Subgroups — #15781553

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88052	C₂	1	1	yes	maximal, characteristic
2	#13115008	C₂	1	1	yes	maximal, characteristic
3	#13112944	C₂	1	4	yes	maximal
4	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
6	C₁₃×#500	—	2	1	no	—
7	C₁₃×#500	—	2	1	no	—
8	C₁₃×#500	C₄	1	2	yes	—
9	C₁₃×#500	C₄	1	2	yes	—
10	C₂×C₁₃×#107	—	2	1	no	—
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88059	—	2	1	no	—
14	C₂×#88059	C₂²	1	2	yes	—
15	C₂×#88052	C₂²	1	2	yes	—
16	#11718585	—	2	1	no	—
17	#11718053	—	2	1	no	—
18	C₄×#6090	—	2	1	no	—
19	C₂²×Q₈×C₁₃	—	4	1	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	—	4	1	no	—
25	C₁₃×C₄⋊Q₈	—	4	1	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×#6099	—	2	2	no	—
30	C₁₃×#107	—	2	2	no	—
31	C₁₃×#107	—	2	2	no	—
32	C₁₃×#107	—	2	2	no	—
33	C₁₃×#107	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
38	#88059	—	2	4	no	—
39	#88059	—	4	2	no	—
40	#88052	—	2	2	no	—
41	#88052	Q₈	1	4	yes	complemented
42	C₂×#6090	—	2	2	no	—
43	C₄⋊#1055	—	4	1	no	—
44	C₄⋊#1055	—	4	1	no	—
45	#13290	—	13	1	no	maximal, Sylow
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	—	4	2	no	—
48	C₂×Q₈×C₁₃	—	8	4	no	—
49	C₂²×C₄×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₃	#41	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	2	4	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃	D₈	1	1	yes	characteristic
68	C₄²×C₁₃	D₈	1	1	yes	characteristic
69	#6099	—	4	2	no	—
70	#6099	—	4	2	no	—
71	C₂×#1055	—	4	2	no	—
72	#6090	—	2	4	no	—
73	C₂×#500	D₁₃	1	1	yes	characteristic
74	C₄×M₃₂	—	13	1	no	—
75	#1848	—	13	1	no	—
76	#1387	—	13	4	no	—
77	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
78	Q₈×C₁₃	—	8	4	no	—
79	Q₈×C₁₃	—	8	4	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	4	4	no	—
83	C₂×C₄×C₁₃	—	8	4	no	—
84	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₈×C₁₃	#103	1	2	yes	—
95	C₈×C₁₃	#103	1	2	yes	—
96	C₈×C₁₃	C₄×Q₈	1	2	yes	—
97	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
98	#1055	—	8	2	no	—
99	C₂×C₄⋊Q₈	—	2	1	no	—
100	C₂×M₃₂	—	13	2	no	—
101	#500	—	2	1	no	—
102	#500	—	2	1	no	—
103	#500	Dic₁₃	1	2	yes	—
104	#500	Dic₁₃	1	2	yes	—
105	C₂×#108	—	13	2	no	—
106	C₂×#108	—	26	1	no	—
107	C₂×#107	—	2	1	no	—
108	C₂×#107	C₂×D₁₃	1	2	yes	—
109	#423	—	26	1	no	—
110	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
111	#346	—	26	1	no	—
112	C₄×C₁₆	—	26	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
116	C₂²×C₁₃	#487	1	1	yes	characteristic
117	C₂²×C₁₃	#498	1	1	yes	characteristic
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	8	4	no	—
121	C₄×C₁₃	#361	1	2	yes	—
122	C₄×C₁₃	#362	1	2	yes	—
123	C₄×C₁₃	#391	1	1	yes	characteristic
124	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
125	C₄×C₁₃	#486	1	1	yes	characteristic
126	C₄×C₁₃	#500	1	1	yes	characteristic
127	C₂²×Q₈	—	4	1	no	—
128	C₂×M₁₆	—	26	2	no	—
129	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
138	M₃₂	—	13	4	no	—
139	M₃₂	—	26	2	no	—
140	C₂×C₁₆	—	26	2	no	—
141	#108	—	26	4	no	—
142	#108	—	52	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	—	2	2	no	—
146	#107	C₂×Dic₁₃	1	4	yes	—
147	C₄⋊C₈	—	52	1	no	—
148	C₄⋊C₈	—	52	1	no	—
149	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
150	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂×C₁₃	—	2	1	no	—
154	C₂×C₁₃	—	2	1	no	—
155	C₂×C₁₃	#1990	1	1	yes	characteristic
156	C₂×C₁₃	C₂×#361	1	1	yes	characteristic
157	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	8	4	no	—
161	C₂²×C₄	—	4	2	no	—
162	C₂²×C₄	#6105	1	1	yes	characteristic
163	C₂²×C₄	#6105	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
165	M₁₆	—	52	2	no	—
166	M₁₆	—	52	2	no	—
167	C₂×C₈	—	52	2	no	—
168	C₂×C₈	#6106	1	2	yes	—
169	C₂×C₈	#6107	1	2	yes	—
170	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
172	C₂×C₈	Q₈×D₁₃	1	2	yes	—
173	C₂×C₈	#6131	1	2	yes	—
174	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄²	#6104	1	1	yes	characteristic
181	C₄²	#6104	1	1	yes	characteristic
182	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
184	C₁₆	—	26	4	no	—
185	C₁₃	#13290	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₂×#6105	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	8	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#88072	1	1	yes	characteristic
194	C₂×C₄	#88072	1	1	yes	characteristic
195	C₂×C₄	#88072	1	1	yes	characteristic
196	C₂×C₄	#88072	1	1	yes	characteristic
197	C₂×C₄	C₂×#6104	1	1	yes	characteristic
198	C₂×C₄	C₂×#6106	1	1	yes	characteristic
199	C₂×C₄	C₂×#6107	1	1	yes	characteristic
200	C₂×C₄	#88198	1	1	yes	characteristic
201	C₂×C₄	#88201	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
203	C₈	—	104	2	no	—
204	C₈	#88075	1	2	yes	—
205	C₈	#88075	1	2	yes	—
206	C₈	Q₈×Dic₁₃	1	2	yes	—
207	C₈	#88200	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#11718740	1	1	yes	characteristic
211	C₂²	#11718754	1	1	yes	characteristic
212	C₂²	C₂×#88072	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	4	no	—
216	C₄	#11718155	1	2	yes	—
217	C₄	#11718157	1	2	yes	—
218	C₄	#11718739	1	1	yes	characteristic
219	C₄	#11718749	1	1	yes	characteristic
220	C₄	C₂×#88075	1	1	yes	characteristic
221	C₄	#11718891	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#13115798	1	1	yes	characteristic
225	C₂	C₂×#11718155	1	1	yes	characteristic
226	C₂	C₂×#11718157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#13290
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
41. #88052	Q₈	semidirect
185. C₁₃	#13290	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88052	2
#13115008	2
#13112944	2
C₂×C₁₃×#500	2
#13290	13