Subgroups — C₂×#11719270 #13128355

Order 1664.

1252 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88193	C₂	1	2	yes	maximal, complemented
3	C₂×#88184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88184	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88180	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719270	C₂	1	16	yes	maximal, direct product factor
8	C₂×#88152	C₂	1	2	yes	maximal, complemented
9	C₂×#88137	C₂	1	2	yes	maximal, complemented
10	C₂×#88135	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88124	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₃	C₂²	1	2	yes	—
17	C₂²×C₂²⋊D₁₃	C₂²	1	2	yes	—
18	C₂³×Dic₁₃	C₂²	1	2	yes	—
19	C₂²×D₅₂	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
21	C₄×D₄×C₁₃	C₂²	1	8	yes	complemented
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
27	#88193	C₂²	1	16	yes	complemented
28	#88184	C₂²	1	8	yes	complemented
29	#88184	C₂²	1	8	yes	complemented
30	C₂×#6103	C₂²	1	2	yes	—
31	C₂×#6103	C₂²	1	2	yes	—
32	C₂×#6103	C₂²	1	2	yes	—
33	#88180	C₂²	1	8	yes	complemented
34	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
38	C₂×C₄⋊Dic₁₃	C₂²	1	8	yes	complemented
39	C₂×#6101	C₂²	1	2	yes	—
40	C₂×#6101	C₂²	1	2	yes	—
41	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
42	#88152	C₂²	1	16	yes	complemented
43	#88137	C₂²	1	16	yes	complemented
44	#88135	C₂²	1	16	yes	complemented
45	C₄×D₅₂	C₂²	1	8	yes	complemented
46	#88124	C₂²	1	8	yes	complemented
47	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
49	C₂³×D₁₃	C₂³	1	2	yes	—
50	C₂×D₄×C₁₃	—	2	4	no	—
51	C₂×D₄×C₁₃	C₂³	1	4	yes	—
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	—	2	8	no	—
62	C₂×C₂²⋊D₁₃	—	4	4	no	—
63	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
65	C₂²×Dic₁₃	—	4	4	no	—
66	C₂²×Dic₁₃	C₂³	1	2	yes	—
67	C₂²×Dic₁₃	C₂³	1	2	yes	—
68	C₂²×Dic₁₃	C₂³	1	2	yes	—
69	C₂²×Dic₁₃	C₂³	1	8	yes	—
70	C₂×D₅₂	—	2	4	no	—
71	C₂×D₅₂	C₂³	1	4	yes	—
72	C₂×C₄×D₁₃	—	4	4	no	—
73	C₂×C₄×D₁₃	C₂³	1	8	yes	—
74	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
75	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
76	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
77	C₄²×C₁₃	C₂³	1	4	yes	—
78	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
79	#6103	C₂³	1	8	yes	—
80	#6103	C₂³	1	8	yes	—
81	#6103	C₂³	1	8	yes	—
82	C₄⋊Dic₁₃	C₂³	1	4	yes	—
83	C₄⋊Dic₁₃	C₂³	1	4	yes	—
84	C₄⋊Dic₁₃	C₂³	1	4	yes	—
85	C₄⋊Dic₁₃	C₂³	1	8	yes	—
86	#6101	C₂³	1	8	yes	—
87	#6101	C₂³	1	8	yes	—
88	C₄×Dic₁₃	C₂³	1	8	yes	—
89	C₂×#553	—	13	1	no	maximal, Sylow
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	—	2	2	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	#41	1	4	yes	—
96	C₂³×C₁₃	C₂⁴	1	2	yes	—
97	C₂³×C₁₃	C₂⁴	1	2	yes	—
98	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₂²×D₁₃	—	2	4	no	—
100	C₂²×D₁₃	—	2	4	no	—
101	C₂²×D₁₃	—	2	4	no	—
102	C₂²×D₁₃	C₂⁴	1	4	yes	—
103	D₄×C₁₃	—	2	8	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	8	no	—
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	#41	1	4	yes	—
111	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
113	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
