Subgroups — C₂×#11719395 #13128366

Order 1664.

1200 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719395	C₂	1	16	yes	maximal, direct product factor
3	C₂×#88151	C₂	1	2	yes	maximal, complemented
4	C₂×#88150	C₂	1	2	yes	maximal, complemented
5	C₂×#88150	C₂	1	2	yes	maximal, complemented
6	C₂×#88147	C₂	1	1	yes	maximal, characteristic
7	C₂×#88147	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₄×D₁₃	C₂	1	2	yes	maximal, complemented
9	C₂×#88144	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88142	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₅₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
17	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₆	C₂²	1	2	yes	—
19	C₁₃×#125	C₂²	1	8	yes	complemented
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
25	C₂×#6103	C₂²	1	2	yes	—
26	C₂×#6103	C₂²	1	2	yes	—
27	C₂×#6103	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
31	C₂×#6101	C₂²	1	1	yes	characteristic
32	C₂×#6101	C₂²	1	1	yes	characteristic
33	C₂×#6101	C₂²	1	2	yes	—
34	C₂×#6101	C₂²	1	2	yes	—
35	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
37	#88151	C₂²	1	16	yes	complemented
38	#88150	C₂²	1	16	yes	complemented
39	#88150	C₂²	1	16	yes	complemented
40	#88147	C₂²	1	8	yes	—
41	#88147	C₂²	1	8	yes	—
42	C₄⋊C₄×D₁₃	C₂²	1	16	yes	complemented
43	#88144	C₂²	1	8	yes	complemented
44	#88142	C₂²	1	8	yes	complemented
45	C₄×D₅₂	C₂²	1	8	yes	—
46	C₄×Dic₂₆	C₂²	1	8	yes	complemented
47	C₂³×D₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂³	1	2	yes	—
57	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂×D₅₂	Q₈	1	8	yes	complemented
59	C₂×D₅₂	C₂³	1	4	yes	—
60	C₂×C₄×D₁₃	—	2	8	no	—
61	C₂×C₄×D₁₃	—	4	4	no	—
62	C₂×C₄×D₁₃	—	4	4	no	—
63	C₂×C₄×D₁₃	C₂³	1	8	yes	—
64	C₂×C₄×D₁₃	C₂³	1	8	yes	—
65	C₂×C₄×D₁₃	C₂³	1	8	yes	—
66	C₂×Dic₂₆	—	2	4	no	—
67	C₂×Dic₂₆	—	4	4	no	—
68	C₂×Dic₂₆	C₂³	1	4	yes	—
69	C₂×Dic₂₆	C₂³	1	8	yes	—
70	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
73	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
74	C₄²×C₁₃	C₂³	1	4	yes	—
75	#6103	C₂³	1	8	yes	—
76	#6103	C₂³	1	8	yes	—
77	#6103	C₂³	1	8	yes	—
78	C₄⋊Dic₁₃	C₂³	1	4	yes	—
79	C₄⋊Dic₁₃	C₂³	1	4	yes	—
80	C₄⋊Dic₁₃	C₂³	1	8	yes	—
81	#6101	C₂³	1	4	yes	—
82	#6101	C₂³	1	4	yes	—
83	#6101	C₂³	1	8	yes	—
84	#6101	C₂³	1	8	yes	—
85	C₄×Dic₁₃	C₂³	1	4	yes	—
86	C₄×Dic₁₃	C₂³	1	4	yes	—
87	C₂×#554	—	13	1	no	maximal, Sylow
88	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂²×D₁₃	—	2	4	no	—
90	C₂²×D₁₃	—	2	4	no	—
91	C₂²×D₁₃	C₂×Q₈	1	8	yes	—
92	C₂²×D₁₃	C₂⁴	1	4	yes	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
98	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	2	no	—
105	C₂×Dic₁₃	—	2	4	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	#41	1	4	yes	—
108	C₂×Dic₁₃	C₂⁴	1	2	yes	—
109	C₂×Dic₁₃	C₂⁴	1	2	yes	—
110	C₂×Dic₁₃	C₂⁴	1	2	yes	—
