Subgroups — #15781036

Order 3328.

310 conjugacy classes of subgroups.

The subgroup lattice has 214 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	#13114996	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	—	2	1	no	—
11	C₂×#88058	C₂²	1	2	yes	—
12	#11718762	—	2	1	no	—
13	#11718596	—	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×C₄⋊Dic₁₃	—	4	2	no	—
24	C₂×#6099	—	2	1	no	—
25	C₁₃×M₃₂	—	2	2	no	—
26	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×C₁₆	—	2	2	no	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	Q₈	1	2	yes	—
30	#88059	—	2	1	no	—
31	#88059	Q₈	1	2	yes	—
32	#88058	—	2	2	no	—
33	#88058	—	2	2	no	—
34	#88058	—	4	1	no	—
35	#88058	—	4	1	no	—
36	#88058	D₄	1	1	yes	characteristic
37	#88058	D₄	1	1	yes	characteristic
38	#88057	—	2	1	no	—
39	#88055	—	4	1	no	—
40	C₈×Dic₁₃	—	4	1	no	—
41	#13181	—	13	1	no	maximal, Sylow
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	4	2	no	—
45	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
48	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂²⋊Dic₁₃	—	4	1	no	—
58	C₄⋊Dic₁₃	—	2	1	no	—
59	C₄⋊Dic₁₃	—	2	1	no	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	C₄⋊Dic₁₃	—	4	2	no	—
62	C₄⋊Dic₁₃	—	8	4	no	—
63	C₄×Dic₁₃	—	4	1	no	—
64	#6099	—	2	2	no	—
65	#6099	—	4	1	no	—
66	C₂×#1055	—	4	1	no	—
67	C₁₃×C₁₆	—	2	4	no	—
68	#2159	—	13	2	no	—
69	C₂×#364	D₁₃	1	1	yes	characteristic
70	#1845	—	13	1	no	—
71	#1772	—	13	1	no	—
72	#1387	—	13	2	no	—
73	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
83	C₂×Dic₁₃	—	4	1	no	—
84	C₂×Dic₁₃	—	4	1	no	—
85	C₂×Dic₁₃	—	8	4	no	—
86	C₂×Dic₁₃	—	8	2	no	—
87	C₈×C₁₃	#107	1	2	yes	—
88	C₈×C₁₃	#107	1	2	yes	—
89	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
90	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
91	C₈×C₁₃	#125	1	1	yes	characteristic
92	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
93	#1055	—	4	2	no	—
94	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
95	C₂×#108	—	13	1	no	—
96	#427	—	13	1	no	—
97	C₂×#107	—	13	2	no	—
98	C₂×#107	—	26	1	no	—
99	#405	—	26	1	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	#384	—	26	1	no	—
102	#367	—	26	2	no	—
103	#366	—	26	2	no	—
104	#364	Dic₁₃	1	4	yes	complemented
105	#339	—	26	1	no	—
106	C₂²×C₁₃	—	2	1	no	—
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	#428	1	1	yes	characteristic
109	C₂²×C₁₃	#459	1	1	yes	characteristic
110	C₂²×C₁₃	#477	1	1	yes	characteristic
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	#389	1	1	yes	characteristic
113	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
114	C₄×C₁₃	#462	1	1	yes	characteristic
115	C₄×C₁₃	#474	1	1	yes	characteristic
116	Dic₁₃	—	8	1	no	—
117	Dic₁₃	—	8	1	no	—
118	Dic₁₃	—	16	2	no	—
119	C₂×M₁₆	—	26	1	no	—
120	C₂²×C₈	D₅₂	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
122	#117	—	26	1	no	—
123	C₂×C₄⋊C₄	—	26	2	no	—
124	C₂×C₄⋊C₄	—	52	2	no	—
125	C₂×C₄²	D₅₂	1	1	yes	characteristic
126	M₃₂	—	2	2	no	—
127	M₃₂	C₂×Dic₁₃	1	4	yes	—
128	C₂×C₁₆	—	2	2	no	—
129	#108	—	13	2	no	—
130	#108	—	26	1	no	—
131	#107	—	13	1	no	—
132	#107	—	13	1	no	—
133	#107	—	26	2	no	—
134	#107	—	26	2	no	—
135	#107	—	52	1	no	—
136	#107	—	52	1	no	—
137	#106	—	26	1	no	—
138	#97	—	52	1	no	—
139	C₄×C₈	—	52	1	no	—
140	C₄×C₈	Dic₂₆	1	2	yes	—
141	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
142	C₂×C₁₃	—	2	1	no	—
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	#1947	1	1	yes	characteristic
145	C₂×C₁₃	#2213	1	1	yes	characteristic
146	C₂×C₁₃	#2214	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	52	2	no	—
149	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
150	M₁₆	—	26	2	no	—
151	M₁₆	—	52	1	no	—
152	C₂×C₈	—	52	1	no	—
153	C₂×C₈	D₁₀₄	1	1	yes	characteristic
154	C₂×C₈	D₁₀₄	1	1	yes	characteristic
155	C₂×C₈	Dic₅₂	1	2	yes	—
156	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
157	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
158	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
159	C₂×C₈	#6129	1	1	yes	characteristic
160	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
161	C₂×C₈	#6131	1	1	yes	characteristic
162	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
163	C₄⋊C₄	—	26	1	no	—
164	C₄⋊C₄	—	26	1	no	—
165	C₄⋊C₄	—	52	2	no	—
166	C₄⋊C₄	—	52	2	no	—
167	C₄⋊C₄	—	104	4	no	—
168	C₂²⋊C₄	—	52	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	52	1	no	—
171	C₄²	C₄⋊Dic₁₃	1	2	yes	—
172	C₁₆	—	2	4	no	—
173	C₁₃	#13181	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88162	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	52	1	no	—
179	C₂×C₄	—	52	1	no	—
180	C₂×C₄	—	104	4	no	—
181	C₂×C₄	—	104	2	no	—
182	C₂×C₄	#88149	1	1	yes	characteristic
183	C₂×C₄	#88151	1	1	yes	characteristic
184	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
186	C₂×C₄	#88163	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
188	C₈	—	52	2	no	—
189	C₈	#88058	1	2	yes	—
190	C₈	#88058	1	2	yes	—
191	C₈	#88142	1	1	yes	characteristic
192	C₈	#88144	1	1	yes	characteristic
193	C₈	D₄×Dic₁₃	1	1	yes	characteristic
194	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	#11718504	1	1	yes	characteristic
198	C₂²	#11718511	1	1	yes	characteristic
199	C₂²	#11718782	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	104	1	no	—
202	C₄	—	104	1	no	—
203	C₄	—	208	2	no	—
204	C₄	#11718507	1	1	yes	characteristic
205	C₄	#11718510	1	1	yes	characteristic
206	C₄	#11718646	1	1	yes	characteristic
207	C₄	C₂×#88058	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#13115595	1	1	yes	characteristic
211	C₂	#13116913	1	1	yes	characteristic
212	C₂	#13116918	1	1	yes	characteristic

	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	#13181
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
104. #364	Dic₁₃	semidirect
173. C₁₃	#13181	semidirect

Maximal subgroups

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

subgroup	index
#13116624	2
#13114996	2
#13114715	2
#13112945	2
C₂×C₁₃×#364	2
#13181	13