Subgroups — #15761718

Order 3328.

546 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114988	C₂	1	1	yes	maximal, characteristic
2	#13114582	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#1055	C₂	1	4	yes	maximal
4	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₁₃×#391	C₄	1	2	yes	—
8	C₁₃×#389	—	2	2	no	—
9	C₁₃×#384	C₄	1	1	yes	characteristic
10	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
11	C₄³×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	—	2	2	no	—
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄×#1055	C₂²	1	2	yes	—
15	Q₈⋊#1055	—	2	8	no	—
16	#11718116	—	2	2	no	—
17	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₁₃×C₄⋊Q₈	—	2	4	no	—
19	C₄×Q₈×C₁₃	—	2	4	no	—
20	C₄×Q₈×C₁₃	—	2	4	no	—
21	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃	D₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×#1055	—	2	2	no	—
32	C₂²×#1055	—	2	2	no	—
33	C₁₃×#95	—	2	4	no	—
34	C₄⋊#1055	—	2	4	no	—
35	C₄⋊#1055	—	4	4	no	—
36	C₄×#1055	—	2	2	no	—
37	C₄×#1055	D₄	1	4	yes	—
38	#9643	—	13	1	no	maximal, Sylow
39	C₂×Q₈×C₁₃	—	2	2	no	—
40	C₂×Q₈×C₁₃	—	2	4	no	—
41	C₂×Q₈×C₁₃	—	2	4	no	—
42	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₁₃×C₄⋊C₄	—	2	8	no	—
51	C₁₃×C₄⋊C₄	—	2	4	no	—
52	C₁₃×C₄⋊C₄	—	4	4	no	—
53	C₁₃×C₄⋊C₄	—	4	4	no	—
54	C₄²×C₁₃	—	2	2	no	—
55	C₄²×C₁₃	—	2	2	no	—
56	C₄²×C₁₃	—	4	2	no	—
57	C₄²×C₁₃	—	4	2	no	—
58	C₄²×C₁₃	—	4	2	no	—
59	C₄²×C₁₃	SD₁₆	1	4	yes	—
60	C₄²×C₁₃	Q₁₆	1	4	yes	—
61	C₄²×C₁₃	C₂×D₄	1	2	yes	—
62	C₄²×C₁₃	#41	1	2	yes	—
63	C₂×#1055	—	2	4	no	—
64	C₂×#1055	—	4	4	no	—
65	C₂×#1055	—	4	2	no	—
66	C₂×#1055	—	4	4	no	—
67	C₄×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
68	#1842	—	13	1	no	—
69	#1744	—	13	1	no	—
70	C₂×Q₈⋊C₈	—	13	4	no	—
71	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	Q₈×C₁₃	—	2	8	no	—
73	Q₈×C₁₃	—	4	4	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	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	—	4	2	no	—
81	C₂×C₄×C₁₃	—	4	2	no	—
82	C₂×C₄×C₁₃	—	4	2	no	—
83	C₂×C₄×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	#103	1	4	yes	—
85	C₂×C₄×C₁₃	#103	1	4	yes	—
86	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₁₃	#121	1	2	yes	—
88	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₁₃	#131	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
94	#1055	—	4	4	no	—
95	#1055	—	8	2	no	—
96	C₂×C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
97	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
98	C₂×C₄⋊C₈	—	13	2	no	—
99	C₂×C₄⋊C₈	—	26	2	no	—
100	C₂×C₄×C₈	—	13	2	no	—
101	#391	Dic₁₃	1	2	yes	—
102	#389	—	2	2	no	—
103	#384	Dic₁₃	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	2	2	no	—
105	C₄³	C₂×D₁₃	1	1	yes	characteristic
106	#336	—	26	2	no	—
107	Q₈⋊C₈	—	26	8	no	—
108	C₂²×C₁₃	#390	1	1	yes	characteristic
109	C₂²×C₁₃	#407	1	1	yes	characteristic
110	C₂²×C₁₃	#408	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×#103	1	2	yes	—
112	C₂²×C₁₃	#421	1	2	yes	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	4	no	—
115	C₄×C₁₃	—	2	4	no	—
116	C₄×C₁₃	—	4	4	no	—
117	C₄×C₁₃	—	8	2	no	—
118	C₄×C₁₃	#435	1	4	yes	—
119	C₄×C₁₃	#487	1	2	yes	—
120	C₄×C₁₃	#492	1	1	yes	characteristic
121	C₄×C₁₃	#494	1	1	yes	characteristic
122	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
123	C₂²×C₈	—	26	2	no	—
124	C₂²×C₈	—	26	2	no	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄×Q₈	—	2	4	no	—
127	C₄×Q₈	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
136	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
137	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
138	C₄⋊C₈	—	26	4	no	—
139	C₄⋊C₈	—	52	4	no	—
140	C₄×C₈	—	13	4	no	—
141	C₄×C₈	—	26	2	no	—
142	#95	—	2	4	no	—
143	C₂×C₁₃	#1481	1	2	yes	—
144	C₂×C₁₃	#1487	1	2	yes	—
145	C₂×C₁₃	#1955	1	1	yes	characteristic
146	C₂×C₁₃	#1956	1	1	yes	characteristic
147	C₂×C₁₃	#1963	1	1	yes	characteristic
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	4	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	#6099	1	4	yes	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
155	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
156	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂×C₈	—	26	4	no	—
160	C₂×C₈	—	52	4	no	—
161	C₂×C₈	—	52	2	no	—
162	C₂×C₈	—	52	4	no	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	#6106	1	4	yes	—
173	C₄²	#6107	1	4	yes	—
174	C₄²	D₄×D₁₃	1	2	yes	—
175	C₄²	#6129	1	2	yes	—
176	C₁₃	#9643	1	1	yes	characteristic, complemented, Sylow
177	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
178	Q₈	—	2	8	no	—
179	Q₈	—	4	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	#88075	1	4	yes	—
191	C₂×C₄	#88075	1	4	yes	—
192	C₂×C₄	C₂×#6099	1	2	yes	—
193	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
194	C₂×C₄	#88190	1	1	yes	characteristic
195	C₂×C₄	#88192	1	2	yes	—
196	C₂×C₄	#88194	1	1	yes	characteristic
197	C₂×C₄	C₂×#6106	1	2	yes	—
198	C₂×C₄	C₂×#6107	1	2	yes	—
199	C₂×C₄	#88202	1	2	yes	—
200	C₈	—	52	4	no	—
201	C₈	—	104	2	no	—
202	C₂²	#11718868	1	1	yes	characteristic
203	C₂²	#11718882	1	1	yes	characteristic
204	C₂²	C₂×#88075	1	2	yes	—
205	C₂²	#11718895	1	1	yes	characteristic
206	C₂²	#11718897	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	4	no	—
210	C₄	—	4	4	no	—
211	C₄	—	8	2	no	—
212	C₄	#11718672	1	4	yes	—
213	C₄	#11718742	1	2	yes	—
214	C₄	#11718746	1	1	yes	characteristic
215	C₄	#11718753	1	1	yes	characteristic
216	C₂	#13113239	1	2	yes	—
217	C₂	#13113265	1	2	yes	—
218	C₂	#13115635	1	1	yes	characteristic
219	C₂	#13115639	1	1	yes	characteristic
220	C₂	#13115669	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#9643
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
176. C₁₃	#9643	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
#13114582	2
C₂×Q₈⋊#1055	2
C₄×C₁₃×C₄⋊Q₈	2
#9643	13