111	C₂×Dic₁₃	C₂⁴	1	4	yes	—
112	C₂×Dic₁₃	C₂⁴	1	2	yes	—
113	D₅₂	C₂×Q₈	1	16	yes	complemented
114	C₄×D₁₃	—	2	16	no	—
115	C₄×D₁₃	—	4	8	no	—
116	C₄×D₁₃	—	4	8	no	—
117	Dic₂₆	—	2	8	no	—
118	Dic₂₆	—	4	8	no	—
119	#554	—	13	16	no	—
120	C₂×C₄⋊Q₈	—	13	1	no	—
121	C₂×#125	—	13	1	no	—
122	C₂×#125	D₁₃	1	1	yes	characteristic, complemented
123	C₂×C₂²⋊Q₈	—	13	2	no	—
124	C₂×C₂²⋊Q₈	—	13	2	no	—
125	C₂×C₂²⋊Q₈	—	13	2	no	—
126	C₂×C₄×Q₈	—	13	1	no	—
127	C₂×C₄×D₄	—	13	1	no	—
128	C₂×C₄×D₄	—	13	1	no	—
129	C₂×C₄×D₄	—	13	1	no	—
130	C₂²×C₄⋊C₄	—	13	2	no	—
131	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
132	C₂²×C₁₃	C₂×#41	1	2	yes	—
133	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
134	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
135	C₂×D₁₃	—	2	16	no	—
136	C₂×D₁₃	—	2	4	no	—
137	C₂×D₁₃	—	2	4	no	—
138	C₂×D₁₃	C₂²×Q₈	1	8	yes	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	4	no	—
141	C₄×C₁₃	—	2	4	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	C₂²×Q₈	1	4	yes	—
144	Dic₁₃	—	2	2	no	—
145	Dic₁₃	—	2	2	no	—
146	Dic₁₃	—	2	4	no	—
147	Dic₁₃	—	2	2	no	—
148	Dic₁₃	C₂×#41	1	4	yes	—
149	C₂²×Q₈	—	13	1	no	—
150	C₂²×Q₈	—	13	2	no	—
151	C₂²×D₄	—	13	1	no	—
152	C₂³×C₄	—	13	2	no	—
153	C₂³×C₄	—	13	2	no	—
154	C₂³×C₄	—	13	2	no	—
155	C₄⋊Q₈	—	13	8	no	—
156	#125	—	13	8	no	—
157	#125	C₂×D₁₃	1	8	yes	complemented
158	C₂²⋊Q₈	—	13	16	no	—
159	C₂²⋊Q₈	—	13	16	no	—
160	C₂²⋊Q₈	—	13	16	no	—
161	C₄×Q₈	—	13	8	no	—
162	C₄×D₄	—	13	8	no	—
163	C₄×D₄	—	13	8	no	—
164	C₄×D₄	—	13	8	no	—
165	C₂×C₄⋊C₄	—	13	1	no	—
166	C₂×C₄⋊C₄	—	13	2	no	—
167	C₂×C₄⋊C₄	—	13	2	no	—
168	C₂×C₄⋊C₄	—	13	1	no	—
169	C₂×C₄⋊C₄	—	13	1	no	—
170	C₂×C₄⋊C₄	—	13	1	no	—
171	C₂×C₄⋊C₄	—	13	2	no	—
172	C₂×C₄⋊C₄	—	13	16	no	—
173	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
177	C₂×C₂²⋊C₄	—	13	2	no	—
178	C₂×C₂²⋊C₄	—	13	2	no	—
179	C₂×C₂²⋊C₄	—	13	2	no	—
180	C₂×C₄²	—	13	1	no	—
181	C₂×C₄²	—	13	1	no	—
182	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
183	C₂×C₁₃	#554	1	4	yes	complemented
184	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
185	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
186	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
187	D₁₃	—	2	8	no	—
188	C₂⁴	—	13	2	no	—
189	C₂×Q₈	—	13	4	no	—
190	C₂×Q₈	—	13	8	no	—
191	C₂×Q₈	—	26	4	no	—
192	C₂×Q₈	—	52	4	no	—
193	C₂×D₄	—	13	8	no	—
194	C₂×D₄	—	13	4	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	—	13	8	no	—
201	C₂²×C₄	—	13	8	no	—
202	C₂²×C₄	—	13	8	no	—
203	C₂²×C₄	—	26	8	no	—
204	C₂²×C₄	—	52	4	no	—
205	C₂²×C₄	—	52	4	no	—
206	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
209	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
210	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
211	C₄⋊C₄	—	13	4	no	—
212	C₄⋊C₄	—	13	8	no	—
213	C₄⋊C₄	—	13	8	no	—
214	C₄⋊C₄	—	13	4	no	—
215	C₄⋊C₄	—	13	4	no	—
216	C₄⋊C₄	—	13	4	no	—
