Subgroups — #15781031

Order 3328.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116624	C₂	1	2	yes	maximal
2	#13115000	C₂	1	1	yes	maximal, characteristic
3	#13114715	C₂	1	1	yes	maximal, characteristic
4	#13112945	C₂	1	2	yes	maximal
5	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88059	C₂²	1	1	yes	characteristic
9	#11718767	C₂²	1	1	yes	characteristic
10	C₂×#88058	C₂²	1	2	yes	—
11	C₂×#88057	—	2	1	no	—
12	#11718762	—	2	1	no	—
13	#11718597	—	2	1	no	—
14	C₁₃×#364	C₄	1	4	yes	complemented
15	#11718115	—	2	1	no	—
16	#11718097	—	2	2	no	—
17	#11718096	—	2	2	no	—
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	#88180	—	2	1	no	—
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×#6099	—	2	1	no	—
24	C₁₃×M₃₂	—	2	2	no	—
25	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×C₁₆	—	2	2	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	Q₈	1	2	yes	—
29	#88071	—	4	2	no	—
30	#88059	—	2	1	no	—
31	#88059	Q₈	1	2	yes	—
32	#88058	—	2	2	no	—
33	#88058	—	2	2	no	—
34	#88058	D₄	1	2	yes	—
35	#88057	—	2	1	no	—
36	#88057	—	4	1	no	—
37	#88057	—	4	1	no	—
38	#88055	—	4	1	no	—
39	C₈×Dic₁₃	—	4	1	no	—
40	#13180	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	—	2	1	no	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
48	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
54	C₄²×C₁₃	—	2	1	no	—
55	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂²⋊Dic₁₃	—	4	1	no	—
57	C₄⋊Dic₁₃	—	2	2	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	C₄⋊Dic₁₃	—	4	2	no	—
60	C₄×Dic₁₃	—	4	1	no	—
61	#6099	—	2	2	no	—
62	#6099	—	4	1	no	—
63	C₂×#1055	—	4	1	no	—
64	C₁₃×C₁₆	—	2	4	no	—
65	#2159	—	13	2	no	—
66	C₂×#364	D₁₃	1	1	yes	characteristic
67	#1846	—	13	1	no	—
68	#1772	—	13	1	no	—
69	#1387	—	13	2	no	—
70	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
79	C₂×Dic₁₃	—	4	2	no	—
80	C₂×Dic₁₃	—	8	4	no	—
81	C₂×Dic₁₃	—	8	2	no	—
82	C₈×C₁₃	#107	1	2	yes	—
83	C₈×C₁₃	#107	1	2	yes	—
84	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
85	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₁₃	#125	1	1	yes	characteristic
87	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
88	#1055	—	4	2	no	—
89	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
90	C₂×#108	—	13	1	no	—
91	#427	—	13	1	no	—
92	C₂×#107	—	13	2	no	—
93	C₂×#106	—	26	1	no	—
94	#405	—	26	1	no	—
95	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
96	#382	—	26	1	no	—
97	#367	—	26	2	no	—
98	#366	—	26	2	no	—
99	#364	Dic₁₃	1	4	yes	complemented
100	#339	—	26	1	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	#428	1	1	yes	characteristic
103	C₂²×C₁₃	#459	1	1	yes	characteristic
104	C₂²×C₁₃	#477	1	1	yes	characteristic
105	C₄×C₁₃	—	4	2	no	—
106	C₄×C₁₃	#389	1	1	yes	characteristic
107	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
108	C₄×C₁₃	#462	1	1	yes	characteristic
109	C₄×C₁₃	#474	1	1	yes	characteristic
110	Dic₁₃	—	8	2	no	—
111	Dic₁₃	—	16	2	no	—
112	C₂×M₁₆	—	26	1	no	—
113	C₂²×C₈	D₅₂	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
115	#117	—	26	1	no	—
116	C₂×C₄⋊C₄	—	26	2	no	—
117	C₂×C₄²	D₅₂	1	1	yes	characteristic
118	M₃₂	—	2	2	no	—
119	M₃₂	C₂×Dic₁₃	1	4	yes	—
120	C₂×C₁₆	—	2	2	no	—
121	#108	—	13	2	no	—
122	#108	—	26	1	no	—
123	#107	—	13	2	no	—
124	#107	—	26	2	no	—
125	#107	—	26	2	no	—
126	#106	—	26	1	no	—
127	#106	—	52	1	no	—
128	#106	—	52	1	no	—
129	#97	—	52	1	no	—
130	C₄×C₈	—	52	1	no	—
131	C₄×C₈	Dic₂₆	1	2	yes	—
132	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
133	#95	—	52	2	no	—
134	C₂×C₁₃	—	2	1	no	—
135	C₂×C₁₃	—	2	1	no	—
136	C₂×C₁₃	#1947	1	1	yes	characteristic
137	C₂×C₁₃	#2212	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	52	2	no	—
140	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
141	M₁₆	—	26	2	no	—
142	M₁₆	—	52	1	no	—
143	C₂×C₈	—	52	1	no	—
144	C₂×C₈	D₁₀₄	1	1	yes	characteristic
145	C₂×C₈	D₁₀₄	1	1	yes	characteristic
146	C₂×C₈	Dic₅₂	1	2	yes	—
147	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
148	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
149	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
150	C₂×C₈	#6129	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
152	C₂×C₈	#6131	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
154	C₄⋊C₄	—	26	2	no	—
155	C₄⋊C₄	—	52	2	no	—
156	C₄⋊C₄	—	52	2	no	—
157	C₂²⋊C₄	—	52	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	52	1	no	—
160	C₄²	C₄⋊Dic₁₃	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₁₃	#13180	1	1	yes	characteristic, complemented, Sylow
163	C₂³	#88162	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	52	2	no	—
167	C₂×C₄	—	104	4	no	—
168	C₂×C₄	—	104	2	no	—
169	C₂×C₄	#88149	1	1	yes	characteristic
170	C₂×C₄	#88151	1	1	yes	characteristic
171	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
173	C₂×C₄	#88163	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
175	C₈	—	52	2	no	—
176	C₈	#88058	1	2	yes	—
177	C₈	#88058	1	2	yes	—
178	C₈	#88142	1	1	yes	characteristic
179	C₈	#88144	1	1	yes	characteristic
180	C₈	D₄×Dic₁₃	1	1	yes	characteristic
181	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#11718504	1	1	yes	characteristic
184	C₂²	#11718511	1	1	yes	characteristic
185	C₂²	#11718782	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	104	2	no	—
188	C₄	—	208	2	no	—
189	C₄	#11718507	1	1	yes	characteristic
190	C₄	#11718510	1	1	yes	characteristic
191	C₄	#11718646	1	1	yes	characteristic
192	C₄	C₂×#88058	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#13115595	1	1	yes	characteristic
196	C₂	#13116906	1	2	yes	—

	subgroup	G/H
center	C₂²	#11718504
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#364	C₂
socle	C₂²×C₁₃	#459
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#364	D₁₃
hypocenter	C₁₃	#13180
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
14. C₁₃×#364	C₄	semidirect
99. #364	Dic₁₃	semidirect
162. C₁₃	#13180	semidirect

Maximal subgroups

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

subgroup	index
#13116624	2
#13115000	2
#13114715	2
#13112945	2
C₂×C₁₃×#364	2
#13180	13