Subgroups — #15761602

Order 3328.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115595	C₂	1	2	yes	maximal
2	#13114996	C₂	1	1	yes	maximal, characteristic
3	#13114703	C₂	1	1	yes	maximal, characteristic
4	#13112720	C₂	1	2	yes	maximal
5	C₂×C₁₃×#334	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	—	2	1	no	—
9	C₂×#88058	—	2	1	no	—
10	C₂×#88058	—	2	2	no	—
11	C₂×#88055	—	2	1	no	—
12	#11718596	C₂²	1	2	yes	—
13	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
15	C₁₃×#334	C₄	1	4	yes	complemented
16	#11718143	—	2	2	no	—
17	#11718142	—	2	1	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	—	2	1	no	—
28	C₂²×#1055	—	2	1	no	—
29	C₁₃×C₄⋊C₈	—	2	2	no	—
30	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃	Q₈	1	2	yes	—
33	#88071	—	2	1	no	—
34	#88058	—	4	4	no	—
35	#88058	—	4	2	no	—
36	#88058	—	4	2	no	—
37	#88055	—	4	2	no	—
38	C₄⋊#1055	—	2	1	no	—
39	C₄⋊#1055	Q₈	1	2	yes	—
40	#9620	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	1	no	—
45	C₂²×Dic₁₃	—	2	1	no	—
46	C₂²×Dic₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	—	4	2	no	—
51	C₂×C₈×C₁₃	—	4	2	no	—
52	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₃	D₈	1	2	yes	—
55	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₃	#41	1	1	yes	characteristic
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄⋊Dic₁₃	—	2	2	no	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	C₄⋊Dic₁₃	—	8	2	no	—
67	C₄×Dic₁₃	—	4	1	no	—
68	C₄×Dic₁₃	—	4	1	no	—
69	C₂×#1055	—	2	2	no	—
70	C₂×#1055	—	4	1	no	—
71	C₂×#1055	—	4	1	no	—
72	#1947	—	13	2	no	—
73	#1845	—	13	1	no	—
74	#1770	—	13	1	no	—
75	C₂×#334	D₁₃	1	1	yes	characteristic
76	#1296	—	13	2	no	—
77	C₂³×C₁₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#137	1	2	yes	—
92	C₂×Dic₁₃	—	4	1	no	—
93	C₂×Dic₁₃	—	4	1	no	—
94	C₂×Dic₁₃	—	4	1	no	—
95	C₂×Dic₁₃	—	4	1	no	—
96	C₂×Dic₁₃	—	4	1	no	—
97	C₂×Dic₁₃	—	4	1	no	—
98	C₂×Dic₁₃	—	8	2	no	—
99	C₂×Dic₁₃	—	8	2	no	—
100	C₂×Dic₁₃	—	8	2	no	—
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	—	4	4	no	—
103	C₈×C₁₃	#107	1	4	yes	complemented
104	#1055	—	4	2	no	—
105	C₂×#107	—	26	2	no	—
106	C₂×#107	—	26	1	no	—
107	C₂×#107	—	26	1	no	—
108	C₂×C₄⋊C₈	—	13	1	no	—
109	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
110	C₂×#97	—	26	1	no	—
111	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
112	#384	—	13	2	no	—
113	C₄×C₄⋊C₄	—	13	1	no	—
114	#340	—	26	2	no	—
115	#336	—	26	1	no	—
116	#334	Dic₁₃	1	4	yes	complemented
117	C₂²×C₁₃	#389	1	1	yes	characteristic
118	C₂²×C₁₃	#459	1	1	yes	characteristic
119	C₂²×C₁₃	#462	1	1	yes	characteristic
120	C₂²×C₁₃	#469	1	1	yes	characteristic
121	C₂²×C₁₃	#470	1	1	yes	characteristic
122	C₂²×C₁₃	#472	1	1	yes	characteristic
123	C₂²×C₁₃	#473	1	1	yes	characteristic
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
127	C₄×C₁₃	#428	1	1	yes	characteristic
128	C₄×C₁₃	#440	1	2	yes	—
129	C₄×C₁₃	#474	1	1	yes	characteristic
130	C₄×C₁₃	#477	1	1	yes	characteristic
131	C₄×C₁₃	#501	1	2	yes	—
132	Dic₁₃	—	8	1	no	—
133	Dic₁₃	—	8	1	no	—
134	Dic₁₃	—	16	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	26	1	no	—
137	C₂²×C₈	D₅₂	1	1	yes	characteristic
138	C₂²×C₈	D₅₂	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	13	1	no	—
140	C₂×C₄⋊C₄	—	13	1	no	—
141	C₂×C₄⋊C₄	—	26	2	no	—
142	C₂×C₄⋊C₄	—	26	2	no	—
143	C₂×C₄⋊C₄	—	26	2	no	—
144	C₂×C₄²	—	26	1	no	—
145	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
146	#107	—	52	4	no	—
147	#107	—	52	2	no	—
148	#107	—	52	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	13	2	no	—
151	C₄⋊C₈	—	26	1	no	—
152	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
153	#97	—	52	2	no	—
154	C₄×C₈	Dic₂₆	1	2	yes	—
