Subgroups — #15761977

Order 3328.

498 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	#13114988	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718138	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	1	no	—
10	C₁₃×#391	C₄	1	4	yes	—
11	C₁₃×#382	—	2	2	no	—
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×C₄×#1055	—	2	1	no	—
17	#11718139	—	2	4	no	—
18	#11718138	—	2	4	no	—
19	#11718116	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
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	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
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	C₄×#1055	—	4	2	no	—
45	#9697	—	13	1	no	maximal, Sylow
46	C₂×Q₈×C₁₃	—	2	4	no	—
47	C₂×Q₈×C₁₃	—	4	4	no	—
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂×Q₈×C₁₃	—	4	1	no	—
51	C₂×Q₈×C₁₃	—	4	1	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₁₃×C₄⋊C₄	—	4	1	no	—
62	C₁₃×C₄⋊C₄	—	4	1	no	—
63	C₁₃×C₄⋊C₄	—	4	1	no	—
64	C₁₃×C₄⋊C₄	—	4	1	no	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	2	1	no	—
67	C₄²×C₁₃	—	2	1	no	—
68	C₄²×C₁₃	—	4	2	no	—
69	C₄²×C₁₃	SD₁₆	1	4	yes	—
70	C₄²×C₁₃	Q₁₆	1	4	yes	—
71	C₄²×C₁₃	C₂×D₄	1	2	yes	—
72	C₄²×C₁₃	#41	1	2	yes	—
73	C₂×#1055	—	2	4	no	—
74	C₂×#1055	—	4	2	no	—
75	C₂×#1055	—	4	2	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	—	4	4	no	—
78	#2862	D₁₃	1	1	yes	characteristic
79	#1842	—	13	1	no	—
80	C₄×C₄⋊C₈	—	13	1	no	—
81	C₂×#333	—	13	2	no	—
82	C₂×#332	—	13	2	no	—
83	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₁₃	—	8	4	no	—
85	Q₈×C₁₃	—	8	2	no	—
86	Q₈×C₁₃	—	8	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	—	4	4	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	#103	1	4	yes	—
98	C₂×C₄×C₁₃	#103	1	4	yes	—
99	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₃	#121	1	2	yes	—
101	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
108	#1055	—	4	4	no	—
109	#1055	—	8	2	no	—
110	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
112	C₂×C₄⋊C₈	—	13	2	no	—
113	C₂×C₄⋊C₈	—	13	2	no	—
114	C₂×C₄⋊C₈	—	26	2	no	—
115	C₂×C₄×C₈	—	26	1	no	—
116	#391	—	2	1	no	—
117	#391	—	2	1	no	—
118	#391	Dic₁₃	1	4	yes	—
119	#382	—	2	2	no	—
120	C₄³	C₂×D₁₃	1	1	yes	characteristic
121	#336	—	26	1	no	—
122	#333	—	26	4	no	—
123	#332	—	26	4	no	—
124	C₂²×C₁₃	#390	1	1	yes	characteristic
125	C₂²×C₁₃	#413	1	2	yes	—
126	C₂²×C₁₃	C₂×#103	1	2	yes	—
127	C₂²×C₁₃	#416	1	1	yes	characteristic
128	C₂²×C₁₃	#417	1	1	yes	characteristic
129	C₄×C₁₃	—	2	4	no	—
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	—	8	2	no	—
133	C₄×C₁₃	—	8	2	no	—
134	C₄×C₁₃	#463	1	2	yes	—
135	C₄×C₁₃	#464	1	2	yes	—
136	C₄×C₁₃	#487	1	2	yes	—
137	C₄×C₁₃	#492	1	1	yes	characteristic
138	C₄×C₁₃	#494	1	1	yes	characteristic
139	C₂²×Q₈	—	2	1	no	—
140	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
141	C₂²×C₈	—	26	2	no	—
142	C₂²×C₈	—	26	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
151	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
152	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
155	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
156	C₄⋊C₈	—	13	4	no	—
157	C₄⋊C₈	—	26	2	no	—
158	C₄⋊C₈	—	26	2	no	—
159	C₄⋊C₈	—	26	2	no	—
160	C₄⋊C₈	—	52	4	no	—
161	C₄×C₈	—	52	2	no	—
162	#95	—	2	4	no	—
163	#95	—	2	4	no	—
164	C₂×C₁₃	#1542	1	2	yes	—
165	C₂×C₁₃	#1550	1	2	yes	—
166	C₂×C₁₃	#1956	1	1	yes	characteristic
167	C₂×C₁₃	#1967	1	1	yes	characteristic
168	C₂×C₁₃	#1983	1	1	yes	characteristic
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
182	C₂×C₈	—	26	4	no	—
183	C₂×C₈	—	52	2	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	—	52	2	no	—
186	C₂×C₈	—	52	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	4	2	no	—
197	C₄²	#6106	1	4	yes	—
198	C₄²	#6107	1	4	yes	—
199	C₄²	D₄×D₁₃	1	2	yes	—
200	C₄²	#6129	1	2	yes	—
201	C₁₃	#9697	1	1	yes	characteristic, complemented, Sylow
202	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
203	Q₈	—	8	4	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	#88075	1	4	yes	—
217	C₂×C₄	#88075	1	4	yes	—
218	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
219	C₂×C₄	#88190	1	1	yes	characteristic
220	C₂×C₄	#88192	1	2	yes	—
221	C₂×C₄	#88194	1	1	yes	characteristic
222	C₂×C₄	C₂×#6106	1	2	yes	—
223	C₂×C₄	C₂×#6107	1	2	yes	—
224	C₂×C₄	#88203	1	1	yes	characteristic
225	C₂×C₄	#88204	1	2	yes	—
226	C₂×C₄	#88205	1	1	yes	characteristic
227	C₈	—	52	4	no	—
228	C₈	—	104	2	no	—
229	C₂²	#11718882	1	1	yes	characteristic
230	C₂²	C₂×#88075	1	2	yes	—
231	C₂²	#11718893	1	1	yes	characteristic
232	C₂²	#11718894	1	1	yes	characteristic
233	C₂²	#11718898	1	2	yes	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	2	no	—
239	C₄	#11718715	1	2	yes	—
240	C₄	#11718718	1	2	yes	—
241	C₄	#11718742	1	2	yes	—
242	C₄	#11718746	1	1	yes	characteristic
243	C₄	#11718753	1	1	yes	characteristic
244	C₂	#13113493	1	2	yes	—
245	C₂	#13113526	1	2	yes	—
246	C₂	#13115639	1	1	yes	characteristic
247	C₂	#13115686	1	1	yes	characteristic
248	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	#9697
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
201. C₁₃	#9697	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
C₄×C₄⋊#1055	2
C₂×#11718138	2
C₂×#11718139	2
C₁₃×#2862	2
#9697	13