Subgroups — C₂×#13123445 #16726615

Order 3328.

2586 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719524	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88202	C₂	1	1	yes	maximal, characteristic
3	#13123445	C₂	1	16	yes	maximal, direct product factor
4	C₂×#11718785	C₂	1	6	yes	maximal, complemented
5	C₂²×#88062	C₂	1	6	yes	maximal, complemented
6	C₂×C₁₃×#568	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
8	C₁₃×#568	—	2	4	no	—
9	C₁₃×#568	C₂²	1	8	yes	—
10	C₂×C₁₃×#131	—	2	4	no	—
11	C₂²×C₁₃×M₁₆	C₂²	1	6	yes	—
12	#11719524	—	2	4	no	—
13	#11719524	C₂²	1	8	yes	—
14	C₂×#88202	—	2	4	no	—
15	C₂×#88202	C₄	1	8	yes	—
16	C₂×#88202	C₄	1	8	yes	—
17	C₂×#88202	C₂²	1	8	yes	—
18	C₂²×#6099	C₂²	1	6	yes	—
19	C₂²×#6099	C₂²	1	3	yes	—
20	C₂³×#1055	C₂²	1	3	yes	—
21	#11718785	C₂²	1	48	yes	complemented
22	C₂×#88062	C₂²	1	4	yes	—
23	C₂×#88062	C₂²	1	12	yes	—
24	C₂×#88062	C₂²	1	48	yes	complemented
25	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×#41	D₄	1	4	yes	—
28	C₂×C₁₃×#41	D₄	1	4	yes	—
29	C₂×C₁₃×#41	Q₈	1	4	yes	—
30	C₂×C₁₃×#41	C₂³	1	4	yes	—
31	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
32	C₂²×D₄×C₁₃	Q₈	1	3	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
34	C₁₃×#131	—	2	32	no	—
35	C₂×C₁₃×M₁₆	—	2	6	no	—
36	C₂×C₁₃×M₁₆	—	2	12	no	—
37	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
38	C₂×C₁₃×M₁₆	C₂³	1	24	yes	—
39	C₂²×C₈×C₁₃	—	2	6	no	—
40	#88202	—	2	32	no	—
41	#88202	C₂×C₄	1	64	yes	—
42	C₂×#6099	—	2	6	no	—
43	C₂×#6099	—	2	12	no	—
44	C₂×#6099	C₂×C₄	1	12	yes	—
45	C₂×#6099	C₂×C₄	1	12	yes	—
46	C₂×#6099	C₂³	1	4	yes	—
47	C₂×#6099	C₂³	1	24	yes	—
48	C₂×#6099	C₂³	1	12	yes	—
49	C₂²×#1055	—	2	6	no	—
50	C₂²×#1055	C₂×C₄	1	12	yes	—
51	C₂²×#1055	C₂×C₄	1	12	yes	—
52	C₂²×#1055	C₂³	1	1	yes	characteristic
53	C₂²×#1055	C₂³	1	3	yes	—
54	C₂²×#1055	C₂³	1	12	yes	—
55	#88062	C₂³	1	16	yes	—
56	#88062	C₂³	1	48	yes	—
57	C₂×#2916	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	C₂×Q₈	1	3	yes	—
59	C₁₃×#41	C₄⋊C₄	1	16	yes	—
60	C₁₃×#41	C₄⋊C₄	1	16	yes	—
61	C₁₃×#41	C₂²×C₄	1	16	yes	—
62	C₁₃×#41	C₂×D₄	1	16	yes	—
63	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
64	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
65	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
66	C₂×D₄×C₁₃	C₄⋊C₄	1	12	yes	—
67	C₂×D₄×C₁₃	C₄⋊C₄	1	12	yes	—
68	C₂×D₄×C₁₃	C₂×Q₈	1	12	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
73	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
74	C₂²×C₄×C₁₃	C₂×Q₈	1	6	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
77	C₁₃×M₁₆	—	2	4	no	—
78	C₁₃×M₁₆	—	2	24	no	—
79	C₁₃×M₁₆	C₂⁴	1	8	yes	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	24	no	—
82	#6099	—	2	4	no	—
83	#6099	—	2	24	no	—
84	#6099	C₂²×C₄	1	48	yes	—
85	#6099	C₂⁴	1	8	yes	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	2	24	no	—
88	C₂×#1055	C₂²×C₄	1	2	yes	—
89	C₂×#1055	C₂²×C₄	1	2	yes	—
90	C₂×#1055	C₂²×C₄	1	6	yes	—
91	C₂×#1055	C₂²×C₄	1	6	yes	—
