Subgroups — #13117797

Order 1664.

464 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88141	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#11718638	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	#11718635	C₂	1	1	yes	maximal, characteristic
6	#11718594	C₂	1	2	yes	maximal
7	#11718593	C₂	1	2	yes	maximal
8	C₄×#6101	C₂	1	1	yes	maximal, characteristic
9	#11718590	C₂	1	1	yes	maximal, characteristic
10	#11718313	C₂	1	2	yes	maximal
11	#11718312	C₂	1	2	yes	maximal
12	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×#6101	C₂²	1	1	yes	characteristic
19	C₂×#6101	C₂²	1	1	yes	characteristic
20	C₂×#6101	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	#88141	C₄	1	8	yes	complemented
26	C₁₃×#95	C₂²	1	2	yes	—
27	#88071	C₂²	1	1	yes	characteristic
28	#88071	C₂²	1	2	yes	—
29	#88071	C₂²	1	2	yes	—
30	#88071	C₂²	1	2	yes	—
31	#88071	C₂²	1	2	yes	—
32	#88071	C₂²	1	2	yes	—
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂³	1	2	yes	—
43	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₆	—	2	2	no	—
48	C₂×Dic₂₆	—	2	2	no	—
49	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	—	2	4	no	—
54	C₄⋊Dic₁₃	—	2	2	no	—
55	#6101	—	2	2	no	—
56	#6101	—	2	2	no	—
57	#6101	C₂×C₄	1	8	yes	—
58	C₄×Dic₁₃	—	2	2	no	—
59	C₄×Dic₁₃	—	2	4	no	—
60	C₄×Dic₁₃	C₂×C₄	1	4	yes	—
61	C₄×Dic₁₃	C₂×C₄	1	8	yes	—
62	#2356	—	13	1	no	maximal, Sylow
63	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	2	1	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₃	#41	1	2	yes	—
77	C₂×C₄×C₁₃	#41	1	2	yes	—
78	C₂×C₄×C₁₃	#41	1	2	yes	—
79	C₂×Dic₁₃	—	2	1	no	—
80	C₂×Dic₁₃	—	2	1	no	—
81	C₂×Dic₁₃	—	2	1	no	—
82	C₂×Dic₁₃	—	2	2	no	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	1	no	—
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	2	1	no	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
95	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
96	C₂×Dic₁₃	#41	1	2	yes	—
97	Dic₂₆	—	2	8	no	—
98	C₂×C₄×Q₈	—	13	1	no	—
99	#391	—	13	1	no	—
100	#391	—	13	1	no	—
101	#387	—	13	2	no	—
102	#387	—	13	2	no	—
103	#383	—	13	2	no	—
104	#382	—	13	1	no	—
105	C₄×C₄⋊C₄	—	13	1	no	—
106	C₄×C₄⋊C₄	—	13	1	no	—
107	C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
108	#376	—	13	2	no	—
109	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₄×C₁₃	—	2	4	no	—
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	4	1	no	—
119	C₄×C₁₃	—	4	2	no	—
120	C₄×C₁₃	#117	1	4	yes	—
121	Dic₁₃	—	2	4	no	—
122	Dic₁₃	—	2	2	no	—
123	Dic₁₃	—	2	2	no	—
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	—	4	1	no	—
126	Dic₁₃	—	4	2	no	—
127	C₂²×Q₈	—	13	1	no	—
128	C₄×Q₈	—	13	8	no	—
129	C₂×C₄⋊C₄	—	13	2	no	—
130	C₂×C₄⋊C₄	—	13	1	no	—
131	C₂×C₄⋊C₄	—	13	1	no	—
132	C₂×C₄⋊C₄	—	13	1	no	—
133	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
135	C₂×C₄²	—	13	2	no	—
136	C₂×C₄²	—	13	1	no	—
137	C₂×C₄²	—	13	2	no	—
138	C₂×C₄²	—	13	1	no	—
139	C₂×C₄²	C₂×D₁₃	1	2	yes	—
140	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
