Subgroups — #13118608

Order 1664.

460 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#11718891	C₂	1	1	yes	maximal, characteristic
4	#11718891	C₂	1	1	yes	maximal, characteristic
5	#11718891	C₂	1	1	yes	maximal, characteristic
6	#11718647	C₂	1	2	yes	maximal
7	#11718644	C₂	1	2	yes	maximal
8	#11718639	C₂	1	2	yes	maximal
9	#11718639	C₂	1	2	yes	maximal
10	C₄×#6101	C₂	1	1	yes	maximal, characteristic
11	#11718589	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	Q₈×Dic₁₃	—	2	4	no	—
19	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×#6101	C₂²	1	2	yes	—
23	C₂×#6101	C₂²	1	1	yes	characteristic
24	C₂×#6101	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
28	#88071	C₂²	1	2	yes	—
29	#88071	C₂²	1	1	yes	characteristic
30	#88071	C₂²	1	1	yes	characteristic
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	2	yes	—
36	#88071	C₂²	1	2	yes	—
37	C₂×Q₈×C₁₃	—	2	2	no	—
38	C₂×Q₈×C₁₃	—	2	2	no	—
39	C₂×Q₈×C₁₃	—	2	2	no	—
40	C₂×Q₈×C₁₃	—	2	2	no	—
41	C₂×Q₈×C₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₃	D₄	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	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	2	yes	—
51	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₄²×C₁₃	—	2	2	no	—
57	C₄⋊Dic₁₃	—	2	4	no	—
58	C₄⋊Dic₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	—	2	2	no	—
60	#6101	—	2	2	no	—
61	#6101	—	2	4	no	—
62	#6101	Q₈	1	4	yes	complemented
63	C₄×Dic₁₃	—	2	4	no	—
64	C₄×Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	—	2	2	no	—
66	#2446	—	13	1	no	maximal, Sylow
67	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
68	Q₈×C₁₃	—	2	8	no	—
69	Q₈×C₁₃	—	4	4	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	#41	1	2	yes	—
83	C₂×C₄×C₁₃	#41	1	2	yes	—
84	C₂×C₄×C₁₃	#41	1	2	yes	—
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	1	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	2	no	—
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	2	no	—
96	C₂×Dic₁₃	—	2	1	no	—
97	C₂×Dic₁₃	—	2	1	no	—
98	C₂×Dic₁₃	—	2	2	no	—
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
102	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
103	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
104	C₂×C₄×Q₈	—	13	1	no	—
105	#400	—	13	2	no	—
106	#398	—	13	2	no	—
107	#391	—	13	1	no	—
108	#391	—	13	1	no	—
109	#391	—	13	1	no	—
110	#389	—	13	1	no	—
111	#387	—	13	2	no	—
112	#387	—	13	2	no	—
113	C₄×C₄⋊C₄	—	13	1	no	—
114	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
120	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
121	C₄×C₁₃	—	2	4	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	#123	1	4	yes	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	2	2	no	—
128	Dic₁₃	—	4	1	no	—
129	Dic₁₃	—	4	2	no	—
130	Dic₁₃	—	4	1	no	—
131	Dic₁₃	—	4	2	no	—
132	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
133	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	4	no	—
135	C₄×Q₈	—	26	4	no	—
136	C₂×C₄⋊C₄	—	13	1	no	—
137	C₂×C₄⋊C₄	—	13	1	no	—
138	C₂×C₄⋊C₄	—	13	2	no	—
139	C₂×C₄⋊C₄	—	13	1	no	—
140	C₂×C₄⋊C₄	—	13	2	no	—
141	C₂×C₄⋊C₄	—	13	1	no	—
142	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
143	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
144	C₂×C₄²	—	13	1	no	—
145	C₂×C₄²	—	13	2	no	—
146	C₂×C₄²	—	13	1	no	—
147	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
148	#95	—	13	1	no	—
149	#95	—	13	1	no	—
150	#95	—	13	2	no	—
151	#95	—	13	2	no	—
152	#95	—	13	2	no	—
153	#95	—	13	2	no	—
154	#95	—	13	2	no	—
155	#95	—	13	2	no	—
156	#95	—	13	2	no	—
157	C₂×C₁₃	C₂×#123	1	1	yes	characteristic
158	C₂×C₁₃	#533	1	1	yes	characteristic
159	C₂×C₁₃	#541	1	1	yes	characteristic
160	C₂×C₁₃	#548	1	1	yes	characteristic
161	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
162	C₂×C₁₃	#556	1	1	yes	characteristic
163	C₂×C₁₃	#557	1	1	yes	characteristic
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	C₂²⋊D₁₃	1	4	yes	—
170	C₂²×C₄	—	13	1	no	—
171	C₂²×C₄	—	13	1	no	—
172	C₂²×C₄	—	13	1	no	—
173	C₂²×C₄	—	13	2	no	—
174	C₂²×C₄	—	13	1	no	—
175	C₂²×C₄	—	13	2	no	—
176	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
177	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
180	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	13	4	no	—
184	C₄⋊C₄	—	26	2	no	—
185	C₄⋊C₄	—	26	2	no	—
186	C₄⋊C₄	—	26	4	no	—
187	C₄⋊C₄	—	26	4	no	—
188	C₄⋊C₄	—	26	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	26	2	no	—
191	C₄²	—	26	4	no	—
192	C₄²	—	26	2	no	—
193	C₁₃	#2446	1	1	yes	characteristic, complemented, Sylow
194	C₂³	C₂³×D₁₃	1	1	yes	characteristic
195	Q₈	—	2	8	no	—
196	Q₈	—	4	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	13	2	no	—
208	C₂×C₄	—	26	1	no	—
209	C₂×C₄	—	26	1	no	—
210	C₂×C₄	—	26	1	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	—	26	1	no	—
216	C₂×C₄	—	26	2	no	—
217	C₂×C₄	—	26	1	no	—
218	C₂×C₄	—	26	1	no	—
219	C₂×C₄	—	26	2	no	—
220	C₂×C₄	—	26	1	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	1	no	—
224	C₂×C₄	#6129	1	2	yes	—
225	C₂×C₄	#6129	1	2	yes	—
226	C₂×C₄	Q₈×D₁₃	1	2	yes	—
227	C₂×C₄	#6131	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
229	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
230	C₂²	C₂×#6129	1	1	yes	characteristic
231	C₂²	C₂×#6129	1	1	yes	characteristic
232	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
233	C₂²	C₂×#6131	1	1	yes	characteristic
234	C₂²	#88254	1	1	yes	characteristic
235	C₂²	#88254	1	1	yes	characteristic
236	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	26	2	no	—
242	C₄	—	26	2	no	—
243	C₄	—	52	1	no	—
244	C₄	—	52	2	no	—
245	C₄	—	52	1	no	—
246	C₄	—	52	2	no	—
247	C₄	#88189	1	4	yes	—
248	C₂	#11719426	1	1	yes	characteristic
249	C₂	#11719433	1	1	yes	characteristic
250	C₂	#11719436	1	1	yes	characteristic
251	C₂	#11719437	1	1	yes	characteristic
252	C₂	C₂×#88189	1	1	yes	characteristic
253	C₂	Q₈×C₂²⋊D₁₃	1	1	yes	characteristic
254	C₂	#11719522	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2446
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
62. #6101	Q₈	semidirect
193. C₁₃	#2446	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×Q₈×Dic₁₃	2
#11718891	2
#11718647	2
#11718644	2
#11718639	2
C₄×#6101	2
#11718589	2
#2446	13