Subgroups — #13117890

Order 1664.

552 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#88198	C₂	1	1	yes	maximal, characteristic
3	C₂×#88141	C₂	1	1	yes	maximal, characteristic
4	#11718891	C₂	1	1	yes	maximal, characteristic
5	#11718639	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	#11718633	C₂	1	2	yes	maximal
8	#11718594	C₂	1	2	yes	maximal
9	C₄×#6101	C₂	1	2	yes	maximal
10	C₄×#6101	C₂	1	1	yes	maximal, characteristic
11	#11718590	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₃	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
19	#88198	C₄	1	8	yes	complemented
20	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×#6101	C₂²	1	2	yes	—
22	C₂×#6101	C₂²	1	1	yes	characteristic
23	C₂×#6101	C₂²	1	2	yes	—
24	C₂×#6101	C₂²	1	2	yes	—
25	C₂×#6101	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
30	#88141	—	2	4	no	—
31	#88071	C₂²	1	2	yes	—
32	#88071	C₂²	1	2	yes	—
33	#88071	C₂²	1	1	yes	characteristic
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	1	yes	characteristic
36	#88071	C₂²	1	2	yes	—
37	#88071	C₂²	1	2	yes	—
38	C₂×Q₈×C₁₃	—	2	2	no	—
39	C₂×Q₈×C₁₃	—	2	2	no	—
40	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂³	1	2	yes	—
47	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂×Dic₂₆	—	2	2	no	—
53	C₂×Dic₂₆	—	2	2	no	—
54	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
57	C₄²×C₁₃	—	2	4	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	—	2	2	no	—
60	#6101	—	2	4	no	—
61	#6101	—	2	2	no	—
62	#6101	C₂×C₄	1	8	yes	—
63	#6101	C₂×C₄	1	8	yes	—
64	#6101	Q₈	1	4	yes	complemented
65	C₄×Dic₁₃	—	2	4	no	—
66	C₄×Dic₁₃	—	2	2	no	—
67	C₄×Dic₁₃	—	2	4	no	—
68	C₄×Dic₁₃	C₂×C₄	1	4	yes	—
69	#2365	—	13	1	no	maximal, Sylow
70	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
71	Q₈×C₁₃	—	2	8	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
84	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
86	C₂×C₄×C₁₃	#41	1	2	yes	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	1	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	1	no	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
101	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
102	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
103	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
104	C₂×Dic₁₃	#41	1	2	yes	—
105	Dic₂₆	—	2	8	no	—
106	C₂×C₄⋊Q₈	—	13	1	no	—
107	C₂×C₄×Q₈	—	13	1	no	—
108	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic
109	#391	—	13	1	no	—
110	#391	—	13	1	no	—
111	#389	—	13	2	no	—
112	#387	—	13	2	no	—
113	#387	—	13	2	no	—
114	C₄×C₄⋊C₄	—	13	1	no	—
115	C₄×C₄⋊C₄	—	13	1	no	—
116	C₄×C₄⋊C₄	—	13	2	no	—
117	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₄×C₁₃	—	2	4	no	—
125	C₄×C₁₃	—	2	4	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	4	1	no	—
128	C₄×C₁₃	C₄×D₄	1	4	yes	—
129	Dic₁₃	—	2	4	no	—
130	Dic₁₃	—	2	2	no	—
131	Dic₁₃	—	2	2	no	—
132	Dic₁₃	—	4	1	no	—
133	Dic₁₃	—	4	2	no	—
134	Dic₁₃	C₄×Q₈	1	4	yes	complemented
135	C₂²×Q₈	—	13	1	no	—
136	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	13	8	no	—
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	26	4	no	—
140	C₂×C₄⋊C₄	—	13	2	no	—
141	C₂×C₄⋊C₄	—	13	2	no	—
142	C₂×C₄⋊C₄	—	13	1	no	—
143	C₂×C₄⋊C₄	—	13	2	no	—
144	C₂×C₄⋊C₄	—	13	1	no	—
145	C₂×C₄⋊C₄	—	13	1	no	—
146	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
