Subgroups — #15756580

Order 3328.

422 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	#13114988	C₂	1	2	yes	maximal
2	#13112671	C₂	1	4	yes	maximal
3	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	—	2	2	no	—
10	C₂×C₄⋊#1055	—	2	2	no	—
11	C₂×C₄⋊#1055	C₂²	1	4	yes	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂²×#1055	—	2	4	no	—
22	C₄×C₈×C₁₃	—	2	4	no	—
23	C₄×C₈×C₁₃	—	2	2	no	—
24	C₄×C₈×C₁₃	—	2	4	no	—
25	C₄×C₈×C₁₃	C₈	1	4	yes	complemented
26	C₄⋊#1055	—	2	8	no	—
27	C₄⋊#1055	—	4	4	no	—
28	C₄⋊#1055	—	4	4	no	—
29	#8784	—	13	1	no	maximal, Sylow
30	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	—	2	2	no	—
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
43	C₄²×C₁₃	—	2	2	no	—
44	C₄²×C₁₃	—	2	2	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	C₂×C₈	1	2	yes	—
47	C₄²×C₁₃	M₁₆	1	2	yes	—
48	C₄²×C₁₃	D₈	1	2	yes	—
49	C₄²×C₁₃	SD₁₆	1	4	yes	—
50	C₄²×C₁₃	Q₁₆	1	2	yes	—
51	C₄²×C₁₃	#41	1	4	yes	—
52	C₂×#1055	—	4	4	no	—
53	C₂×#1055	—	4	4	no	—
54	C₂×#1055	—	4	4	no	—
55	#1842	—	13	2	no	—
56	C₄²×C₈	D₁₃	1	1	yes	characteristic
57	#1277	—	13	4	no	—
58	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃	—	2	1	no	—
60	C₂×C₄×C₁₃	—	2	1	no	—
61	C₂×C₄×C₁₃	—	2	1	no	—
62	C₂×C₄×C₁₃	—	2	1	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
66	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
67	C₂×C₄×C₁₃	#106	1	2	yes	—
68	C₂×C₄×C₁₃	#106	1	2	yes	—
69	C₂×C₄×C₁₃	#107	1	2	yes	—
70	C₂×C₄×C₁₃	#107	1	2	yes	—
71	C₂×C₄×C₁₃	#108	1	2	yes	—
72	C₂×C₄×C₁₃	#108	1	2	yes	—
73	C₂×C₄×C₁₃	#117	1	2	yes	—
74	C₂×C₄×C₁₃	#124	1	2	yes	—
75	C₂×C₄×C₁₃	#125	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
82	C₈×C₁₃	—	2	4	no	—
83	C₈×C₁₃	—	2	4	no	—
84	#1055	—	8	4	no	—
85	C₂×C₄⋊C₈	—	13	4	no	—
86	C₂×C₄⋊C₈	—	26	2	no	—
87	C₂×C₄⋊C₈	—	26	2	no	—
88	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
89	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
90	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
91	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
92	C₄³	C₂×D₁₃	1	1	yes	characteristic
93	C₂²×C₁₃	#382	1	1	yes	characteristic
94	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂²×C₁₃	#423	1	1	yes	characteristic
96	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
97	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
98	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
99	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
100	C₄×C₁₃	—	2	2	no	—
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	#334	1	4	yes	—
104	C₄×C₁₃	#335	1	4	yes	—
105	C₄×C₁₃	#431	1	2	yes	—
106	C₄×C₁₃	#432	1	2	yes	—
107	C₄×C₁₃	#486	1	2	yes	—
108	C₄×C₁₃	#487	1	2	yes	—
109	C₂²×C₈	—	26	4	no	—
110	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
111	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
112	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
113	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
114	C₂×C₄²	D₅₂	1	1	yes	characteristic
115	C₂×C₄²	D₅₂	1	1	yes	characteristic
116	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
118	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
119	C₄⋊C₈	—	26	8	no	—
120	C₄⋊C₈	—	52	4	no	—
121	C₄⋊C₈	—	52	4	no	—
122	C₄×C₈	—	2	4	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	4	no	—
125	C₄×C₈	#1055	1	4	yes	complemented
126	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
127	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
128	C₂×C₁₃	#1566	1	1	yes	characteristic
129	C₂×C₁₃	#1567	1	1	yes	characteristic
130	C₂×C₁₃	#1831	1	1	yes	characteristic
131	C₂×C₁₃	#1834	1	1	yes	characteristic
132	C₂×C₁₃	#1837	1	1	yes	characteristic
133	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
134	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
135	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
136	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	52	4	no	—
145	C₂×C₈	—	52	4	no	—
146	C₂×C₈	—	52	4	no	—
147	C₂×C₈	C₂×#1055	1	4	yes	—
148	C₂×C₈	#6099	1	4	yes	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	#6095	1	4	yes	—
153	C₄²	D₁₀₄	1	2	yes	—
154	C₄²	Dic₅₂	1	2	yes	—
155	C₄²	C₂×#1055	1	2	yes	—
156	C₄²	#6099	1	2	yes	—
157	C₄²	#6127	1	4	yes	—
158	C₁₃	#8784	1	1	yes	characteristic, complemented, Sylow
159	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	C₄⋊#1055	1	2	yes	—
167	C₂×C₄	C₄⋊#1055	1	2	yes	—
168	C₂×C₄	#88057	1	2	yes	—
169	C₂×C₄	#88057	1	2	yes	—
170	C₂×C₄	#88058	1	2	yes	—
171	C₂×C₄	#88058	1	2	yes	—
172	C₂×C₄	#88059	1	2	yes	—
173	C₂×C₄	#88059	1	2	yes	—
174	C₂×C₄	#88124	1	2	yes	—
175	C₂×C₄	#88129	1	2	yes	—
176	C₂×C₄	C₂×#6095	1	2	yes	—
177	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
178	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
179	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
180	C₂×C₄	C₂×#6099	1	2	yes	—
181	C₂×C₄	C₂×#6099	1	1	yes	characteristic
182	C₂×C₄	#88180	1	2	yes	—
183	C₈	—	2	4	no	—
184	C₈	—	2	4	no	—
185	C₈	—	104	4	no	—
186	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
187	C₂²	#11718585	1	1	yes	characteristic
188	C₂²	#11718597	1	1	yes	characteristic
189	C₂²	C₂×#88057	1	1	yes	characteristic
190	C₂²	C₂×#88058	1	1	yes	characteristic
191	C₂²	C₂×#88059	1	1	yes	characteristic
192	C₂²	C₂×#88059	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	#11718049	1	4	yes	—
197	C₄	#11718050	1	4	yes	—
198	C₄	#11718342	1	2	yes	—
199	C₄	#11718355	1	2	yes	—
200	C₄	#11718586	1	2	yes	—
201	C₄	#11718587	1	2	yes	—
202	C₂	C₂×#11718049	1	1	yes	characteristic
203	C₂	C₂×#11718050	1	1	yes	characteristic
204	C₂	#13113592	1	1	yes	characteristic
205	C₂	#13113595	1	1	yes	characteristic
206	C₂	#13114953	1	1	yes	characteristic
207	C₂	#13114964	1	1	yes	characteristic
208	C₂	#13114974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#8784
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
25. C₄×C₈×C₁₃	C₈	semidirect
125. C₄×C₈	#1055	semidirect
158. C₁₃	#8784	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
#13112671	2
C₄²×C₈×C₁₃	2
#8784	13