115	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
116	C₂²⋊D₁₃	—	2	16	no	—
117	C₂²⋊D₁₃	—	4	8	no	—
118	C₂×Dic₁₃	—	2	4	no	—
119	C₂×Dic₁₃	—	2	4	no	—
120	C₂×Dic₁₃	—	2	4	no	—
121	C₂×Dic₁₃	—	4	8	no	—
122	C₂×Dic₁₃	C₂⁴	1	4	yes	—
123	C₂×Dic₁₃	C₂⁴	1	4	yes	—
124	C₂×Dic₁₃	C₂⁴	1	4	yes	—
125	D₅₂	—	2	8	no	—
126	C₄×D₁₃	—	4	8	no	—
127	#553	—	13	16	no	—
128	C₂×#126	—	13	2	no	—
129	C₂×#125	—	13	1	no	—
130	C₂×#123	—	13	2	no	—
131	C₂×#121	—	13	2	no	—
132	C₂×#121	—	13	1	no	—
133	C₂×#121	—	13	1	no	—
134	C₂×C₄×D₄	—	13	2	no	—
135	C₂×C₄×D₄	—	13	1	no	—
136	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic, complemented
137	C₂×#117	—	13	1	no	—
138	C₂²×C₄⋊C₄	—	13	1	no	—
139	C₂²×C₁₃	—	2	8	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	4	4	no	—
146	C₂²×C₁₃	C₂×#41	1	4	yes	—
147	C₂²×C₁₃	C₂×#41	1	2	yes	—
148	C₂²×C₁₃	C₂×#41	1	2	yes	—
149	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
150	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
151	C₂×D₁₃	—	2	4	no	—
152	C₂×D₁₃	—	2	4	no	—
153	C₂×D₁₃	—	2	4	no	—
154	C₂×D₁₃	—	4	8	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	C₂×#41	1	4	yes	—
160	Dic₁₃	—	2	4	no	—
161	Dic₁₃	—	2	4	no	—
162	Dic₁₃	—	2	4	no	—
163	C₂²×D₄	—	13	2	no	—
164	C₂²×D₄	—	13	2	no	—
165	C₂²×D₄	—	13	1	no	—
166	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
167	C₂³×C₄	—	13	2	no	—
168	C₂³×C₄	—	13	2	no	—
169	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
170	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
171	#126	—	13	16	no	—
172	#125	—	13	8	no	—
173	#123	—	13	16	no	—
174	#121	—	13	16	no	—
175	#121	—	13	8	no	—
176	#121	—	13	8	no	—
177	C₄×D₄	—	13	16	no	—
178	C₄×D₄	—	13	8	no	—
179	C₄×D₄	C₂×D₁₃	1	8	yes	complemented
180	#117	—	13	8	no	—
181	C₂×C₄⋊C₄	—	13	1	no	—
182	C₂×C₄⋊C₄	—	13	1	no	—
183	C₂×C₄⋊C₄	—	13	2	no	—
184	C₂×C₄⋊C₄	—	13	8	no	—
185	C₂×C₄⋊C₄	—	13	1	no	—
186	C₂×C₄⋊C₄	—	13	2	no	—
187	C₂×C₄⋊C₄	—	13	2	no	—
188	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	—	13	2	no	—
190	C₂×C₂²⋊C₄	—	13	2	no	—
191	C₂×C₂²⋊C₄	—	13	2	no	—
192	C₂×C₂²⋊C₄	—	13	2	no	—
193	C₂×C₂²⋊C₄	C₂×D₁₃	1	1	yes	characteristic
194	C₂×C₂²⋊C₄	C₂×D₁₃	1	1	yes	characteristic
195	C₂×C₄²	—	13	2	no	—
196	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
197	C₂×C₁₃	—	2	4	no	—
198	C₂×C₁₃	—	4	2	no	—
199	C₂×C₁₃	#553	1	4	yes	complemented
200	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
201	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
202	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
203	D₁₃	—	4	4	no	—
204	C₂⁴	—	13	2	no	—
205	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
206	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	13	8	no	—
209	C₂×D₄	—	13	8	no	—
210	C₂×D₄	—	13	4	no	—
211	C₂×D₄	—	26	8	no	—
212	C₂×D₄	—	26	4	no	—
213	C₂×D₄	—	52	4	no	—
214	C₂×D₄	C₂²×D₁₃	1	4	yes	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	8	no	—
221	C₂²×C₄	—	13	8	no	—
222	C₂²×C₄	—	52	4	no	—
223	C₂²×C₄	—	52	4	no	—
224	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
225	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
