Subgroups — #15756664

Order 3328.

422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114953	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	2	yes	maximal
3	C₂×#11718049	C₂	1	2	yes	maximal
4	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
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	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₄	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	C₄³×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	C₂²	1	4	yes	—
13	#11718116	—	2	4	no	—
14	#11718050	—	2	4	no	—
15	#11718049	—	2	4	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
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	4	no	—
26	C₄×C₈×C₁₃	—	2	2	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	C₄×C₈×C₁₃	—	2	2	no	—
29	C₄×C₈×C₁₃	C₈	1	4	yes	complemented
30	C₄×C₈×C₁₃	D₄	1	2	yes	—
31	C₄×C₈×C₁₃	D₄	1	2	yes	—
32	C₄×C₈×C₁₃	Q₈	1	2	yes	—
33	C₄×C₈×C₁₃	Q₈	1	2	yes	—
34	C₄⋊#1055	—	2	4	no	—
35	C₄⋊#1055	—	2	4	no	—
36	#8807	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
53	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
54	C₄²×C₁₃	—	2	4	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	—	2	1	no	—
57	C₄²×C₁₃	—	2	4	no	—
58	C₄²×C₁₃	C₂×C₈	1	2	yes	—
59	C₄²×C₁₃	M₁₆	1	2	yes	—
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	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×#1055	—	4	4	no	—
65	C₂×#1055	—	4	8	no	—
66	#1831	—	13	2	no	—
67	C₄²×C₈	D₁₃	1	1	yes	characteristic
68	C₂×#335	—	13	2	no	—
69	C₂×#334	—	13	2	no	—
70	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#108	1	4	yes	—
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#135	1	4	yes	—
89	C₈×C₁₃	—	2	2	no	—
90	C₈×C₁₃	—	2	2	no	—
91	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
92	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
93	#1055	—	8	4	no	—
94	C₂×C₄⋊C₈	—	13	4	no	—
95	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
96	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
97	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
98	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	C₄³	C₂×D₁₃	1	1	yes	characteristic
102	#336	—	26	4	no	—
103	#335	—	26	4	no	—
104	#334	—	26	4	no	—
105	C₂²×C₁₃	#384	1	1	yes	characteristic
106	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₂²×C₁₃	#423	1	1	yes	characteristic
108	C₂²×C₁₃	#427	1	2	yes	—
109	C₂²×C₁₃	C₂×#108	1	2	yes	—
110	C₄×C₁₃	—	2	4	no	—
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	2	1	no	—
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₄×C₁₃	#423	1	1	yes	characteristic
117	C₄×C₁₃	#423	1	1	yes	characteristic
118	C₄×C₁₃	#495	1	2	yes	—
119	C₄×C₁₃	#499	1	2	yes	—
120	C₂²×C₈	—	26	4	no	—
121	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
122	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
125	C₂×C₄²	Dic₂₆	1	2	yes	—
126	C₂×C₄²	D₅₂	1	2	yes	—
127	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
128	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
130	C₄⋊C₈	—	26	4	no	—
131	C₄⋊C₈	—	26	4	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	#1055	1	4	yes	complemented
136	C₄×C₈	Dic₂₆	1	2	yes	—
137	C₄×C₈	Dic₂₆	1	2	yes	—
138	C₄×C₈	D₅₂	1	2	yes	—
139	C₄×C₈	D₅₂	1	2	yes	—
140	C₂×C₁₃	#1564	1	2	yes	—
141	C₂×C₁₃	#1568	1	2	yes	—
142	C₂×C₁₃	#1842	1	1	yes	characteristic
143	C₂×C₁₃	#1846	1	1	yes	characteristic
144	C₂×C₁₃	#1848	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
146	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
147	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	52	4	no	—
155	C₂×C₈	—	52	8	no	—
156	C₂×C₈	C₂×#1055	1	2	yes	—
157	C₂×C₈	C₂×#1055	1	2	yes	—
158	C₂×C₈	#6099	1	2	yes	—
159	C₂×C₈	#6099	1	2	yes	—
160	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
161	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
162	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
163	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	4	no	—
168	C₄²	C₂×#1055	1	2	yes	—
169	C₄²	#6099	1	2	yes	—
170	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
171	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
172	C₄²	C₂×D₅₂	1	1	yes	characteristic
173	C₄²	C₂×D₅₂	1	1	yes	characteristic
174	C₁₃	#8807	1	1	yes	characteristic, complemented, Sylow
175	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	C₄⋊#1055	1	4	yes	—
182	C₂×C₄	#88059	1	4	yes	—
183	C₂×C₄	#88123	1	1	yes	characteristic
184	C₂×C₄	#88123	1	1	yes	characteristic
185	C₂×C₄	#88123	1	1	yes	characteristic
186	C₂×C₄	#88128	1	1	yes	characteristic
187	C₂×C₄	#88158	1	4	yes	—
188	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
189	C₂×C₄	C₂×#6099	1	1	yes	characteristic
190	C₂×C₄	C₂×#6099	1	1	yes	characteristic
191	C₂×C₄	C₂×#6099	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	104	4	no	—
197	C₈	C₄⋊#1055	1	4	yes	—
198	C₈	C₄⋊#1055	1	4	yes	—
199	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
200	C₂²	#11718585	1	1	yes	characteristic
201	C₂²	#11718596	1	1	yes	characteristic
202	C₂²	#11718767	1	2	yes	—
203	C₂²	C₂×#88059	1	2	yes	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	4	no	—
207	C₄	—	2	4	no	—
208	C₄	#11718344	1	2	yes	—
209	C₄	#11718357	1	2	yes	—
210	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
211	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
212	C₄	#11718585	1	1	yes	characteristic
213	C₄	#11718585	1	1	yes	characteristic
214	C₂	#13113584	1	2	yes	—
215	C₂	#13113599	1	2	yes	—
216	C₂	#13114988	1	1	yes	characteristic
217	C₂	#13115000	1	1	yes	characteristic
218	C₂	#13115008	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	#8807
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
29. C₄×C₈×C₁₃	C₈	semidirect
135. C₄×C₈	#1055	semidirect
174. C₁₃	#8807	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
C₂×#11718050	2
C₂×#11718049	2
C₄²×C₈×C₁₃	2
#8807	13