Subgroups — C₂×#11719390 #13128407

Order 1664.

1048 conjugacy classes of subgroups.

The subgroup lattice has 260 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
2	#11719390	C₂	1	16	yes	maximal, direct product factor
3	C₂×#88144	C₂	1	1	yes	maximal, characteristic
4	C₂×#88143	C₂	1	2	yes	maximal
5	C₂×#88143	C₂	1	2	yes	maximal
6	C₂×#88142	C₂	1	1	yes	maximal, characteristic
7	C₂×#88142	C₂	1	2	yes	maximal
8	C₂×#88141	C₂	1	1	yes	maximal, characteristic
9	C₂×#88141	C₂	1	2	yes	maximal
10	C₂×#88141	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₆	C₂²	1	2	yes	—
15	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
16	C₁₃×#125	C₂²	1	8	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
25	C₂×#6101	C₂²	1	1	yes	characteristic
26	C₂×#6101	C₂²	1	2	yes	—
27	C₂×#6101	C₂²	1	1	yes	characteristic
28	C₂×#6101	C₂²	1	2	yes	—
29	C₂×#6101	C₂²	1	2	yes	—
30	C₂×#6101	C₂²	1	2	yes	—
31	C₂×#6101	C₂²	1	2	yes	—
32	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
34	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
37	#88144	C₂²	1	8	yes	—
38	#88143	C₂²	1	16	yes	—
39	#88143	C₂²	1	16	yes	—
40	#88142	C₂²	1	8	yes	—
41	#88142	C₂²	1	16	yes	—
42	#88141	C₂²	1	8	yes	—
43	#88141	C₂²	1	16	yes	—
44	#88141	C₂²	1	8	yes	—
45	C₄×Dic₂₆	C₂²	1	8	yes	—
46	C₄×Dic₂₆	C₂²	1	8	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂³	1	2	yes	—
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×Dic₂₆	—	2	4	no	—
59	C₂×Dic₂₆	—	2	8	no	—
60	C₂×Dic₂₆	Q₈	1	8	yes	complemented
61	C₂×Dic₂₆	C₂³	1	4	yes	—
62	C₂×Dic₂₆	C₂³	1	8	yes	—
63	C₂×Dic₂₆	C₂³	1	4	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
65	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
68	C₄²×C₁₃	C₂³	1	4	yes	—
69	C₄⋊Dic₁₃	C₂³	1	4	yes	—
70	C₄⋊Dic₁₃	C₂³	1	4	yes	—
71	C₄⋊Dic₁₃	C₂³	1	8	yes	—
72	#6101	C₂³	1	4	yes	—
73	#6101	C₂³	1	8	yes	—
74	#6101	C₂³	1	4	yes	—
75	#6101	C₂³	1	8	yes	—
76	#6101	C₂³	1	8	yes	—
77	#6101	C₂³	1	8	yes	—
78	#6101	C₂³	1	8	yes	—
79	C₄×Dic₁₃	C₂³	1	4	yes	—
80	C₄×Dic₁₃	C₂³	1	8	yes	—
81	C₄×Dic₁₃	C₂³	1	4	yes	—
82	C₄×Dic₁₃	C₂³	1	8	yes	—
83	C₄×Dic₁₃	C₂³	1	8	yes	—
84	C₂×#557	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	4	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	4	no	—
90	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
91	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
92	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
93	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
94	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
95	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	4	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
101	C₂×Dic₁₃	#41	1	4	yes	—
102	C₂×Dic₁₃	C₂⁴	1	2	yes	—
103	C₂×Dic₁₃	C₂⁴	1	2	yes	—
104	C₂×Dic₁₃	C₂⁴	1	4	yes	—
105	C₂×Dic₁₃	C₂⁴	1	2	yes	—
106	C₂×Dic₁₃	C₂⁴	1	4	yes	—
107	C₂×Dic₁₃	C₂⁴	1	2	yes	—
108	Dic₂₆	—	2	8	no	—
109	Dic₂₆	—	2	16	no	—
110	Dic₂₆	C₂×Q₈	1	16	yes	complemented
111	#557	—	13	16	no	—
112	C₂×C₄⋊Q₈	—	13	2	no	—
113	C₂×C₄⋊Q₈	—	13	1	no	—
114	C₂×#125	—	13	2	no	—
115	C₂×#125	—	13	2	no	—
116	C₂×#125	—	13	1	no	—
117	C₂×#125	D₁₃	1	1	yes	characteristic
118	C₂×C₄×Q₈	—	13	1	no	—
119	C₂×C₄×Q₈	—	13	2	no	—
120	C₂×C₄×Q₈	—	13	1	no	—
121	C₂×C₄×Q₈	—	13	1	no	—
122	C₂×C₄×Q₈	—	13	1	no	—
123	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
124	C₂²×C₁₃	C₂×#41	1	2	yes	—
125	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
126	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	—	2	2	no	—
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	C₂²×Q₈	1	4	yes	—
132	Dic₁₃	—	2	2	no	—
133	Dic₁₃	—	2	4	no	—
134	Dic₁₃	—	2	2	no	—
135	Dic₁₃	—	2	2	no	—
136	Dic₁₃	C₂²×Q₈	1	8	yes	—
137	Dic₁₃	C₂×#41	1	4	yes	—
138	C₂²×Q₈	—	13	1	no	—
139	C₂²×Q₈	—	13	2	no	—
140	C₂²×Q₈	—	13	1	no	—
141	C₄⋊Q₈	—	13	16	no	—
142	C₄⋊Q₈	—	13	8	no	—
