Subgroups — #15761479

Order 3328.

378 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115552	C₂	1	2	yes	maximal
2	C₄×#88058	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
4	#13112675	C₂	1	2	yes	maximal
5	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	C₄	1	2	yes	—
9	C₂×#88054	—	2	2	no	—
10	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#335	—	2	2	no	—
14	#11718143	C₄	1	1	yes	characteristic
15	#11718143	C₄	1	1	yes	characteristic
16	#11718142	—	2	2	no	—
17	#11718046	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₁₃	—	2	2	no	—
25	C₂²×#1055	—	2	1	no	—
26	C₂²×#1055	D₄	1	1	yes	characteristic
27	C₂²×#1055	D₄	1	1	yes	characteristic
28	C₁₃×C₄⋊C₈	—	2	2	no	—
29	C₁₃×C₄⋊C₈	—	2	2	no	—
30	C₄×C₈×C₁₃	D₄	1	2	yes	—
31	C₄×C₈×C₁₃	Q₈	1	2	yes	—
32	#88071	—	2	2	no	—
33	#88058	—	2	4	no	—
34	#88054	—	4	4	no	—
35	C₄×#1055	—	2	1	no	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	C₄×#1055	Q₈	1	2	yes	—
39	#9598	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	2	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	—	4	2	no	—
49	C₂×C₈×C₁₃	—	4	2	no	—
50	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
52	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₁₃	#41	1	1	yes	characteristic
56	C₄⋊Dic₁₃	—	2	2	no	—
57	C₄⋊Dic₁₃	—	2	2	no	—
58	C₄⋊Dic₁₃	—	2	2	no	—
59	C₄⋊Dic₁₃	M₁₆	1	4	yes	—
60	C₄×Dic₁₃	—	4	2	no	—
61	C₄×Dic₁₃	—	4	2	no	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	2	1	no	—
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	—	2	2	no	—
67	C₂×#1055	—	4	2	no	—
68	C₂×#1055	D₈	1	2	yes	—
69	C₂×#1055	Q₁₆	1	2	yes	—
70	#1939	—	13	2	no	—
71	C₄×#107	—	13	1	no	—
72	C₂×#335	D₁₃	1	1	yes	characteristic
73	C₂×#322	—	13	1	no	—
74	#1279	—	13	2	no	—
75	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	#131	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
89	C₂×Dic₁₃	—	2	4	no	—
90	C₂×Dic₁₃	—	4	2	no	—
91	C₂×Dic₁₃	—	4	2	no	—
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	—	4	2	no	—
94	C₂×Dic₁₃	—	4	2	no	—
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	—	8	2	no	—
98	#1055	—	2	2	no	—
99	#1055	—	2	2	no	—
100	#1055	—	4	2	no	—
101	C₂×#107	—	13	2	no	—
102	C₂×C₄⋊C₈	—	26	2	no	—
103	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	13	2	no	—
108	#340	—	13	1	no	—
109	#340	—	13	1	no	—
110	#336	—	26	2	no	—
111	#335	—	2	2	no	—
112	#322	—	26	2	no	—
113	C₂²×C₁₃	#389	1	1	yes	characteristic
114	C₂²×C₁₃	#435	1	2	yes	—
115	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
116	C₂²×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
117	C₂²×C₁₃	#444	1	2	yes	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	#423	1	1	yes	characteristic
121	C₄×C₁₃	#424	1	1	yes	characteristic
122	C₄×C₁₃	#446	1	2	yes	—
123	C₄×C₁₃	#466	1	1	yes	characteristic
124	C₄×C₁₃	#467	1	1	yes	characteristic
125	C₄×C₁₃	#499	1	1	yes	characteristic
126	C₄×C₁₃	#500	1	1	yes	characteristic
127	Dic₁₃	—	4	4	no	—
128	Dic₁₃	—	8	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	1	no	—
132	C₂²×C₈	—	26	1	no	—
133	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
134	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	13	2	no	—
136	C₂×C₄⋊C₄	—	13	2	no	—