92	C₂×#1055	C₂²×C₄	1	48	yes	—
93	C₂×#1055	C₂⁴	1	2	yes	—
94	C₂×#1055	C₂⁴	1	6	yes	—
95	C₂×#568	—	13	1	no	—
96	C₂×#568	D₁₃	1	1	yes	characteristic
97	C₂²×#131	—	13	1	no	—
98	#2916	—	13	16	no	—
99	C₂×#430	—	13	6	no	—
100	C₂²×#108	—	13	6	no	—
101	C₂³×C₁₃	—	4	6	no	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
103	C₂³×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
104	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
106	Q₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
107	D₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
111	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
120	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
121	C₈×C₁₃	—	2	8	no	—
122	#1055	—	2	8	no	—
123	#1055	C₂³×C₄	1	4	yes	—
124	#1055	C₂³×C₄	1	12	yes	—
125	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
126	#568	—	2	4	no	—
127	#568	—	13	8	no	—
128	#568	—	26	4	no	—
129	#568	C₂×D₁₃	1	8	yes	—
130	C₂×#131	—	2	4	no	—
131	C₂×#131	—	13	8	no	—
132	C₂×#131	—	13	8	no	—
133	C₂×#131	—	13	8	no	—
134	C₂×#131	—	26	4	no	—
135	C₂²×M₁₆	—	13	3	no	—
136	C₂²×M₁₆	—	13	6	no	—
137	C₂²×M₁₆	C₂×D₁₃	1	6	yes	—
138	C₂³×C₈	—	13	3	no	—
139	#430	—	13	48	no	—
140	C₂×#108	—	13	4	no	—
141	C₂×#108	—	13	12	no	—
142	C₂×#108	—	13	48	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	12	no	—
145	C₂²×C₁₃	—	4	6	no	—
146	C₂²×C₁₃	—	4	6	no	—
147	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
149	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
151	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
154	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
155	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
156	C₂×#41	Dic₂₆	1	4	yes	—
157	C₂×#41	C₄×D₁₃	1	4	yes	—
158	C₂×#41	C₄×D₁₃	1	4	yes	—
159	C₂×#41	C₂²⋊D₁₃	1	4	yes	—
160	C₂×#41	C₂²⋊D₁₃	1	4	yes	—
161	C₂×#41	C₂²×D₁₃	1	4	yes	—
162	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
163	C₂²×D₄	Dic₂₆	1	3	yes	—
164	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
165	#131	—	2	32	no	—
166	#131	—	13	64	no	—
167	#131	—	26	32	no	—
168	C₂×M₁₆	—	2	6	no	—
169	C₂×M₁₆	—	2	12	no	—
170	C₂×M₁₆	—	13	12	no	—
171	C₂×M₁₆	—	13	12	no	—
172	C₂×M₁₆	—	13	12	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂×M₁₆	—	13	24	no	—
175	C₂×M₁₆	—	26	6	no	—
176	C₂×M₁₆	—	26	12	no	—
177	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
178	C₂×M₁₆	C₂²×D₁₃	1	24	yes	—
179	C₂²×C₈	—	2	6	no	—
180	C₂²×C₈	—	13	12	no	—
181	C₂²×C₈	—	13	12	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₂²×C₈	—	13	3	no	—
184	C₂²×C₈	—	13	12	no	—
185	C₂²×C₈	—	26	6	no	—
186	#108	—	13	16	no	—
187	#108	—	13	48	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	1	no	—
190	C₂×C₁₃	—	2	12	no	—
191	C₂×C₁₃	#2916	1	2	yes	complemented
192	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
193	C₂⁴	C₂×Dic₂₆	1	3	yes	—
194	#41	#6101	1	16	yes	—
195	#41	#6101	1	16	yes	—
196	#41	C₂×C₄×D₁₃	1	16	yes	—
197	#41	C₂×C₂²⋊D₁₃	1	16	yes	—
