Subgroups — C₂×#11719234 #13128347

Order 1664.

1124 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	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#88178	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719234	C₂	1	16	yes	maximal, direct product factor
5	C₂×#88151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊C₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88144	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88137	C₂	1	2	yes	maximal
10	C₂×#88133	C₂	1	2	yes	maximal, complemented
11	C₂×#88126	C₂	1	2	yes	maximal, complemented
12	C₂×#88124	C₂	1	2	yes	maximal, complemented
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
18	C₁₃×#117	C₂²	1	8	yes	complemented
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
25	C₄×C₂²⋊D₁₃	C₂²	1	8	yes	—
26	C₂×#6103	C₂²	1	1	yes	characteristic
27	C₂×#6103	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
30	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
31	#88178	C₂²	1	8	yes	complemented
32	C₂×#6101	C₂²	1	1	yes	characteristic
33	C₂×#6101	C₂²	1	2	yes	—
34	C₂×#6101	C₂²	1	2	yes	—
35	C₂×#6101	C₂²	1	2	yes	—
36	C₂×#6101	C₂²	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
39	#88151	C₂²	1	8	yes	complemented
40	C₄⋊C₄×D₁₃	C₂²	1	8	yes	complemented
41	#88144	C₂²	1	8	yes	complemented
42	#88141	C₂²	1	8	yes	complemented
43	#88137	C₂²	1	16	yes	—
44	#88133	C₂²	1	16	yes	complemented
45	#88126	C₂²	1	16	yes	complemented
46	#88124	C₂²	1	16	yes	complemented
47	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
48	C₂³×D₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₃	—	2	4	no	—
56	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
57	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂³	1	2	yes	—
60	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂³	1	2	yes	—
62	C₂×C₄×D₁₃	—	2	8	no	—
63	C₂×C₄×D₁₃	—	2	4	no	—
64	C₂×C₄×D₁₃	C₂³	1	8	yes	—
65	C₂×C₄×D₁₃	C₂³	1	4	yes	—
66	C₂×Dic₂₆	—	2	4	no	—
67	C₂×Dic₂₆	C₂³	1	4	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
69	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
70	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
71	C₄²×C₁₃	C₂³	1	8	yes	—
72	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
73	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
74	#6103	C₂³	1	4	yes	—
75	#6103	C₂³	1	8	yes	—
76	C₄⋊Dic₁₃	C₂³	1	8	yes	—
77	C₄⋊Dic₁₃	C₂³	1	8	yes	—
78	C₄⋊Dic₁₃	C₂³	1	4	yes	—
79	#6101	C₂³	1	4	yes	—
80	#6101	C₂³	1	8	yes	—
81	#6101	C₂³	1	8	yes	—
82	#6101	C₂³	1	8	yes	—
83	#6101	C₂³	1	8	yes	—
84	C₄×Dic₁₃	C₂³	1	8	yes	—
85	C₄×Dic₁₃	C₂³	1	4	yes	—
86	C₂×#552	—	13	1	no	maximal, Sylow
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	2	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	C₂⁴	1	2	yes	—
91	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₂²×D₁₃	—	2	2	no	—
93	C₂²×D₁₃	—	2	2	no	—
94	C₂²×D₁₃	#41	1	4	yes	—
95	C₂²×D₁₃	C₂⁴	1	2	yes	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	8	no	—
100	C₂×C₄×C₁₃	#41	1	4	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
