Subgroups — #16015512

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718741	C₂	1	4	yes	maximal
2	#13123199	C₂	1	1	yes	maximal, characteristic
3	#13115002	C₂	1	4	yes	maximal
4	#13114995	C₂	1	2	yes	maximal
5	#13114992	C₂	1	2	yes	maximal
6	C₄×#88043	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×#88124	C₂²	1	2	yes	—
11	C₁₃×#384	—	2	2	no	—
12	C₁₃×#384	C₂²	1	2	yes	—
13	C₁₃×#384	C₂²	1	2	yes	—
14	C₄³×C₁₃	C₂²	1	1	yes	characteristic
15	#11718741	—	2	16	no	—
16	C₂×#88048	C₂²	1	8	yes	—
17	C₂×#88047	C₂²	1	8	yes	—
18	#11718597	C₂²	1	2	yes	—
19	#11718596	C₂²	1	1	yes	characteristic
20	#11718596	C₂²	1	1	yes	characteristic
21	#11718589	—	2	2	no	—
22	#11718589	—	2	2	no	—
23	C₂×#88043	C₂²	1	2	yes	—
24	C₂×#88043	C₂²	1	2	yes	—
25	C₂×C₄×#1055	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₃	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	—	2	8	no	—
28	C₁₃×C₄⋊Q₈	—	4	8	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
32	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
42	C₂×#6101	—	2	2	no	—
43	C₂×#6101	—	2	2	no	—
44	C₂²×#1055	C₂³	1	4	yes	—
45	#88124	—	2	8	no	—
46	#88071	—	2	4	no	—
47	#88048	—	2	16	no	—
48	#88047	—	2	16	no	—
49	#88043	D₄	1	8	yes	—
50	#88043	Q₈	1	8	yes	complemented
51	C₄×#1055	Q₈	1	8	yes	complemented
52	#33316	—	13	1	no	maximal, Sylow
53	C₂×Q₈×C₁₃	—	4	8	no	—
54	C₂×Q₈×C₁₃	—	4	4	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₁₃	—	2	2	no	—
62	C₂²×Dic₁₃	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	8	no	—
64	C₁₃×C₄⋊C₄	—	4	8	no	—
65	C₁₃×C₄⋊C₄	—	4	4	no	—
66	C₁₃×C₄⋊C₄	—	4	8	no	—
67	C₄²×C₁₃	—	2	4	no	—
68	C₄²×C₁₃	—	2	4	no	—
69	C₄²×C₁₃	C₂×D₄	1	4	yes	—
70	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
71	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
72	C₄⋊Dic₁₃	—	2	4	no	—
73	C₄⋊Dic₁₃	—	2	4	no	—
74	C₄⋊Dic₁₃	—	4	4	no	—
75	C₄⋊Dic₁₃	—	4	4	no	—
76	#6101	—	4	4	no	—
77	#6101	—	4	4	no	—
78	C₂×#1055	C₂×D₄	1	8	yes	—
79	C₂×#1055	C₂×Q₈	1	8	yes	—
80	C₂×#1055	C₂×Q₈	1	8	yes	—
81	C₂×#501	—	13	4	no	—
82	#2868	D₁₃	1	1	yes	characteristic
83	#2859	—	13	1	no	—
84	#1847	—	13	4	no	—
85	#1845	—	13	2	no	—
86	#1844	—	13	2	no	—
87	C₄×#97	—	13	1	no	—
88	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₁₃	—	8	8	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	4	4	no	—
95	C₂×C₄×C₁₃	—	4	8	no	—
96	C₂×C₄×C₁₃	#127	1	2	yes	—
97	C₂×C₄×C₁₃	#127	1	2	yes	—
98	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
99	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
100	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
101	C₂×C₄×C₁₃	#137	1	4	yes	—
102	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
105	C₂×Dic₁₃	—	4	2	no	—
106	C₂×Dic₁₃	—	4	2	no	—
107	C₂×Dic₁₃	—	4	4	no	—
108	C₂×Dic₁₃	—	4	4	no	—
109	#1055	C₄⋊Q₈	1	16	yes	complemented
110	C₂×C₄⋊Q₈	—	2	2	no	—
111	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
112	C₂×#125	—	13	2	no	—
113	#501	—	26	16	no	—
114	C₂×#107	—	13	8	no	—
115	C₂×#106	—	13	8	no	—
116	C₂×#97	—	13	2	no	—
117	C₂×#97	—	13	2	no	—
118	C₂×C₄×C₈	—	13	2	no	—
119	#389	—	26	2	no	—
120	#389	—	26	2	no	—
121	#384	—	2	2	no	—
122	#384	—	13	1	no	—
123	#384	—	13	1	no	—
124	#384	C₂×D₁₃	1	2	yes	—
125	#384	C₂×D₁₃	1	2	yes	—
126	#382	—	13	2	no	—
127	C₄³	C₂×D₁₃	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
