Subgroups — #15763067

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114953	C₂	1	1	yes	maximal, characteristic
2	#13114689	C₂	1	1	yes	maximal, characteristic
3	#13112719	C₂	1	4	yes	maximal
4	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
6	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₄	1	2	yes	—
8	C₁₃×#384	—	2	1	no	—
9	C₁₃×#384	—	2	1	no	—
10	C₁₃×#384	C₄	1	2	yes	—
11	C₁₃×#384	C₂²	1	2	yes	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₂×#88043	—	2	1	no	—
16	#11718116	—	2	1	no	—
17	#11718116	—	2	1	no	—
18	#11718116	—	2	1	no	—
19	C₂²×Q₈×C₁₃	—	2	2	no	—
20	C₁₃×C₄⋊Q₈	—	4	4	no	—
21	C₁₃×C₄⋊Q₈	—	4	4	no	—
22	C₁₃×C₄⋊Q₈	—	4	4	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×#1055	—	2	2	no	—
37	C₂²×#1055	—	2	2	no	—
38	C₄⋊#1055	—	2	2	no	—
39	C₄⋊#1055	—	2	4	no	—
40	C₄⋊#1055	Q₈	1	4	yes	complemented
41	#88043	—	4	2	no	—
42	#9903	—	13	1	no	maximal, Sylow
43	C₂×Q₈×C₁₃	—	4	4	no	—
44	C₂×Q₈×C₁₃	—	4	4	no	—
45	C₂×Q₈×C₁₃	—	4	4	no	—
46	C₂²×C₄×C₁₃	—	2	4	no	—
47	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₁₃×C₄⋊C₄	—	4	4	no	—
55	C₁₃×C₄⋊C₄	—	4	4	no	—
56	C₁₃×C₄⋊C₄	—	4	4	no	—
57	C₁₃×C₄⋊C₄	—	4	4	no	—
58	C₁₃×C₄⋊C₄	—	4	4	no	—
59	C₁₃×C₄⋊C₄	—	4	4	no	—
60	C₄²×C₁₃	—	2	2	no	—
61	C₄²×C₁₃	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	SD₁₆	1	2	yes	—
66	C₄²×C₁₃	Q₁₆	1	2	yes	—
67	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
68	C₄²×C₁₃	#41	1	2	yes	—
69	C₂×#1055	—	2	4	no	—
70	C₂×#1055	—	4	2	no	—
71	C₂×#1055	—	4	2	no	—
72	C₂×#1055	—	4	2	no	—
73	C₂×#1055	—	4	2	no	—
74	C₂×#1055	—	4	2	no	—
75	#2868	D₁₃	1	1	yes	characteristic
76	#1831	—	13	1	no	—
77	#1767	—	13	1	no	—
78	#1296	—	13	4	no	—
79	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	Q₈×C₁₃	—	8	8	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	4	4	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	#100	1	2	yes	—
95	C₂×C₄×C₁₃	#101	1	2	yes	—
96	C₂×C₄×C₁₃	#103	1	2	yes	—
97	C₂×C₄×C₁₃	#103	1	2	yes	—
98	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₁₃	#137	1	2	yes	—
107	#1055	—	4	4	no	—
108	#1055	—	8	2	no	—
109	C₂×C₄⋊Q₈	—	2	1	no	—
110	C₂×C₄⋊Q₈	—	2	1	no	—
111	C₂×C₄⋊Q₈	Dic₁₃	1	2	yes	—
112	C₂×C₄⋊C₈	—	13	2	no	—
113	C₂×C₄⋊C₈	—	13	2	no	—
114	C₂×#97	—	26	1	no	—
115	#384	—	2	1	no	—
116	#384	—	2	1	no	—
117	#384	Dic₁₃	1	2	yes	—
118	#384	C₂×D₁₃	1	2	yes	—
119	C₄³	C₂×D₁₃	1	1	yes	characteristic
120	#336	—	26	1	no	—
121	#336	—	26	1	no	—
122	#336	—	26	1	no	—
123	C₂²×C₁₃	#391	1	1	yes	characteristic
124	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
125	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
127	C₂²×C₁₃	#418	1	1	yes	characteristic
128	C₂²×C₁₃	#418	1	1	yes	characteristic
129	C₂²×C₁₃	#419	1	1	yes	characteristic
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	8	4	no	—
136	C₄×C₁₃	#476	1	2	yes	—
137	C₄×C₁₃	#477	1	2	yes	—