217	C₄⋊C₄	—	13	8	no	—
218	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
219	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
220	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
221	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
222	C₂²⋊C₄	—	13	8	no	—
223	C₂²⋊C₄	—	13	8	no	—
224	C₂²⋊C₄	—	13	8	no	—
225	C₄²	—	13	4	no	—
226	C₄²	—	13	4	no	—
227	C₄²	C₂²×D₁₃	1	4	yes	—
228	C₁₃	C₂×#554	1	1	yes	characteristic, complemented, Sylow
229	C₂³	—	13	8	no	—
230	C₂³	—	13	4	no	—
231	C₂³	—	26	4	no	—
232	C₂³	—	26	4	no	—
233	C₂³	C₂³×D₁₃	1	1	yes	characteristic
234	Q₈	—	26	8	no	—
235	Q₈	—	52	8	no	—
236	D₄	—	13	16	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	13	4	no	—
242	C₂×C₄	—	13	2	no	—
243	C₂×C₄	—	13	2	no	—
244	C₂×C₄	—	13	2	no	—
245	C₂×C₄	—	13	2	no	—
246	C₂×C₄	—	13	4	no	—
247	C₂×C₄	—	26	16	no	—
248	C₂×C₄	—	26	2	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	26	4	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	52	8	no	—
253	C₂×C₄	—	52	8	no	—
254	C₂×C₄	Q₈×D₁₃	1	4	yes	—
255	C₂×C₄	C₂³×D₁₃	1	2	yes	—
256	C₂×C₄	C₂³×D₁₃	1	2	yes	—
257	C₂×C₄	C₂³×D₁₃	1	4	yes	—
258	C₂×C₄	C₂³×D₁₃	1	4	yes	—
259	C₂×C₄	C₂³×D₁₃	1	2	yes	—
260	C₂²	—	13	8	no	—
261	C₂²	—	26	4	no	—
262	C₂²	—	26	4	no	—
263	C₂²	—	26	16	no	—
264	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
265	C₂²	#41×D₁₃	1	2	yes	—
266	C₂²	#88256	1	2	yes	—
267	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	13	4	no	—
273	C₄	—	26	2	no	—
274	C₄	—	26	2	no	—
275	C₄	—	26	4	no	—
276	C₄	—	26	2	no	—
277	C₄	C₂×Q₈×D₁₃	1	4	yes	—
278	C₂	—	26	8	no	—
279	C₂	#11719395	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
281	C₂	C₂×#41×D₁₃	1	1	yes	characteristic
282	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×C₁₃×#125	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	C₂×#554
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#125	C₂	semidirect
2. #11719395	C₂	direct (N × H)
3. C₂×#88151	C₂	semidirect
4. C₂×#88150	C₂	semidirect
5. C₂×#88150	C₂	semidirect
8. C₂×C₄⋊C₄×D₁₃	C₂	semidirect
9. C₂×#88144	C₂	semidirect
10. C₂×#88142	C₂	semidirect
12. C₂×C₄×Dic₂₆	C₂	semidirect
19. C₁₃×#125	C₂²	semidirect
37. #88151	C₂²	semidirect
38. #88150	C₂²	semidirect
39. #88150	C₂²	semidirect
42. C₄⋊C₄×D₁₃	C₂²	semidirect
43. #88144	C₂²	semidirect
44. #88142	C₂²	semidirect
46. C₄×Dic₂₆	C₂²	semidirect
58. C₂×D₅₂	Q₈	semidirect
113. D₅₂	C₂×Q₈	semidirect
122. C₂×#125	D₁₃	semidirect
157. #125	C₂×D₁₃	semidirect
183. C₂×C₁₃	#554	semidirect
228. C₁₃	C₂×#554	semidirect
279. C₂	#11719395	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×#125	2
#11719395	2
C₂×#88151	2
C₂×#88150	2
C₂×#88147	2
C₂×C₄⋊C₄×D₁₃	2
C₂×#88144	2
C₂×#88142	2
C₂×C₄×D₅₂	2
C₂×C₄×Dic₂₆	2
C₂×#554	13

Subgroups — C2×#11719395 #13128366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719395 #13128366