105	C₂²⋊D₁₃	—	2	8	no	—
106	C₂×Dic₁₃	—	2	4	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	—	2	4	no	—
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	C₂⁴	1	2	yes	—
112	C₂×Dic₁₃	C₂⁴	1	2	yes	—
113	C₂×Dic₁₃	C₂⁴	1	4	yes	—
114	C₂×Dic₁₃	C₂⁴	1	2	yes	—
115	C₂×Dic₁₃	C₂⁴	1	4	yes	—
116	C₄×D₁₃	—	2	16	no	—
117	C₄×D₁₃	—	2	8	no	—
118	Dic₂₆	—	2	8	no	—
119	#552	—	13	16	no	—
120	C₂×#126	—	13	2	no	—
121	C₂×#125	—	13	1	no	—
122	C₂×#125	—	13	2	no	—
123	C₂×#123	—	13	2	no	—
124	C₂×C₂²⋊Q₈	—	13	1	no	—
125	C₂×C₂²⋊Q₈	—	13	1	no	—
126	C₂×C₄×Q₈	—	13	1	no	—
127	C₂×C₄×D₄	—	13	1	no	—
128	C₂×#117	—	13	2	no	—
129	C₂×#117	D₁₃	1	1	yes	characteristic, complemented
130	C₂²×C₄⋊C₄	—	13	1	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	4	4	no	—
135	C₂²×C₁₃	C₂×#41	1	2	yes	—
136	C₂²×C₁₃	C₂×#41	1	2	yes	—
137	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
138	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
139	C₂×D₁₃	—	2	8	no	—
140	C₂×D₁₃	—	2	2	no	—
141	C₂×D₁₃	—	2	2	no	—
142	C₂×D₁₃	C₂×#41	1	4	yes	—
143	C₄×C₁₃	—	2	4	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	C₂×#41	1	4	yes	—
147	Dic₁₃	—	2	4	no	—
148	Dic₁₃	—	2	2	no	—
149	Dic₁₃	—	2	4	no	—
150	Dic₁₃	—	2	2	no	—
151	Dic₁₃	—	2	2	no	—
152	C₂²×Q₈	—	13	1	no	—
153	C₂²×D₄	—	13	1	no	—
154	C₂³×C₄	—	13	2	no	—
155	C₂³×C₄	—	13	1	no	—
156	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
157	#126	—	13	16	no	—
158	#125	—	13	8	no	—
159	#125	—	13	16	no	—
160	#123	—	13	16	no	—
161	C₂²⋊Q₈	—	13	8	no	—
162	C₂²⋊Q₈	—	13	8	no	—
163	C₄×Q₈	—	13	8	no	—
164	C₄×D₄	—	13	8	no	—
165	#117	—	13	16	no	—
166	#117	C₂×D₁₃	1	8	yes	complemented
167	C₂×C₄⋊C₄	—	13	1	no	—
168	C₂×C₄⋊C₄	—	13	2	no	—
169	C₂×C₄⋊C₄	—	13	2	no	—
170	C₂×C₄⋊C₄	—	13	2	no	—
171	C₂×C₄⋊C₄	—	13	1	no	—
172	C₂×C₄⋊C₄	—	13	2	no	—
173	C₂×C₄⋊C₄	—	13	2	no	—
174	C₂×C₄⋊C₄	—	13	2	no	—
175	C₂×C₄⋊C₄	—	13	8	no	—
176	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	—	13	1	no	—
179	C₂×C₂²⋊C₄	—	13	2	no	—
180	C₂×C₂²⋊C₄	—	13	1	no	—
181	C₂×C₂²⋊C₄	—	13	2	no	—
182	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
183	C₂×C₄²	—	13	2	no	—
184	C₂×C₄²	—	13	1	no	—
185	C₂×C₄²	C₂×D₁₃	1	2	yes	—
186	C₂×C₁₃	—	4	2	no	—
187	C₂×C₁₃	#552	1	4	yes	complemented
188	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
189	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
190	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
191	D₁₃	—	2	4	no	—
192	C₂⁴	—	13	1	no	—
193	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
194	C₂×Q₈	—	13	4	no	—
195	C₂×Q₈	—	26	4	no	—
196	C₂×D₄	—	13	4	no	—
197	C₂×D₄	—	26	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	—	13	2	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	2	no	—
204	C₂²×C₄	—	13	8	no	—
205	C₂²×C₄	—	13	4	no	—
206	C₂²×C₄	—	26	8	no	—
207	C₂²×C₄	—	26	4	no	—
208	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
209	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
210	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
212	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
