Subgroups — #15756741

Order 3328.

322 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115008	C₂	1	1	yes	maximal, characteristic
2	#13114974	C₂	1	1	yes	maximal, characteristic
3	#13113595	C₂	1	1	yes	maximal, characteristic
4	#13113595	C₂	1	1	yes	maximal, characteristic
5	#13113592	C₂	1	1	yes	maximal, characteristic
6	#13113592	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88059	—	2	2	no	—
12	#11718585	C₂²	1	1	yes	characteristic
13	#11718585	C₂²	1	1	yes	characteristic
14	#11718585	C₂²	1	1	yes	characteristic
15	#11718585	C₂²	1	1	yes	characteristic
16	C₁₃×#322	C₄	1	2	yes	—
17	C₁₃×#322	C₄	1	2	yes	—
18	#11718144	—	2	2	no	—
19	#11718050	—	2	2	no	—
20	#11718050	—	2	2	no	—
21	#11718049	—	2	2	no	—
22	#11718049	—	2	2	no	—
23	C₂²×C₈×C₁₃	D₄	1	2	yes	—
24	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×#6099	—	2	1	no	—
28	C₂×#6099	—	2	1	no	—
29	C₂×#6099	—	2	1	no	—
30	C₂×#6099	—	2	1	no	—
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₃	D₄	1	2	yes	—
36	C₄×C₈×C₁₃	D₄	1	2	yes	—
37	C₄×C₈×C₁₃	Q₈	1	2	yes	—
38	#88059	—	4	4	no	—
39	C₄⋊#1055	—	2	1	no	—
40	C₄⋊#1055	—	2	1	no	—
41	C₄⋊#1055	—	2	1	no	—
42	C₄⋊#1055	—	2	1	no	—
43	C₄⋊#1055	—	2	1	no	—
44	C₄⋊#1055	—	2	1	no	—
45	C₄⋊#1055	—	2	1	no	—
46	C₄⋊#1055	—	2	1	no	—
47	#8828	—	13	1	no	maximal, Sylow
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	4	no	—
54	C₂×C₈×C₁₃	—	4	2	no	—
55	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
57	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
58	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	#6099	—	4	2	no	—
64	#6099	—	4	2	no	—
65	#6099	—	4	2	no	—
66	#6099	—	4	2	no	—
67	C₂×#1055	—	4	1	no	—
68	C₂×#1055	—	4	1	no	—
69	C₂×#1055	—	4	1	no	—
70	C₂×#1055	—	4	1	no	—
71	#1848	—	13	1	no	—
72	#1837	—	13	1	no	—
73	#1567	—	13	1	no	—
74	#1567	—	13	1	no	—
75	#1566	—	13	1	no	—
76	#1566	—	13	1	no	—
77	C₂×#322	D₁₃	1	1	yes	characteristic
78	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄×C₁₃	#137	1	2	yes	—
89	C₈×C₁₃	—	2	4	no	—
90	C₈×C₁₃	—	2	2	no	—
91	C₈×C₁₃	—	2	4	no	—
92	C₈×C₁₃	#108	1	4	yes	complemented
93	#1055	—	8	1	no	—
94	#1055	—	8	1	no	—
95	#1055	—	8	1	no	—
96	#1055	—	8	1	no	—
97	C₂×#108	—	26	2	no	—
98	#423	—	13	1	no	—
99	#423	—	13	1	no	—
100	#423	—	13	1	no	—
101	#423	—	13	1	no	—
102	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	#341	—	26	2	no	—
106	#335	—	26	2	no	—
107	#335	—	26	2	no	—
108	#334	—	26	2	no	—
109	#334	—	26	2	no	—
110	#322	Dic₁₃	1	2	yes	—
111	#322	Dic₁₃	1	2	yes	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	#384	1	1	yes	characteristic
115	C₂²×C₁₃	#501	1	2	yes	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	#428	1	2	yes	—
119	C₄×C₁₃	C₂×#108	1	2	yes	—
120	C₄×C₁₃	#430	1	2	yes	—
121	C₄×C₁₃	#496	1	1	yes	characteristic
122	C₄×C₁₃	#497	1	1	yes	characteristic
123	C₂×M₁₆	—	26	1	no	—
124	C₂×M₁₆	—	26	1	no	—
125	C₂×M₁₆	—	26	1	no	—
126	C₂×M₁₆	—	26	1	no	—
127	C₂²×C₈	Dic₂₆	1	2	yes	—
128	C₂²×C₈	Dic₂₆	1	2	yes	—
129	C₂²×C₈	D₅₂	1	2	yes	—
130	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
131	#108	—	52	4	no	—
132	C₄⋊C₈	—	26	1	no	—
133	C₄⋊C₈	—	26	1	no	—
134	C₄⋊C₈	—	26	1	no	—
135	C₄⋊C₈	—	26	1	no	—
136	C₄⋊C₈	—	26	1	no	—
137	C₄⋊C₈	—	26	1	no	—
138	C₄⋊C₈	—	26	1	no	—
139	C₄⋊C₈	—	26	1	no	—
140	C₄×C₈	Dic₂₆	1	2	yes	—
141	C₄×C₈	D₅₂	1	2	yes	—
142	C₄×C₈	D₅₂	1	2	yes	—
143	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
144	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
145	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
146	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
147	C₂×C₁₃	—	2	2	no	—
148	C₂×C₁₃	#1847	1	1	yes	characteristic
149	C₂×C₁₃	#1848	1	1	yes	characteristic
150	C₂×C₁₃	#1849	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
154	M₁₆	—	52	2	no	—
155	M₁₆	—	52	2	no	—
156	M₁₆	—	52	2	no	—
157	M₁₆	—	52	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	52	1	no	—
163	C₂×C₈	—	52	1	no	—
164	C₂×C₈	—	52	1	no	—
165	C₂×C₈	—	52	1	no	—
166	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
167	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
168	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
169	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
170	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
171	C₄²	C₂×D₅₂	1	1	yes	characteristic
172	C₄²	C₂×D₅₂	1	1	yes	characteristic
173	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
174	C₁₃	#8828	1	1	yes	characteristic, complemented, Sylow
175	C₂³	#88123	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	#88123	1	1	yes	characteristic
179	C₂×C₄	#88123	1	1	yes	characteristic
180	C₂×C₄	#88128	1	1	yes	characteristic
181	C₂×C₄	#88162	1	2	yes	—
182	C₂×C₄	#88163	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	2	4	no	—
188	C₈	—	2	4	no	—
189	C₈	—	104	1	no	—
190	C₈	—	104	1	no	—
191	C₈	—	104	1	no	—
192	C₈	—	104	1	no	—
193	C₈	#88059	1	4	yes	complemented
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	#11718363	1	2	yes	—
197	C₂²	#11718596	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	#11718373	1	1	yes	characteristic
201	C₄	#11718376	1	1	yes	characteristic
202	C₄	C₂×#88059	1	2	yes	—
203	C₄	#11718782	1	2	yes	—
204	C₄	#11718786	1	2	yes	—
205	C₂	—	2	2	no	—
206	C₂	#13115004	1	1	yes	characteristic
207	C₂	#13115008	1	1	yes	characteristic
208	C₂	#13115011	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂²×C₁₃	#384
prime radical	C₂³×C₁₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#8828
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
92. C₈×C₁₃	#108	semidirect
174. C₁₃	#8828	semidirect
193. C₈	#88059	semidirect

Maximal subgroups

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

subgroup	index
#13115008	2
#13114974	2
#13113595	2
#13113592	2
C₂×C₁₃×#322	2
#8828	13