂²×C₄	#6101	1	2	yes	—
196	C₂²×C₄	#6101	1	1	yes	characteristic
197	C₂²×C₄	#6101	1	2	yes	—
198	C₂²×C₄	#6101	1	1	yes	characteristic
199	C₂²×C₄	#6101	1	2	yes	—
200	C₂²×C₄	#6104	1	4	yes	—
201	C₂²×C₄	#6107	1	4	yes	—
202	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
203	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
204	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
205	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
208	M₁₆	—	4	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	4	no	—
211	C₂×C₈	—	13	8	no	—
212	C₂×C₈	—	26	8	no	—
213	C₂×C₈	—	26	4	no	—
214	C₂×C₈	—	52	4	no	—
215	C₂×C₈	—	52	8	no	—
216	C₄²	#6101	1	2	yes	—
217	C₄²	#6101	1	2	yes	—
218	C₄²	#6101	1	2	yes	—
219	C₄²	C₂×Dic₂₆	1	2	yes	—
220	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
221	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
222	C₄²	C₂×C₄×D₁₃	1	2	yes	—
223	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
224	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	C₄²	C₂³×D₁₃	1	1	yes	characteristic
226	C₁₃	C₂×#1571	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#88047	1	4	yes	—
230	C₂³	C₂×#6101	1	2	yes	—
231	C₂³	C₂×#6101	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	#88047	1	8	yes	—
240	C₂×C₄	#88047	1	4	yes	—
241	C₂×C₄	#88161	1	4	yes	—
242	C₂×C₄	C₂×#6101	1	2	yes	—
243	C₂×C₄	C₂×#6101	1	1	yes	characteristic
244	C₂×C₄	C₂×#6101	1	2	yes	—
245	C₂×C₄	C₂×#6101	1	1	yes	characteristic
246	C₂×C₄	C₂×#6101	1	2	yes	—
247	C₂×C₄	C₂×#6101	1	1	yes	characteristic
248	C₂×C₄	C₂×#6104	1	2	yes	—
249	C₂×C₄	C₂×#6104	1	4	yes	—
250	C₂×C₄	C₂×#6107	1	4	yes	—
251	C₂×C₄	C₂×#6107	1	2	yes	—
252	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
255	C₈	—	4	4	no	—
256	C₈	—	26	8	no	—
257	C₈	—	52	4	no	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	C₂×#88047	1	4	yes	—
262	C₂²	C₂×#88047	1	2	yes	—
263	C₂²	#11718780	1	2	yes	—
264	C₂²	#11718785	1	2	yes	—
265	C₂²	C₂²×#6101	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	C₂×#88047	1	4	yes	—
271	C₄	C₂×#88047	1	2	yes	—
272	C₄	#11718780	1	2	yes	—
273	C₄	C₂×#88161	1	2	yes	—
274	C₄	C₂²×#6104	1	1	yes	characteristic
275	C₄	C₂²×#6107	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	#13113612	1	4	yes	direct product factor
278	C₂	C₂×#11718780	1	1	yes	characteristic
279	C₂	C₂²×#88047	1	1	yes	characteristic
280	C₂	C₂×#11718785	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#423	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#423	D₁₃
hypocenter	C₁₃	C₂×#1571
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #13113612	C₂	direct (N × H)
4. C₂×#11718071	C₂	semidirect
22. #11718071	C₂²	semidirect
187. C₂×C₁₃	#1571	semidirect
226. C₁₃	C₂×#1571	semidirect
277. C₂	#13113612	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×#11718586	2
C₂²×C₄⋊#1055	2
#13113612	2
C₂×#11718071	2
C₂×C₁₃×#423	2
C₂×#1571	13

Subgroups — C2×#13113612 #15831981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113612 #15831981