213	C₄⋊C₄	—	13	4	no	—
214	C₄⋊C₄	—	13	8	no	—
215	C₄⋊C₄	—	13	8	no	—
216	C₄⋊C₄	—	13	8	no	—
217	C₄⋊C₄	—	13	4	no	—
218	C₄⋊C₄	—	13	8	no	—
219	C₄⋊C₄	—	13	8	no	—
220	C₄⋊C₄	—	13	8	no	—
221	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
222	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
223	C₂²⋊C₄	—	13	4	no	—
224	C₂²⋊C₄	—	13	8	no	—
225	C₂²⋊C₄	—	13	4	no	—
226	C₂²⋊C₄	—	13	8	no	—
227	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
228	C₄²	—	13	8	no	—
229	C₄²	—	13	4	no	—
230	C₄²	C₂²×D₁₃	1	8	yes	—
231	C₁₃	C₂×#552	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	13	4	no	—
236	C₂³	—	13	2	no	—
237	C₂³	—	26	2	no	—
238	C₂³	—	26	2	no	—
239	C₂³	C₂³×D₁₃	1	2	yes	—
240	C₂³	C₂³×D₁₃	1	1	yes	characteristic
241	Q₈	—	26	8	no	—
242	D₄	—	26	8	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	13	2	no	—
248	C₂×C₄	—	13	4	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	4	no	—
252	C₂×C₄	—	26	16	no	—
253	C₂×C₄	—	26	2	no	—
254	C₂×C₄	—	26	4	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	8	no	—
259	C₂×C₄	#6127	1	4	yes	—
260	C₂×C₄	C₂³×D₁₃	1	4	yes	—
261	C₂×C₄	C₂³×D₁₃	1	2	yes	—
262	C₂×C₄	C₂³×D₁₃	1	4	yes	—
263	C₂×C₄	C₂³×D₁₃	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	13	4	no	—
269	C₂²	—	26	8	no	—
270	C₂²	—	26	2	no	—
271	C₂²	—	26	2	no	—
272	C₂²	C₂×#6127	1	2	yes	—
273	C₂²	#41×D₁₃	1	2	yes	—
274	C₂²	#88257	1	2	yes	—
275	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
276	C₄	—	2	2	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	4	no	—
279	C₄	—	26	2	no	—
280	C₄	—	26	4	no	—
281	C₄	—	26	4	no	—
282	C₄	—	26	2	no	—
283	C₄	—	26	2	no	—
284	C₄	C₂×#6127	1	4	yes	—
285	C₂	—	4	2	no	—
286	C₂	—	26	4	no	—
287	C₂	#11719234	1	4	yes	direct product factor
288	C₂	C₂²×#6127	1	1	yes	characteristic
289	C₂	C₂×#41×D₁₃	1	1	yes	characteristic
290	C₂	C₂×#88257	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#117	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×C₂²⋊D₁₃	C₂²
hypercenter	C₂×#117	D₁₃
hypocenter	C₁₃	C₂×#552
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#117	C₂	semidirect
3. C₂×#88178	C₂	semidirect
4. #11719234	C₂	direct (N × H)
5. C₂×#88151	C₂	semidirect
6. C₂×C₄⋊C₄×D₁₃	C₂	semidirect
7. C₂×#88144	C₂	semidirect
8. C₂×#88141	C₂	semidirect
10. C₂×#88133	C₂	semidirect
11. C₂×#88126	C₂	semidirect
12. C₂×#88124	C₂	semidirect
18. C₁₃×#117	C₂²	semidirect
31. #88178	C₂²	semidirect
39. #88151	C₂²	semidirect
40. C₄⋊C₄×D₁₃	C₂²	semidirect
41. #88144	C₂²	semidirect
42. #88141	C₂²	semidirect
44. #88133	C₂²	semidirect
45. #88126	C₂²	semidirect
46. #88124	C₂²	semidirect
129. C₂×#117	D₁₃	semidirect
166. #117	C₂×D₁₃	semidirect
187. C₂×C₁₃	#552	semidirect
231. C₁₃	C₂×#552	semidirect
287. C₂	#11719234	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×C₁₃×#117	2
C₂×C₄×C₂²⋊D₁₃	2
C₂×#88178	2
#11719234	2
C₂×#88151	2
C₂×C₄⋊C₄×D₁₃	2
C₂×#88144	2
C₂×#88141	2
C₂×#88137	2
C₂×#88133	2
C₂×#88126	2
C₂×#88124	2
C₂×#552	13

Subgroups — C2×#11719234 #13128347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719234 #13128347