Subgroups — #15762727

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115798	C₂	1	2	yes	maximal
2	#13115639	C₂	1	2	yes	maximal
3	#13114964	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₁₃×#500	—	2	2	no	—
8	C₁₃×#500	C₄	1	4	yes	complemented
9	C₂×C₁₃×#107	—	2	2	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88075	—	2	4	no	—
12	#11718596	C₂²	1	2	yes	—
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	#11718143	—	2	2	no	—
16	#11718121	—	2	4	no	—
17	#11718046	—	2	2	no	—
18	C₂²×Q₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	2	yes	—
20	C₁₃×C₄⋊Q₈	—	2	2	no	—
21	C₁₃×C₄⋊Q₈	—	2	2	no	—
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	C₂×C₄⋊Dic₁₃	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	C₂²×#1055	D₄	1	2	yes	—
31	C₂²×#1055	D₄	1	2	yes	—
32	C₁₃×#107	—	2	8	no	—
33	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₃	D₄	1	2	yes	—
35	#88075	—	4	8	no	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	C₄×#1055	—	2	1	no	—
39	C₄×#1055	—	2	1	no	—
40	#9831	—	13	1	no	maximal, Sylow
41	C₂×Q₈×C₁₃	—	2	4	no	—
42	C₂×Q₈×C₁₃	—	4	2	no	—
43	C₂×Q₈×C₁₃	—	4	4	no	—
44	C₂×Q₈×C₁₃	—	4	4	no	—
45	C₂²×C₄×C₁₃	—	4	2	no	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	—	4	2	no	—
50	C₂×C₈×C₁₃	—	2	4	no	—
51	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
52	C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₃	#41	1	1	yes	characteristic
59	C₄²×C₁₃	#41	1	1	yes	characteristic
60	C₄⋊Dic₁₃	—	4	4	no	—
61	C₄⋊Dic₁₃	—	4	4	no	—
62	C₄⋊Dic₁₃	—	4	4	no	—
63	C₂×#1055	—	2	2	no	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	4	no	—
67	C₂×#1055	D₈	1	4	yes	—
68	C₂×#500	D₁₃	1	1	yes	characteristic
69	#1990	—	13	2	no	—
70	#1956	—	13	2	no	—
71	#1834	—	13	1	no	—
72	C₂×#322	—	13	1	no	—
73	C₂³×C₁₃	#127	1	1	yes	characteristic
74	Q₈×C₁₃	—	4	8	no	—
75	Q₈×C₁₃	—	8	4	no	—
76	C₂×C₄×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	—	8	2	no	—
78	C₂×C₄×C₁₃	—	8	2	no	—
79	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₁₃	#135	1	2	yes	—
87	C₂×C₄×C₁₃	#137	1	4	yes	—
88	C₂×C₄×C₁₃	#137	1	4	yes	—
89	C₂×Dic₁₃	—	8	2	no	—
90	C₂×Dic₁₃	—	8	2	no	—
91	C₂×Dic₁₃	—	8	2	no	—
92	C₈×C₁₃	—	2	4	no	—
93	#1055	—	2	4	no	—
94	#1055	—	4	2	no	—
95	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
96	#500	—	2	2	no	—
97	#500	Dic₁₃	1	4	yes	complemented
98	C₂×#107	—	2	2	no	—
99	C₂×#103	—	26	4	no	—
100	C₂×C₄×C₈	—	13	1	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
103	#384	—	13	2	no	—
104	#340	—	26	4	no	—
105	#340	—	26	2	no	—
106	#322	—	26	2	no	—
107	C₂²×C₁₃	#390	1	1	yes	characteristic
108	C₂²×C₁₃	#493	1	1	yes	characteristic
109	C₂²×C₁₃	#495	1	1	yes	characteristic
110	C₂²×C₁₃	#497	1	2	yes	—
111	C₂²×C₁₃	#497	1	2	yes	—
112	C₄×C₁₃	—	8	4	no	—
113	C₄×C₁₃	#419	1	2	yes	—
114	C₄×C₁₃	C₄×D₈	1	2	yes	—
115	C₄×C₁₃	#441	1	2	yes	—
116	C₄×C₁₃	#466	1	2	yes	—
117	C₄×C₁₃	#470	1	2	yes	—
118	C₄×C₁₃	#491	1	2	yes	—