198	C₂×Q₈	#6101	1	4	yes	—
199	C₂×Q₈	#6101	1	4	yes	—
200	C₂×Q₈	C₂×Dic₂₆	1	4	yes	—
201	C₂×D₄	#6101	1	12	yes	—
202	C₂×D₄	#6101	1	12	yes	—
203	C₂×D₄	C₂×Dic₂₆	1	12	yes	—
204	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
205	C₂²×C₄	C₂×Dic₂₆	1	6	yes	—
206	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
207	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
210	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
211	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	24	no	—
214	M₁₆	—	13	48	no	—
215	M₁₆	—	13	8	no	—
216	M₁₆	—	26	4	no	—
217	M₁₆	—	26	24	no	—
218	M₁₆	C₂³×D₁₃	1	8	yes	—
219	C₂×C₈	—	2	24	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	13	2	no	—
223	C₂×C₈	—	13	6	no	—
224	C₂×C₈	—	13	6	no	—
225	C₂×C₈	—	13	48	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	13	6	no	—
228	C₂×C₈	—	26	24	no	—
229	C₂×C₈	—	26	4	no	—
230	C₁₃	C₂×#2916	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	4	6	no	—
232	C₂³	C₂×#6101	1	6	yes	—
233	C₂³	C₂×#6101	1	6	yes	—
234	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
235	C₂³	C₂²×Dic₂₆	1	6	yes	—
236	Q₈	C₂×#6101	1	16	yes	—
237	D₄	C₂×#6101	1	48	yes	—
238	C₂×C₄	C₂×#6101	1	6	yes	—
239	C₂×C₄	C₂×#6101	1	6	yes	—
240	C₂×C₄	C₂×#6101	1	12	yes	—
241	C₂×C₄	C₂×#6101	1	12	yes	—
242	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
243	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
244	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₄×D₁₃	1	12	yes	—
247	C₂×C₄	C₂²×C₂²⋊D₁₃	1	12	yes	—
248	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₈	—	13	4	no	—
253	C₈	—	13	12	no	—
254	C₈	—	26	8	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	12	no	—
257	C₂²	—	4	6	no	—
258	C₂²	—	4	6	no	—
259	C₂²	C₂²×#6101	1	1	yes	characteristic
260	C₂²	C₂²×#6101	1	12	yes	—
261	C₂²	C₂²×#6101	1	1	yes	characteristic
262	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
263	C₄	C₂²×#6101	1	1	yes	characteristic
264	C₄	C₂²×#6101	1	1	yes	characteristic
265	C₄	C₂²×#6101	1	12	yes	—
266	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
267	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	2	12	no	—
271	C₂	#13123445	1	2	yes	direct product factor
272	C₂	C₂³×#6101	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#568	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂²×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×#568	D₁₃
hypocenter	C₁₃	C₂×#2916
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. #13123445	C₂	direct (N × H)
4. C₂×#11718785	C₂	semidirect
5. C₂²×#88062	C₂	semidirect
21. #11718785	C₂²	semidirect
24. C₂×#88062	C₂²	semidirect
191. C₂×C₁₃	#2916	semidirect
230. C₁₃	C₂×#2916	semidirect
271. C₂	#13123445	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×#11719524	2
C₂²×#88202	2
#13123445	2
C₂×#11718785	2
C₂²×#88062	2
C₂×C₁₃×#568	2
C₂×#2916	13

Subgroups — C2×#13123445 #16726615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123445 #16726615