141	#95	—	13	1	no	—
142	#95	—	13	2	no	—
143	#95	—	13	2	no	—
144	#95	—	13	2	no	—
145	#95	—	13	1	no	—
146	#95	—	13	2	no	—
147	#95	—	13	2	no	—
148	#95	—	13	2	no	—
149	#95	—	13	2	no	—
150	#95	C₂×D₁₃	1	2	yes	—
151	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
152	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
153	C₂×C₁₃	#519	1	1	yes	characteristic
154	C₂×C₁₃	#552	1	1	yes	characteristic
155	C₂×C₁₃	#552	1	1	yes	characteristic
156	C₂×C₁₃	#556	1	1	yes	characteristic
157	C₂×C₁₃	#556	1	1	yes	characteristic
158	C₂×Q₈	—	13	4	no	—
159	C₂×Q₈	—	26	2	no	—
160	C₂×Q₈	—	26	2	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	—	13	1	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	13	2	no	—
167	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
168	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
170	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	13	8	no	—
174	C₄⋊C₄	—	26	2	no	—
175	C₄⋊C₄	—	26	2	no	—
176	C₄⋊C₄	—	26	2	no	—
177	C₄⋊C₄	C₄×D₁₃	1	4	yes	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	4	no	—
180	C₄²	—	13	4	no	—
181	C₄²	—	13	8	no	—
182	C₄²	—	26	4	no	—
183	C₄²	—	26	2	no	—
184	C₁₃	#2356	1	1	yes	characteristic, complemented, Sylow
185	C₂³	C₂³×D₁₃	1	1	yes	characteristic
186	Q₈	—	26	8	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	13	2	no	—
199	C₂×C₄	—	13	4	no	—
200	C₂×C₄	—	13	2	no	—
201	C₂×C₄	—	26	1	no	—
202	C₂×C₄	—	26	2	no	—
203	C₂×C₄	—	26	2	no	—
204	C₂×C₄	—	26	1	no	—
205	C₂×C₄	—	26	1	no	—
206	C₂×C₄	—	26	1	no	—
207	C₂×C₄	—	26	1	no	—
208	C₂×C₄	—	26	1	no	—
209	C₂×C₄	—	26	1	no	—
210	C₂×C₄	—	26	2	no	—
211	C₂×C₄	—	26	1	no	—
212	C₂×C₄	—	26	1	no	—
213	C₂×C₄	—	26	2	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
216	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
217	C₂×C₄	#6127	1	2	yes	—
218	C₂×C₄	#6127	1	2	yes	—
219	C₂×C₄	#6129	1	2	yes	—
220	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
221	C₂²	C₂×#6127	1	1	yes	characteristic
222	C₂²	C₂×#6127	1	1	yes	characteristic
223	C₂²	C₂×#6129	1	1	yes	characteristic
224	C₂²	#88254	1	1	yes	characteristic
225	C₂²	#41×D₁₃	1	1	yes	characteristic
226	C₂²	#88257	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	26	4	no	—
232	C₄	—	26	2	no	—
233	C₄	—	26	2	no	—
234	C₄	—	26	2	no	—
235	C₄	—	52	1	no	—
236	C₄	—	52	2	no	—
237	C₄	#88126	1	4	yes	—
238	C₂	C₂×#88126	1	1	yes	characteristic
239	C₂	#11719218	1	1	yes	characteristic
240	C₂	#11719234	1	1	yes	characteristic
241	C₂	C₄×#6129	1	1	yes	characteristic
242	C₂	#11719247	1	1	yes	characteristic
243	C₂	#11719273	1	1	yes	characteristic
244	C₂	#11719277	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#2356
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
25. #88141	C₄	semidirect
184. C₁₃	#2356	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#88141	2
C₄×C₁₃×C₄⋊C₄	2
#11718638	2
C₄⋊C₄×Dic₁₃	2
#11718635	2
#11718594	2
#11718593	2
C₄×#6101	2
#11718590	2
#11718313	2
#11718312	2
#2356	13