119	Dic₁₃	—	16	2	no	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂²×C₈	—	13	2	no	—
122	C₂²×C₈	—	13	2	no	—
123	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	C₂×Dic₁₃	1	4	yes	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	26	2	no	—
131	C₂×C₄⋊C₄	—	26	2	no	—
132	C₂×C₄⋊C₄	—	26	2	no	—
133	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
134	#107	—	2	8	no	—
135	#103	—	52	8	no	—
136	C₄×C₈	—	26	1	no	—
137	C₄×C₈	—	26	1	no	—
138	C₄×C₈	—	26	1	no	—
139	C₄×C₈	—	26	1	no	—
140	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
142	C₂×C₁₃	#1718	1	2	yes	—
143	C₂×C₁₃	#1722	1	2	yes	—
144	C₂×C₁₃	#1960	1	1	yes	characteristic
145	C₂×C₁₃	#1970	1	1	yes	characteristic
146	C₂×C₁₃	#1971	1	1	yes	characteristic
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	52	2	no	—
153	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
154	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	13	4	no	—
158	C₂×C₈	—	26	2	no	—
159	C₂×C₈	—	26	2	no	—
160	C₂×C₈	—	26	4	no	—
161	C₂×C₈	—	26	2	no	—
162	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	52	4	no	—
168	C₄⋊C₄	—	52	4	no	—
169	C₄⋊C₄	—	52	4	no	—
170	C₄²	#6129	1	1	yes	characteristic
171	C₄²	#6129	1	1	yes	characteristic
172	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
173	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
174	C₁₃	#9831	1	1	yes	characteristic, complemented, Sylow
175	C₂³	#88194	1	1	yes	characteristic
176	Q₈	—	4	8	no	—
177	Q₈	—	8	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	104	2	no	—
182	C₂×C₄	—	104	2	no	—
183	C₂×C₄	—	104	2	no	—
184	C₂×C₄	D₈×D₁₃	1	2	yes	—
185	C₂×C₄	#88166	1	2	yes	—
186	C₂×C₄	#88172	1	4	yes	—
187	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
188	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄	#88190	1	1	yes	characteristic
190	C₂×C₄	#88192	1	1	yes	characteristic
191	C₂×C₄	#88192	1	1	yes	characteristic
192	C₂×C₄	#88196	1	4	yes	—
193	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
194	C₈	—	2	4	no	—
195	C₈	—	26	4	no	—
196	C₈	—	52	2	no	—
197	C₂²	#11718747	1	2	yes	—
198	C₂²	#11718812	1	1	yes	characteristic
199	C₂²	#11718816	1	1	yes	characteristic
200	C₂²	#11718851	1	2	yes	—
201	C₂²	#11718882	1	1	yes	characteristic
202	C₄	—	8	4	no	—
203	C₄	—	208	2	no	—
204	C₄	#11718743	1	2	yes	—
205	C₄	D₈×Dic₁₃	1	2	yes	—
206	C₄	#11718818	1	2	yes	—
207	C₄	#11718845	1	2	yes	—
208	C₄	#11718850	1	2	yes	—
209	C₄	#11718886	1	2	yes	—
210	C₂	#13114481	1	2	yes	—
211	C₂	#13114506	1	2	yes	—
212	C₂	#13115657	1	1	yes	characteristic
213	C₂	#13115703	1	1	yes	characteristic
214	C₂	#13115708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#9831
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
8. C₁₃×#500	C₄	semidirect
97. #500	Dic₁₃	semidirect
174. C₁₃	#9831	semidirect

Maximal subgroups

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

subgroup	index
#13115798	2
#13115639	2
#13114964	2
C₂×#11718046	2
C₂×C₁₃×#500	2
#9831	13