155	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
156	#95	—	26	1	no	—
157	C₂×C₁₃	#1687	1	1	yes	characteristic
158	C₂×C₁₃	#1689	1	1	yes	characteristic
159	C₂×C₁₃	#1694	1	1	yes	characteristic
160	C₂×C₁₃	#1696	1	1	yes	characteristic
161	C₂×C₁₃	#1944	1	1	yes	characteristic
162	C₂×C₁₃	#1947	1	1	yes	characteristic
163	C₂×C₁₃	#1952	1	1	yes	characteristic
164	C₂²×C₄	—	26	1	no	—
165	C₂²×C₄	—	26	1	no	—
166	C₂²×C₄	—	52	2	no	—
167	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
168	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	#6131	1	1	yes	characteristic
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	26	2	no	—
176	C₂×C₈	—	52	1	no	—
177	C₂×C₈	—	52	1	no	—
178	C₂×C₈	D₁₀₄	1	2	yes	—
179	C₂×C₈	Dic₅₂	1	2	yes	—
180	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
181	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
182	C₄⋊C₄	—	26	2	no	—
183	C₄⋊C₄	—	26	2	no	—
184	C₄⋊C₄	—	52	2	no	—
185	C₄⋊C₄	—	52	2	no	—
186	C₄⋊C₄	—	52	2	no	—
187	C₄⋊C₄	—	52	2	no	—
188	C₄⋊C₄	—	104	2	no	—
189	C₄²	—	52	1	no	—
190	C₄²	—	52	1	no	—
191	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
192	C₄²	#6129	1	1	yes	characteristic
193	C₄²	Q₈×D₁₃	1	1	yes	characteristic
194	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
195	C₁₃	#9620	1	1	yes	characteristic, complemented, Sylow
196	C₂³	#88149	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	52	1	no	—
200	C₂×C₄	—	52	1	no	—
201	C₂×C₄	—	52	1	no	—
202	C₂×C₄	—	52	1	no	—
203	C₂×C₄	—	52	1	no	—
204	C₂×C₄	—	52	1	no	—
205	C₂×C₄	—	104	2	no	—
206	C₂×C₄	—	104	2	no	—
207	C₂×C₄	—	104	2	no	—
208	C₂×C₄	#88142	1	1	yes	characteristic
209	C₂×C₄	#88144	1	1	yes	characteristic
210	C₂×C₄	#88151	1	1	yes	characteristic
211	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
212	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
213	C₂×C₄	#88162	1	1	yes	characteristic
214	C₂×C₄	#88163	1	1	yes	characteristic
215	C₂×C₄	#88168	1	2	yes	—
216	C₂×C₄	#88169	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
218	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
219	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	52	2	no	—
223	C₈	#88058	1	4	yes	complemented
224	C₂²	#11718504	1	1	yes	characteristic
225	C₂²	#11718510	1	1	yes	characteristic
226	C₂²	#11718526	1	1	yes	characteristic
227	C₂²	#11718529	1	1	yes	characteristic
228	C₂²	#11718557	1	1	yes	characteristic
229	C₂²	#11718558	1	1	yes	characteristic
230	C₂²	#11718646	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	104	1	no	—
234	C₄	—	104	1	no	—
235	C₄	—	208	2	no	—
236	C₄	#11718507	1	1	yes	characteristic
237	C₄	#11718511	1	1	yes	characteristic
238	C₄	#11718517	1	2	yes	—
239	C₄	C₂×#88058	1	1	yes	characteristic
240	C₄	#11718782	1	1	yes	characteristic
241	C₄	#11718825	1	2	yes	—
242	C₂	#13114316	1	1	yes	characteristic
243	C₂	#13114326	1	1	yes	characteristic
244	C₂	#13114355	1	1	yes	characteristic
245	C₂	#13114365	1	1	yes	characteristic
246	C₂	#13115580	1	1	yes	characteristic
247	C₂	#13115595	1	1	yes	characteristic
248	C₂	#13115622	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88149
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#334	C₂
socle, prime radical	C₂³×C₁₃	#121
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#334	D₁₃
hypocenter	C₁₃	#9620
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
15. C₁₃×#334	C₄	semidirect
103. C₈×C₁₃	#107	semidirect
116. #334	Dic₁₃	semidirect
195. C₁₃	#9620	semidirect
223. C₈	#88058	semidirect

Maximal subgroups

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

subgroup	index
#13115595	2
#13114996	2
#13114703	2
#13112720	2
C₂×C₁₃×#334	2
#9620	13