Subgroups — #15756669

Order 3328.

418 conjugacy classes of subgroups.

The subgroup lattice has 208 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×#88043	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718071	C₂	1	2	yes	maximal
4	C₂×#11718072	C₂	1	2	yes	maximal
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	—	2	2	no	—
10	C₂×C₄⋊#1055	C₂²	1	2	yes	—
11	C₂×#88043	C₄	1	1	yes	characteristic
12	C₂×#88043	C₄	1	2	yes	—
13	C₂×#88043	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	2	yes	—
15	C₁₃×#336	—	2	2	no	—
16	#11718072	—	2	4	no	—
17	#11718071	—	2	4	no	—
18	C₂²×C₈×C₁₃	—	2	1	no	—
19	C₂²×C₈×C₁₃	—	2	1	no	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	2	yes	—
23	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	2	yes	—
28	C₁₃×C₄⋊C₈	—	2	2	no	—
29	C₁₃×C₄⋊C₈	—	2	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	4	4	no	—
33	#88043	—	2	2	no	—
34	#88043	—	2	4	no	—
35	#88043	C₈	1	4	yes	complemented
36	C₄×#1055	D₄	1	4	yes	—
37	C₄×#1055	Q₈	1	4	yes	—
38	#8808	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₃	—	4	1	no	—
45	C₂×C₈×C₁₃	—	4	1	no	—
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₄²×C₁₃	—	2	2	no	—
49	C₄²×C₁₃	—	2	2	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	—	4	2	no	—
52	C₄²×C₁₃	C₂×C₈	1	2	yes	—
53	C₄²×C₁₃	M₁₆	1	2	yes	—
54	C₄²×C₁₃	C₂×D₄	1	2	yes	—
55	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	4	2	no	—
59	C₂×#1055	—	4	4	no	—
60	C₂×#1055	C₄⋊C₄	1	4	yes	—
61	C₂×#1055	C₄⋊C₄	1	4	yes	—
62	C₂×#1055	C₂×C₈	1	4	yes	—
63	C₂×#1055	M₁₆	1	4	yes	—
64	#1842	—	13	1	no	—
65	#1831	D₁₃	1	1	yes	characteristic
66	C₄×#97	—	13	1	no	—
67	C₂×#335	—	13	2	no	—
68	C₂×#334	—	13	2	no	—
69	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₄×C₁₃	#137	1	2	yes	—
86	C₈×C₁₃	—	8	1	no	—
87	C₈×C₁₃	—	8	1	no	—
88	#1055	—	2	4	no	—
89	#1055	—	8	2	no	—
90	#1055	C₄⋊C₈	1	8	yes	complemented
91	C₂×C₄⋊C₈	—	13	2	no	—
92	C₂×C₄⋊C₈	—	26	2	no	—
93	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
94	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
95	C₂×#97	—	13	1	no	—
96	C₂×#97	—	13	2	no	—
97	C₂×#97	—	13	1	no	—
98	C₂×C₄×C₈	—	13	2	no	—
99	C₄³	C₂×D₁₃	1	1	yes	characteristic
100	#336	—	2	2	no	—
101	#335	—	26	4	no	—
102	#334	—	26	4	no	—
103	C₂²×C₁₃	#384	1	1	yes	characteristic
104	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₂²×C₁₃	#423	1	1	yes	characteristic
106	C₂²×C₁₃	#428	1	2	yes	—
107	C₂²×C₁₃	#430	1	2	yes	—
108	C₄×C₁₃	—	2	4	no	—
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	—	4	2	no	—
111	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
112	C₄×C₁₃	#423	1	2	yes	—
113	C₄×C₁₃	#496	1	1	yes	characteristic
114	C₄×C₁₃	#497	1	1	yes	characteristic
115	C₄×C₁₃	#501	1	2	yes	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	13	2	no	—
119	C₂²×C₈	—	13	2	no	—
120	C₂²×C₈	—	26	2	no	—
121	C₂×C₄²	Dic₂₆	1	2	yes	—
122	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
123	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
124	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
125	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	26	2	no	—
129	C₄⋊C₈	—	26	2	no	—
130	C₄⋊C₈	—	52	4	no	—
131	#97	—	13	4	no	—
132	#97	—	26	4	no	—
133	#97	—	26	2	no	—
134	C₄×C₈	—	13	4	no	—
135	C₄×C₈	—	13	4	no	—
136	C₂×C₁₃	#1565	1	2	yes	—
137	C₂×C₁₃	#1569	1	2	yes	—
138	C₂×C₁₃	#1842	1	1	yes	characteristic
139	C₂×C₁₃	#1847	1	1	yes	characteristic
140	C₂×C₁₃	#1849	1	1	yes	characteristic
141	C₂²×C₄	#6101	1	2	yes	—
142	C₂²×C₄	#6101	1	2	yes	—
143	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	13	4	no	—
151	C₂×C₈	—	13	4	no	—
152	C₂×C₈	—	13	4	no	—
153	C₂×C₈	—	13	4	no	—
154	C₂×C₈	—	26	2	no	—
155	C₂×C₈	—	26	2	no	—
156	C₂×C₈	—	52	2	no	—
157	C₂×C₈	—	52	4	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	4	2	no	—
162	C₄²	C₈×D₁₃	1	2	yes	—
163	C₄²	#6094	1	2	yes	—
164	C₄²	D₄×D₁₃	1	2	yes	—
165	C₄²	Q₈×D₁₃	1	2	yes	—
166	C₁₃	#8808	1	1	yes	characteristic, complemented, Sylow
167	C₂³	C₂×#6101	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	#88054	1	4	yes	—
175	C₂×C₄	#88142	1	2	yes	—
176	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
177	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
178	C₂×C₄	C₂×#6094	1	1	yes	characteristic
179	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
180	C₂×C₄	#88194	1	1	yes	characteristic
181	C₂×C₄	#88198	1	1	yes	characteristic
182	C₂×C₄	#88203	1	2	yes	—
183	C₂×C₄	#88205	1	2	yes	—
184	C₈	—	8	1	no	—
185	C₈	—	8	1	no	—
186	C₈	—	13	8	no	—
187	C₈	—	26	4	no	—
188	C₈	—	104	2	no	—
189	C₂²	#11718637	1	1	yes	characteristic
190	C₂²	#11718656	1	2	yes	—
191	C₂²	C₂×#88054	1	1	yes	characteristic
192	C₂²	#11718779	1	1	yes	characteristic
193	C₂²	#11718785	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	4	2	no	—
197	C₄	D₁₃×C₄⋊C₈	1	2	yes	—
198	C₄	#11718498	1	2	yes	—
199	C₄	#11718719	1	1	yes	characteristic
200	C₄	#11718721	1	1	yes	characteristic
201	C₄	#11718726	1	2	yes	—
202	C₂	#13113589	1	2	yes	—
203	C₂	#13113604	1	2	yes	—
204	C₂	#13114987	1	1	yes	characteristic
205	C₂	#13115005	1	1	yes	characteristic
206	C₂	#13115010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#8808
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
35. #88043	C₈	semidirect
90. #1055	C₄⋊C₈	semidirect
166. C₁₃	#8808	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×#88043	2
C₂×#11718071	2
C₂×#11718072	2
C₁₃×#1831	2
#8808	13