143	#125	—	13	16	no	—
144	#125	—	13	16	no	—
145	#125	—	13	8	no	—
146	#125	C₂×D₁₃	1	8	yes	—
147	C₄×Q₈	—	13	8	no	—
148	C₄×Q₈	—	13	16	no	—
149	C₄×Q₈	—	13	8	no	—
150	C₄×Q₈	—	13	8	no	—
151	C₄×Q₈	—	13	8	no	—
152	C₂×C₄⋊C₄	—	13	1	no	—
153	C₂×C₄⋊C₄	—	13	2	no	—
154	C₂×C₄⋊C₄	—	13	2	no	—
155	C₂×C₄⋊C₄	—	13	1	no	—
156	C₂×C₄⋊C₄	—	13	1	no	—
157	C₂×C₄⋊C₄	—	13	2	no	—
158	C₂×C₄⋊C₄	—	13	2	no	—
159	C₂×C₄⋊C₄	—	13	1	no	—
160	C₂×C₄⋊C₄	—	13	2	no	—
161	C₂×C₄⋊C₄	—	13	2	no	—
162	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
166	C₂×C₄²	—	13	2	no	—
167	C₂×C₄²	—	13	2	no	—
168	C₂×C₄²	—	13	1	no	—
169	C₂×C₄²	—	13	2	no	—
170	C₂×C₄²	—	13	1	no	—
171	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
172	C₂×C₁₃	#557	1	4	yes	complemented
173	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
174	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
175	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
176	C₂×Q₈	—	13	8	no	—
177	C₂×Q₈	—	13	4	no	—
178	C₂×Q₈	—	13	8	no	—
179	C₂×Q₈	—	13	4	no	—
180	C₂×Q₈	—	26	8	no	—
181	C₂×Q₈	—	26	4	no	—
182	C₂²×C₄	—	13	2	no	—
183	C₂²×C₄	—	13	1	no	—
184	C₂²×C₄	—	13	1	no	—
185	C₂²×C₄	—	13	1	no	—
186	C₂²×C₄	—	13	1	no	—
187	C₂²×C₄	—	13	2	no	—
188	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
190	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
191	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
193	C₄⋊C₄	—	13	4	no	—
194	C₄⋊C₄	—	13	8	no	—
195	C₄⋊C₄	—	13	8	no	—
196	C₄⋊C₄	—	13	4	no	—
197	C₄⋊C₄	—	13	4	no	—
198	C₄⋊C₄	—	13	8	no	—
199	C₄⋊C₄	—	13	8	no	—
200	C₄⋊C₄	—	13	4	no	—
201	C₄⋊C₄	—	13	8	no	—
202	C₄⋊C₄	—	13	8	no	—
203	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
204	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
205	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
206	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
207	C₄²	—	13	8	no	—
208	C₄²	—	13	8	no	—
209	C₄²	—	13	4	no	—
210	C₄²	—	13	8	no	—
211	C₄²	—	13	4	no	—
212	C₄²	C₂²×D₁₃	1	4	yes	—
213	C₁₃	C₂×#557	1	1	yes	characteristic, complemented, Sylow
214	C₂³	C₂³×D₁₃	1	1	yes	characteristic
215	Q₈	—	13	16	no	—
216	Q₈	—	26	16	no	—
217	Q₈	—	26	8	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	13	8	no	—
223	C₂×C₄	—	13	4	no	—
224	C₂×C₄	—	13	4	no	—
225	C₂×C₄	—	13	2	no	—
226	C₂×C₄	—	13	2	no	—
227	C₂×C₄	—	13	2	no	—
228	C₂×C₄	—	13	2	no	—
229	C₂×C₄	—	13	4	no	—
230	C₂×C₄	—	26	2	no	—
231	C₂×C₄	—	26	4	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	26	2	no	—
234	C₂×C₄	Q₈×D₁₃	1	4	yes	—
235	C₂×C₄	C₂³×D₁₃	1	2	yes	—
236	C₂×C₄	C₂³×D₁₃	1	4	yes	—
237	C₂×C₄	C₂³×D₁₃	1	4	yes	—
238	C₂×C₄	C₂³×D₁₃	1	2	yes	—
239	C₂×C₄	C₂³×D₁₃	1	2	yes	—
240	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
241	C₂²	#41×D₁₃	1	2	yes	—
242	C₂²	#88257	1	2	yes	—
243	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
244	C₄	—	2	4	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	13	8	no	—
249	C₄	—	13	4	no	—
250	C₄	—	26	2	no	—
251	C₄	—	26	4	no	—
252	C₄	—	26	2	no	—
253	C₄	—	26	2	no	—
254	C₄	C₂×Q₈×D₁₃	1	4	yes	—
255	C₂	#11719390	1	4	yes	direct product factor
256	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
257	C₂	C₂×#41×D₁₃	1	1	yes	characteristic
258	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×#125	C₂
socle, prime radical	C₂³×C₁₃	C₂⁴
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	C₂×#557
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
2. #11719390	C₂	direct (N × H)
60. C₂×Dic₂₆	Q₈	semidirect
110. Dic₂₆	C₂×Q₈	semidirect
172. C₂×C₁₃	#557	semidirect
213. C₁₃	C₂×#557	semidirect
255. C₂	#11719390	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
#11719390	2
C₂×#88144	2
C₂×#88143	2
C₂×#88142	2
C₂×#88141	2
C₂×C₄×Dic₂₆	2
C₂×#557	13

Subgroups — C2×#11719390 #13128407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719390 #13128407