132	C₂²×C₁₃	C₂×#137	1	2	yes	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	4	no	—
136	C₄×C₁₃	—	8	4	no	—
137	C₄×C₁₃	#501	1	8	yes	complemented
138	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
139	Dic₁₃	—	8	2	no	—
140	Dic₁₃	—	8	2	no	—
141	C₂²×Q₈	—	2	2	no	—
142	C₂²×C₈	—	13	4	no	—
143	C₄⋊Q₈	—	2	8	no	—
144	C₄⋊Q₈	—	4	8	no	—
145	#125	—	26	8	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	13	2	no	—
150	C₂×C₄⋊C₄	—	13	2	no	—
151	C₂×C₄⋊C₄	—	26	2	no	—
152	C₂×C₄⋊C₄	—	26	2	no	—
153	C₂×C₄⋊C₄	—	26	2	no	—
154	C₂×C₄⋊C₄	—	26	2	no	—
155	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
156	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
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	#107	—	26	16	no	—
162	#106	—	26	16	no	—
163	#97	—	13	8	no	—
164	#97	—	13	8	no	—
165	C₄×C₈	—	13	8	no	—
166	#95	—	26	4	no	—
167	C₂×C₁₃	#2868	1	1	yes	characteristic
168	C₂×C₁₃	#3151	1	1	yes	characteristic
169	C₂×C₁₃	#3152	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#501	1	2	yes	—
171	C₂×C₁₃	#3165	1	2	yes	—
172	C₂×Q₈	—	4	8	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	26	2	no	—
176	C₂²×C₄	—	26	2	no	—
177	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
181	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
182	C₂×C₈	—	13	8	no	—
183	C₂×C₈	—	13	8	no	—
184	C₂×C₈	—	13	8	no	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	4	8	no	—
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	26	4	no	—
190	C₄⋊C₄	—	26	4	no	—
191	C₄⋊C₄	—	52	4	no	—
192	C₄⋊C₄	—	52	4	no	—
193	C₄⋊C₄	—	52	4	no	—
194	C₄⋊C₄	—	52	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	D₄×D₁₃	1	4	yes	—
198	C₄²	Q₈×D₁₃	1	4	yes	—
199	C₄²	Q₈×D₁₃	1	4	yes	—
200	C₁₃	#33316	1	1	yes	characteristic, complemented, Sylow
201	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
202	Q₈	—	8	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	52	4	no	—
210	C₂×C₄	—	52	2	no	—
211	C₂×C₄	—	52	2	no	—
212	C₂×C₄	—	52	4	no	—
213	C₂×C₄	#88186	1	4	yes	—
214	C₂×C₄	#88194	1	2	yes	—
215	C₂×C₄	#88194	1	2	yes	—
216	C₂×C₄	#88196	1	4	yes	—
217	C₂×C₄	#88198	1	4	yes	—
218	C₂×C₄	#88198	1	4	yes	—
219	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
220	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
221	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
222	C₈	—	13	16	no	—
223	C₂²	C₂×#88186	1	2	yes	—
224	C₂²	C₂×#88194	1	1	yes	characteristic
225	C₂²	C₂×#88196	1	2	yes	—
226	C₂²	C₂×#88198	1	1	yes	characteristic
227	C₂²	C₂×#88198	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	8	4	no	—
232	C₄	—	104	2	no	—
233	C₄	—	104	2	no	—
234	C₄	#11718741	1	8	yes	complemented
235	C₄	D₁₃×C₄⋊Q₈	1	4	yes	—
236	C₂	#13123232	1	1	yes	characteristic
237	C₂	#13125286	1	1	yes	characteristic
238	C₂	#13125293	1	1	yes	characteristic
239	C₂	C₂×#11718741	1	2	yes	—
240	C₂	#13125378	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2868	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	#2868	D₁₃
hypocenter	C₁₃	#33316
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
50. #88043	Q₈	semidirect
51. C₄×#1055	Q₈	semidirect
109. #1055	C₄⋊Q₈	semidirect
137. C₄×C₁₃	#501	semidirect
200. C₁₃	#33316	semidirect
234. C₄	#11718741	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718741	2
#13123199	2
#13115002	2
#13114995	2
#13114992	2
C₄×#88043	2
C₁₃×#2868	2
#33316	13