138	C₄×C₁₃	#489	1	2	yes	—
139	C₄×C₁₃	#501	1	2	yes	—
140	C₂²×Q₈	—	2	2	no	—
141	C₂²×C₈	—	26	2	no	—
142	C₂²×C₈	—	26	2	no	—
143	C₄⋊Q₈	—	4	4	no	—
144	C₄⋊Q₈	—	4	4	no	—
145	C₄⋊Q₈	—	4	4	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
152	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
155	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
159	C₄⋊C₈	—	13	4	no	—
160	C₄⋊C₈	—	26	2	no	—
161	C₄⋊C₈	—	26	4	no	—
162	#97	—	52	2	no	—
163	C₂×C₁₃	#1548	1	1	yes	characteristic
164	C₂×C₁₃	#1549	1	1	yes	characteristic
165	C₂×C₁₃	#1555	1	1	yes	characteristic
166	C₂×C₁₃	#1556	1	1	yes	characteristic
167	C₂×C₁₃	#1987	1	1	yes	characteristic
168	C₂×C₁₃	#1993	1	1	yes	characteristic
169	C₂×C₁₃	#1997	1	1	yes	characteristic
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
181	C₂×C₈	—	26	4	no	—
182	C₂×C₈	—	52	2	no	—
183	C₂×C₈	—	52	2	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	—	52	2	no	—
186	C₂×C₈	—	52	2	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	#6106	1	2	yes	—
199	C₄²	#6107	1	2	yes	—
200	C₄²	Q₈×D₁₃	1	2	yes	—
201	C₄²	#6131	1	2	yes	—
202	C₁₃	#9903	1	1	yes	characteristic, complemented, Sylow
203	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
204	Q₈	—	8	8	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	#88073	1	2	yes	—
219	C₂×C₄	#88075	1	2	yes	—
220	C₂×C₄	#88075	1	2	yes	—
221	C₂×C₄	#88076	1	2	yes	—
222	C₂×C₄	#88186	1	2	yes	—
223	C₂×C₄	#88186	1	2	yes	—
224	C₂×C₄	C₂×#6106	1	1	yes	characteristic
225	C₂×C₄	#88196	1	2	yes	—
226	C₂×C₄	C₂×#6107	1	1	yes	characteristic
227	C₂×C₄	#88198	1	1	yes	characteristic
228	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
229	C₂×C₄	#88200	1	2	yes	—
230	C₂×C₄	#88201	1	1	yes	characteristic
231	C₈	—	52	4	no	—
232	C₈	—	104	2	no	—
233	C₂²	#11718876	1	1	yes	characteristic
234	C₂²	#11718876	1	1	yes	characteristic
235	C₂²	C₂×#88073	1	1	yes	characteristic
236	C₂²	C₂×#88075	1	1	yes	characteristic
237	C₂²	#11718886	1	1	yes	characteristic
238	C₂²	C₂×#88076	1	1	yes	characteristic
239	C₂²	#11718891	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	8	4	no	—
246	C₄	#11718741	1	2	yes	—
247	C₄	#11718745	1	2	yes	—
248	C₄	#11718750	1	2	yes	—
249	C₄	#11718752	1	2	yes	—
250	C₂	#13113517	1	1	yes	characteristic
251	C₂	#13113521	1	1	yes	characteristic
252	C₂	#13113550	1	1	yes	characteristic
253	C₂	#13113553	1	1	yes	characteristic
254	C₂	#13115785	1	1	yes	characteristic
255	C₂	#13115810	1	1	yes	characteristic
256	C₂	#13115826	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2868	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2868	D₁₃
hypocenter	C₁₃	#9903
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
40. C₄⋊#1055	Q₈	semidirect
202. C₁₃	#9903	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
#13114689	2
#13112719	2
C₁₃×#2868	2
#9903	13