226	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
231	C₄⋊C₄	—	13	4	no	—
232	C₄⋊C₄	—	13	4	no	—
233	C₄⋊C₄	—	13	8	no	—
234	C₄⋊C₄	—	13	4	no	—
235	C₄⋊C₄	—	13	8	no	—
236	C₄⋊C₄	—	13	8	no	—
237	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
238	C₂²⋊C₄	—	13	8	no	—
239	C₂²⋊C₄	—	13	8	no	—
240	C₂²⋊C₄	—	13	8	no	—
241	C₂²⋊C₄	—	13	8	no	—
242	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
243	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
244	C₄²	—	13	8	no	—
245	C₄²	C₂²×D₁₃	1	4	yes	—
246	C₁₃	C₂×#553	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	13	4	no	—
253	C₂³	—	26	4	no	—
254	C₂³	—	26	4	no	—
255	C₂³	—	26	4	no	—
256	C₂³	#6127	1	4	yes	—
257	C₂³	C₂³×D₁₃	1	2	yes	—
258	C₂³	C₂³×D₁₃	1	2	yes	—
259	C₂³	C₂³×D₁₃	1	1	yes	characteristic
260	D₄	—	2	8	no	—
261	D₄	—	26	16	no	—
262	D₄	—	26	8	no	—
263	D₄	—	52	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	13	4	no	—
271	C₂×C₄	—	13	4	no	—
272	C₂×C₄	—	13	4	no	—
273	C₂×C₄	—	26	4	no	—
274	C₂×C₄	—	26	4	no	—
275	C₂×C₄	—	26	4	no	—
276	C₂×C₄	—	52	8	no	—
277	C₂×C₄	—	52	8	no	—
278	C₂×C₄	#6129	1	4	yes	—
279	C₂×C₄	C₂³×D₁₃	1	2	yes	—
280	C₂×C₄	C₂³×D₁₃	1	2	yes	—
281	C₂×C₄	C₂³×D₁₃	1	2	yes	—
282	C₂×C₄	C₂³×D₁₃	1	2	yes	—
283	C₂×C₄	C₂³×D₁₃	1	2	yes	—
284	C₂²	—	2	8	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	26	4	no	—
292	C₂²	—	26	4	no	—
293	C₂²	—	26	4	no	—
294	C₂²	—	52	8	no	—
295	C₂²	C₂×#6127	1	4	yes	—
296	C₂²	C₂×#6127	1	2	yes	—
297	C₂²	C₂×#6129	1	2	yes	—
298	C₂²	#88256	1	2	yes	—
299	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	26	4	no	—
305	C₄	—	26	4	no	—
306	C₄	—	26	4	no	—
307	C₄	C₂×#6129	1	4	yes	—
308	C₂	—	2	4	no	—
309	C₂	—	4	2	no	—
310	C₂	—	52	4	no	—
311	C₂	#11719270	1	4	yes	direct product factor
312	C₂	C₂²×#6127	1	1	yes	characteristic
313	C₂	C₂²×#6129	1	1	yes	characteristic
314	C₂	C₂×#88256	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₄×D₄×C₁₃	C₂
socle	C₂³×C₁₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₃
hypocenter	C₁₃	C₂×#553
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₄×C₁₃	C₂	semidirect
2. C₂×#88193	C₂	semidirect
3. C₂×#88184	C₂	semidirect
4. C₂×#88184	C₂	semidirect
5. C₂×#88180	C₂	semidirect
6. C₂²×C₄⋊Dic₁₃	C₂	semidirect
7. #11719270	C₂	direct (N × H)
8. C₂×#88152	C₂	semidirect
9. C₂×#88137	C₂	semidirect
10. C₂×#88135	C₂	semidirect
11. C₂×C₄×D₅₂	C₂	semidirect
12. C₂×#88124	C₂	semidirect
21. C₄×D₄×C₁₃	C₂²	semidirect
27. #88193	C₂²	semidirect
28. #88184	C₂²	semidirect
29. #88184	C₂²	semidirect
33. #88180	C₂²	semidirect
38. C₂×C₄⋊Dic₁₃	C₂²	semidirect
42. #88152	C₂²	semidirect
43. #88137	C₂²	semidirect
44. #88135	C₂²	semidirect
45. C₄×D₅₂	C₂²	semidirect
46. #88124	C₂²	semidirect
136. C₂×C₄×D₄	D₁₃	semidirect
179. C₄×D₄	C₂×D₁₃	semidirect
199. C₂×C₁₃	#553	semidirect
246. C₁₃	C₂×#553	semidirect
311. C₂	#11719270	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×C₄×D₄×C₁₃	2
C₂×#88193	2
C₂×#88184	2
C₂×#88180	2
C₂²×C₄⋊Dic₁₃	2
#11719270	2
C₂×#88152	2
C₂×#88137	2
C₂×#88135	2
C₂×C₄×D₅₂	2
C₂×#88124	2
C₂×#553	13

Subgroups — C2×#11719270 #13128355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719270 #13128355