147	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
148	C₂×C₄²	—	13	1	no	—
149	C₂×C₄²	—	13	2	no	—
150	C₂×C₄²	—	13	1	no	—
151	C₂×C₄²	—	13	2	no	—
152	C₂×C₄²	C₂×D₁₃	1	2	yes	—
153	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
154	#95	—	13	1	no	—
155	#95	—	13	2	no	—
156	#95	—	13	1	no	—
157	#95	—	13	2	no	—
158	#95	—	13	2	no	—
159	#95	—	13	2	no	—
160	#95	—	13	2	no	—
161	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
163	C₂×C₁₃	#519	1	1	yes	characteristic
164	C₂×C₁₃	#547	1	1	yes	characteristic
165	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
166	C₂×C₁₃	#556	1	1	yes	characteristic
167	C₂×C₁₃	#557	1	1	yes	characteristic
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	13	4	no	—
171	C₂×Q₈	—	26	2	no	—
172	C₂×Q₈	—	26	2	no	—
173	C₂×Q₈	C₄×D₁₃	1	4	yes	—
174	C₂²×C₄	—	13	1	no	—
175	C₂²×C₄	—	13	1	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	—	13	2	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
182	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	13	8	no	—
187	C₄⋊C₄	—	13	8	no	—
188	C₄⋊C₄	—	13	4	no	—
189	C₄⋊C₄	—	26	2	no	—
190	C₄⋊C₄	—	26	2	no	—
191	C₄⋊C₄	—	26	4	no	—
192	C₄⋊C₄	C₂²⋊D₁₃	1	4	yes	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	13	4	no	—
196	C₄²	—	26	4	no	—
197	C₄²	—	26	4	no	—
198	C₄²	—	26	2	no	—
199	C₁₃	#2365	1	1	yes	characteristic, complemented, Sylow
200	C₂³	C₂³×D₁₃	1	1	yes	characteristic
201	Q₈	—	2	8	no	—
202	Q₈	—	26	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	13	2	no	—
213	C₂×C₄	—	13	2	no	—
214	C₂×C₄	—	13	2	no	—
215	C₂×C₄	—	13	2	no	—
216	C₂×C₄	—	13	4	no	—
217	C₂×C₄	—	13	2	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	1	no	—
220	C₂×C₄	—	26	1	no	—
221	C₂×C₄	—	26	1	no	—
222	C₂×C₄	—	26	1	no	—
223	C₂×C₄	—	26	2	no	—
224	C₂×C₄	—	26	2	no	—
225	C₂×C₄	—	26	2	no	—
226	C₂×C₄	—	26	2	no	—
227	C₂×C₄	—	26	1	no	—
228	C₂×C₄	—	26	1	no	—
229	C₂×C₄	—	26	1	no	—
230	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
231	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
232	C₂×C₄	#6127	1	2	yes	—
233	C₂×C₄	Q₈×D₁₃	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
235	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
236	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
237	C₂²	C₂×#6127	1	1	yes	characteristic
238	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
239	C₂²	#88254	1	1	yes	characteristic
240	C₂²	#41×D₁₃	1	1	yes	characteristic
241	C₂²	#88257	1	1	yes	characteristic
242	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	4	no	—
246	C₄	—	4	1	no	—
247	C₄	—	13	4	no	—
248	C₄	—	26	4	no	—
249	C₄	—	26	2	no	—
250	C₄	—	26	2	no	—
251	C₄	—	52	2	no	—
252	C₄	—	52	1	no	—
253	C₄	C₄×C₂²⋊D₁₃	1	4	yes	—
254	C₂	#11719272	1	1	yes	characteristic
255	C₂	#11719273	1	1	yes	characteristic
256	C₂	C₄×Q₈×D₁₃	1	1	yes	characteristic
257	C₂	#11719277	1	1	yes	characteristic
258	C₂	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂	#11719529	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₃	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	#2365
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
19. #88198	C₄	semidirect
64. #6101	Q₈	semidirect
134. Dic₁₃	C₄×Q₈	semidirect
199. C₁₃	#2365	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
C₂×#88198	2
C₂×#88141	2
#11718891	2
#11718639	2
C₄⋊C₄×Dic₁₃	2
#11718633	2
#11718594	2
C₄×#6101	2
#11718590	2
#2365	13