Subgroups — #15762028

Order 3328.

486 conjugacy classes of subgroups.

The subgroup lattice has 256 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	#13114689	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718139	C₂	1	2	yes	maximal
4	C₂×#11718139	C₂	1	2	yes	maximal
5	C₁₃×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₁₃×#391	—	2	1	no	—
9	C₁₃×#391	—	2	2	no	—
10	C₁₃×#391	—	2	1	no	—
11	C₁₃×#382	C₄	1	4	yes	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#1055	—	2	2	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₂×C₄⋊#1055	C₂²	1	2	yes	—
16	C₂×#88043	—	2	1	no	—
17	#11718139	—	2	4	no	—
18	#11718139	—	2	4	no	—
19	#11718116	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×Q₈×C₁₃	—	2	1	no	—
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	—	2	2	no	—
25	C₁₃×C₄⋊Q₈	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×#1055	—	2	2	no	—
36	C₂²×#1055	—	2	2	no	—
37	C₁₃×#95	—	2	4	no	—
38	C₁₃×#95	—	2	4	no	—
39	C₄⋊#1055	—	2	2	no	—
40	C₄⋊#1055	—	2	2	no	—
41	C₄⋊#1055	—	2	2	no	—
42	C₄⋊#1055	—	4	4	no	—
43	C₄⋊#1055	D₄	1	4	yes	—
44	#88043	—	4	2	no	—
45	#9708	—	13	1	no	maximal, Sylow
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	—	4	1	no	—
48	C₂×Q₈×C₁₃	—	4	1	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	1	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	—	2	4	no	—
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₄⋊C₄	—	4	2	no	—
64	C₁₃×C₄⋊C₄	—	4	1	no	—
65	C₁₃×C₄⋊C₄	—	4	1	no	—
66	C₄²×C₁₃	—	2	4	no	—
67	C₄²×C₁₃	—	2	2	no	—
68	C₄²×C₁₃	—	2	1	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂×D₄	1	2	yes	—
73	C₄²×C₁₃	#41	1	2	yes	—
74	C₂×#1055	—	2	4	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	—	4	2	no	—
78	C₂×#1055	—	4	4	no	—
79	#2862	D₁₃	1	1	yes	characteristic
80	#1842	—	13	1	no	—
81	#1767	—	13	1	no	—
82	C₂×#332	—	13	2	no	—
83	C₂×#332	—	13	2	no	—
84	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₁₃	—	8	2	no	—
86	Q₈×C₁₃	—	8	2	no	—
87	Q₈×C₁₃	—	8	2	no	—
88	Q₈×C₁₃	—	8	1	no	—
89	Q₈×C₁₃	—	8	1	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	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	#101	1	4	yes	—
102	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	#121	1	2	yes	—
104	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#135	1	2	yes	—
107	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#137	1	2	yes	—
111	#1055	—	4	4	no	—
112	#1055	—	8	2	no	—
113	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
114	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
115	C₂×C₄⋊C₈	—	13	2	no	—
116	C₂×C₄⋊C₈	—	13	2	no	—
117	C₂×C₄⋊C₈	—	26	2	no	—
118	C₂×#97	—	26	1	no	—
119	#391	—	2	1	no	—
120	#391	—	2	1	no	—
121	#391	—	2	2	no	—
122	#382	Dic₁₃	1	4	yes	—
123	C₄³	C₂×D₁₃	1	1	yes	characteristic
124	#336	—	26	1	no	—
125	#332	—	26	4	no	—
126	#332	—	26	4	no	—
127	C₂²×C₁₃	#390	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#101	1	2	yes	—
129	C₂²×C₁₃	#416	1	1	yes	characteristic
130	C₂²×C₁₃	#417	1	2	yes	—
131	C₂²×C₁₃	#417	1	1	yes	characteristic
132	C₄×C₁₃	—	2	4	no	—
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	4	2	no	—
135	C₄×C₁₃	—	8	2	no	—
136	C₄×C₁₃	—	8	2	no	—
137	C₄×C₁₃	#463	1	2	yes	—
138	C₄×C₁₃	#464	1	2	yes	—
139	C₄×C₁₃	#489	1	2	yes	—
140	C₄×C₁₃	#496	1	1	yes	characteristic
141	C₄×C₁₃	#497	1	1	yes	characteristic
142	C₂²×Q₈	—	2	1	no	—
143	C₂²×Q₈	—	2	1	no	—
144	C₂²×C₈	—	26	2	no	—
145	C₂²×C₈	—	26	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
154	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
158	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
159	C₄⋊C₈	—	13	4	no	—
160	C₄⋊C₈	—	26	2	no	—
161	C₄⋊C₈	—	26	2	no	—
162	C₄⋊C₈	—	26	2	no	—
163	C₄⋊C₈	—	52	4	no	—
164	#97	—	52	2	no	—
165	#95	—	2	4	no	—
166	#95	—	2	4	no	—
167	C₂×C₁₃	#1552	1	2	yes	—
168	C₂×C₁₃	#1561	1	2	yes	—
169	C₂×C₁₃	#1959	1	1	yes	characteristic
170	C₂×C₁₃	#1966	1	1	yes	characteristic
171	C₂×C₁₃	#1983	1	1	yes	characteristic
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	1	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
183	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
187	C₂×C₈	—	26	4	no	—
188	C₂×C₈	—	52	2	no	—
189	C₂×C₈	—	52	2	no	—
190	C₂×C₈	—	52	2	no	—
191	C₂×C₈	—	52	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	1	no	—
203	C₄²	D₄×D₁₃	1	2	yes	—
204	C₄²	#6129	1	2	yes	—
205	C₁₃	#9708	1	1	yes	characteristic, complemented, Sylow
206	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
207	Q₈	—	8	2	no	—
208	Q₈	—	8	2	no	—
209	Q₈	—	8	2	no	—
210	Q₈	—	8	1	no	—
211	Q₈	—	8	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	#88076	1	4	yes	—
224	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
225	C₂×C₄	#88190	1	1	yes	characteristic
226	C₂×C₄	#88192	1	2	yes	—
227	C₂×C₄	#88194	1	1	yes	characteristic
228	C₂×C₄	#88203	1	2	yes	—
229	C₂×C₄	#88203	1	1	yes	characteristic
230	C₂×C₄	#88204	1	2	yes	—
231	C₂×C₄	#88205	1	1	yes	characteristic
232	C₂×C₄	#88205	1	2	yes	—
233	C₈	—	52	4	no	—
234	C₈	—	104	2	no	—
235	C₂²	#11718882	1	1	yes	characteristic
236	C₂²	C₂×#88076	1	2	yes	—
237	C₂²	#11718893	1	2	yes	—
238	C₂²	#11718893	1	1	yes	characteristic
239	C₂²	#11718894	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	#11718715	1	2	yes	—
246	C₄	#11718716	1	2	yes	—
247	C₄	#11718718	1	2	yes	—
248	C₄	#11718719	1	1	yes	characteristic
249	C₄	#11718721	1	1	yes	characteristic
250	C₂	#13113536	1	2	yes	—
251	C₂	#13113574	1	2	yes	—
252	C₂	#13115653	1	1	yes	characteristic
253	C₂	#13115681	1	1	yes	characteristic
254	C₂	#13115769	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2862	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2862	D₁₃
hypocenter	C₁₃	#9708
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
205. C₁₃	#9708	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
#13114689	2
C₂×#11718139	2
C₁₃×#2862	2
#9708	13