Subgroups — #15763075

Order 3328.

506 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	#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	2	no	—
23	C₁₃×C₄⋊Q₈	—	4	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	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	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	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×#1055	—	2	2	no	—
38	C₂²×#1055	—	2	2	no	—
39	C₄⋊#1055	—	2	2	no	—
40	C₄⋊#1055	—	2	4	no	—
41	C₄⋊#1055	Q₈	1	4	yes	complemented
42	#88043	—	4	2	no	—
43	#9905	—	13	1	no	maximal, Sylow
44	C₂×Q₈×C₁₃	—	4	4	no	—
45	C₂×Q₈×C₁₃	—	4	4	no	—
46	C₂×Q₈×C₁₃	—	4	4	no	—
47	C₂²×C₄×C₁₃	—	2	4	no	—
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	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
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	2	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₁₃×C₄⋊C₄	—	4	4	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	—	2	2	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	D₈	1	2	yes	—
68	C₄²×C₁₃	SD₁₆	1	2	yes	—
69	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
70	C₄²×C₁₃	#41	1	2	yes	—
71	C₂×#1055	—	2	4	no	—
72	C₂×#1055	—	4	2	no	—
73	C₂×#1055	—	4	2	no	—
74	C₂×#1055	—	4	2	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	4	2	no	—
77	#2868	D₁₃	1	1	yes	characteristic
78	#1831	—	13	1	no	—
79	#1767	—	13	1	no	—
80	#1296	—	13	4	no	—
81	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	Q₈×C₁₃	—	8	8	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	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	—	4	4	no	—
95	C₂×C₄×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	#100	1	2	yes	—
97	C₂×C₄×C₁₃	#101	1	2	yes	—
98	C₂×C₄×C₁₃	#102	1	2	yes	—
99	C₂×C₄×C₁₃	#102	1	2	yes	—
100	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₁₃	#137	1	2	yes	—
108	C₂×C₄×C₁₃	#137	1	2	yes	—
109	#1055	—	4	4	no	—
110	#1055	—	8	2	no	—
111	C₂×C₄⋊Q₈	—	2	1	no	—
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×C₄⋊Q₈	Dic₁₃	1	2	yes	—
114	C₂×C₄⋊C₈	—	13	2	no	—
115	C₂×C₄⋊C₈	—	13	2	no	—
116	C₂×#97	—	26	1	no	—
117	#384	—	2	1	no	—
118	#384	—	2	1	no	—
119	#384	Dic₁₃	1	2	yes	—
120	#384	C₂×D₁₃	1	2	yes	—
121	C₄³	C₂×D₁₃	1	1	yes	characteristic
122	#336	—	26	1	no	—
123	#336	—	26	1	no	—
124	#336	—	26	1	no	—
125	C₂²×C₁₃	#391	1	1	yes	characteristic
126	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
129	C₂²×C₁₃	#418	1	1	yes	characteristic
130	C₂²×C₁₃	#419	1	1	yes	characteristic
131	C₂²×C₁₃	#419	1	1	yes	characteristic
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	8	4	no	—
138	C₄×C₁₃	#474	1	2	yes	—
139	C₄×C₁₃	#475	1	2	yes	—
140	C₄×C₁₃	#489	1	2	yes	—
141	C₄×C₁₃	#501	1	2	yes	—
142	C₂²×Q₈	—	2	2	no	—
143	C₂²×C₈	—	26	2	no	—
144	C₂²×C₈	—	26	2	no	—
145	C₄⋊Q₈	—	4	4	no	—
146	C₄⋊Q₈	—	4	4	no	—
147	C₄⋊Q₈	—	4	2	no	—
148	C₄⋊Q₈	—	4	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
155	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
156	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
162	C₄⋊C₈	—	13	4	no	—
163	C₄⋊C₈	—	26	2	no	—
164	C₄⋊C₈	—	26	4	no	—
165	#97	—	52	2	no	—
166	C₂×C₁₃	#1545	1	1	yes	characteristic
167	C₂×C₁₃	#1546	1	1	yes	characteristic
168	C₂×C₁₃	#1556	1	1	yes	characteristic
169	C₂×C₁₃	#1558	1	1	yes	characteristic
170	C₂×C₁₃	#1988	1	1	yes	characteristic
171	C₂×C₁₃	#1993	1	1	yes	characteristic
172	C₂×C₁₃	#1997	1	1	yes	characteristic
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
184	C₂×C₈	—	26	4	no	—
185	C₂×C₈	—	52	2	no	—
186	C₂×C₈	—	52	2	no	—
187	C₂×C₈	—	52	2	no	—
188	C₂×C₈	—	52	2	no	—
189	C₂×C₈	—	52	2	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	#6104	1	2	yes	—
203	C₄²	#6105	1	2	yes	—
204	C₄²	Q₈×D₁₃	1	2	yes	—
205	C₄²	#6131	1	2	yes	—
206	C₁₃	#9905	1	1	yes	characteristic, complemented, Sylow
207	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
208	Q₈	—	8	8	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	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	#88072	1	2	yes	—
223	C₂×C₄	#88072	1	2	yes	—
224	C₂×C₄	#88073	1	2	yes	—
225	C₂×C₄	#88076	1	2	yes	—
226	C₂×C₄	C₂×#6104	1	1	yes	characteristic
227	C₂×C₄	#88186	1	2	yes	—
228	C₂×C₄	C₂×#6105	1	1	yes	characteristic
229	C₂×C₄	#88196	1	2	yes	—
230	C₂×C₄	#88196	1	2	yes	—
231	C₂×C₄	#88198	1	1	yes	characteristic
232	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
233	C₂×C₄	#88200	1	2	yes	—
234	C₂×C₄	#88201	1	1	yes	characteristic
235	C₈	—	52	4	no	—
236	C₈	—	104	2	no	—
237	C₂²	C₂×#88072	1	1	yes	characteristic
238	C₂²	#11718876	1	1	yes	characteristic
239	C₂²	C₂×#88073	1	1	yes	characteristic
240	C₂²	#11718886	1	1	yes	characteristic
241	C₂²	#11718886	1	1	yes	characteristic
242	C₂²	C₂×#88076	1	1	yes	characteristic
243	C₂²	#11718891	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	8	4	no	—
250	C₄	#11718741	1	2	yes	—
251	C₄	#11718748	1	2	yes	—
252	C₄	#11718750	1	2	yes	—
253	C₄	#11718755	1	2	yes	—
254	C₂	#13113504	1	1	yes	characteristic
255	C₂	#13113508	1	1	yes	characteristic
256	C₂	#13113553	1	1	yes	characteristic
257	C₂	#13113562	1	1	yes	characteristic
258	C₂	#13115789	1	1	yes	characteristic
259	C₂	#13115810	1	1	yes	characteristic
260	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	#9905
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
41. C₄⋊#1055	Q₈	semidirect
206. C₁₃	#9905	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
#9905	13