137	C₂×C₄²	—	26	2	no	—
138	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
139	#107	—	26	4	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	52	4	no	—
143	C₄×C₈	—	13	2	no	—
144	C₄×C₈	—	26	1	no	—
145	C₄×C₈	—	26	1	no	—
146	C₄×C₈	—	26	1	no	—
147	C₄×C₈	Dic₂₆	1	2	yes	—
148	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
149	#95	—	26	2	no	—
150	C₂×C₁₃	#1581	1	2	yes	—
151	C₂×C₁₃	#1584	1	2	yes	—
152	C₂×C₁₃	#1940	1	1	yes	characteristic
153	C₂×C₁₃	#1942	1	1	yes	characteristic
154	C₂×C₁₃	#1946	1	1	yes	characteristic
155	C₂²×C₄	—	26	2	no	—
156	C₂²×C₄	—	26	2	no	—
157	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
158	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
159	C₂²×C₄	#6131	1	1	yes	characteristic
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	26	1	no	—
168	C₂×C₈	—	26	1	no	—
169	C₂×C₈	—	26	1	no	—
170	C₂×C₈	—	26	1	no	—
171	C₂×C₈	—	26	2	no	—
172	C₂×C₈	—	52	2	no	—
173	C₂×C₈	#6101	1	2	yes	—
174	C₂×C₈	#6101	1	2	yes	—
175	C₄⋊C₄	—	13	4	no	—
176	C₄⋊C₄	—	26	2	no	—
177	C₄⋊C₄	—	26	2	no	—
178	C₄⋊C₄	—	26	2	no	—
179	C₄²	—	52	2	no	—
180	C₄²	—	52	2	no	—
181	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
182	C₄²	#6129	1	1	yes	characteristic
183	C₄²	Q₈×D₁₃	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
185	C₁₃	#9598	1	1	yes	characteristic, complemented, Sylow
186	C₂³	#88147	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	26	4	no	—
189	C₂×C₄	—	52	2	no	—
190	C₂×C₄	—	52	2	no	—
191	C₂×C₄	—	52	2	no	—
192	C₂×C₄	—	52	2	no	—
193	C₂×C₄	—	52	2	no	—
194	C₂×C₄	#88141	1	1	yes	characteristic
195	C₂×C₄	#88142	1	1	yes	characteristic
196	C₂×C₄	#88144	1	1	yes	characteristic
197	C₂×C₄	#88155	1	2	yes	—
198	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
199	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
200	C₂×C₄	#88166	1	1	yes	characteristic
201	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
202	C₂×C₄	#88173	1	1	yes	characteristic
203	C₂×C₄	C₂×#6101	1	1	yes	characteristic
204	C₂×C₄	#88192	1	1	yes	characteristic
205	C₂×C₄	#88200	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	8	2	no	—
209	C₈	—	26	2	no	—
210	C₈	—	26	2	no	—
211	C₈	—	52	2	no	—
212	C₂²	#11718497	1	2	yes	—
213	C₂²	#11718533	1	1	yes	characteristic
214	C₂²	#11718534	1	1	yes	characteristic
215	C₂²	#11718555	1	2	yes	—
216	C₂²	#11718634	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	52	4	no	—
220	C₄	—	104	2	no	—
221	C₄	#11718489	1	2	yes	—
222	C₄	#11718535	1	1	yes	characteristic
223	C₄	#11718539	1	1	yes	characteristic
224	C₄	#11718763	1	1	yes	characteristic
225	C₄	#11718779	1	1	yes	characteristic
226	C₄	#11718818	1	1	yes	characteristic
227	C₄	#11718849	1	1	yes	characteristic
228	C₂	#13113664	1	2	yes	—
229	C₂	#13113685	1	2	yes	—
230	C₂	#13115557	1	1	yes	characteristic
231	C₂	#13115567	1	1	yes	characteristic
232	C₂	#13115587	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	#9598
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
185. C₁₃	#9598	semidirect

Maximal subgroups

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

subgroup	index
#13115552	2
C₄×#88058	2
C₂×#11718046	2
#13112675	2
C₂×C₁₃×#335	2
#9598	13