Subgroups — #15761949

Order 3328.

570 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	#13123234	C₂	1	1	yes	maximal, characteristic
2	#13114988	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718073	C₂	1	4	yes	maximal
4	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	C₄	1	4	yes	—
6	C₂×#88123	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	—	2	1	no	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718596	—	2	2	no	—
11	#11718596	—	2	1	no	—
12	#11718596	—	2	1	no	—
13	C₂×C₄⋊#1055	—	2	2	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₁₃×#336	—	2	1	no	—
16	#11718073	—	2	8	no	—
17	C₂²×Dic₂₆	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	4	no	—
25	C₂×C₄⋊Dic₁₃	—	2	4	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	C₂²×#1055	—	2	2	no	—
29	#88123	—	2	4	no	—
30	#88123	—	2	4	no	—
31	#88123	—	4	8	no	—
32	C₁₃×C₄⋊C₈	—	2	2	no	—
33	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
34	C₄×C₈×C₁₃	—	4	2	no	—
35	C₄⋊#1055	—	2	4	no	—
36	C₄⋊#1055	—	4	4	no	—
37	#9691	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	2	4	no	—
44	C₂×Dic₂₆	—	2	8	no	—
45	C₂×Dic₂₆	—	4	8	no	—
46	C₂×C₈×C₁₃	—	2	4	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	—	4	2	no	—
49	C₄²×C₁₃	—	2	4	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	SD₁₆	1	4	yes	—
52	C₄²×C₁₃	SD₁₆	1	2	yes	—
53	C₄²×C₁₃	Q₁₆	1	4	yes	—
54	C₄²×C₁₃	Q₁₆	1	2	yes	—
55	C₄²×C₁₃	C₂×D₄	1	2	yes	—
56	C₄²×C₁₃	#41	1	2	yes	—
57	C₄⋊Dic₁₃	—	4	8	no	—
58	C₄⋊Dic₁₃	—	4	8	no	—
59	C₄⋊Dic₁₃	—	4	4	no	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	C₄⋊Dic₁₃	—	4	2	no	—
62	C₂×#1055	—	4	2	no	—
63	C₂×#1055	—	4	4	no	—
64	#2868	—	13	1	no	—
65	#1842	—	13	1	no	—
66	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
67	C₂×#333	—	13	4	no	—
68	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	#100	1	4	yes	—
73	C₂×C₄×C₁₃	#103	1	4	yes	—
74	C₂×C₄×C₁₃	#103	1	4	yes	—
75	C₂×C₄×C₁₃	#103	1	4	yes	—
76	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₁₃	#121	1	2	yes	—
78	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	#137	1	2	yes	—
85	C₂×Dic₁₃	—	4	4	no	—
86	C₂×Dic₁₃	—	4	8	no	—
87	Dic₂₆	—	8	8	no	—
88	C₈×C₁₃	—	4	4	no	—
89	#1055	—	8	2	no	—
90	C₂×C₄⋊Q₈	—	13	4	no	—
91	C₂×C₄⋊Q₈	—	13	2	no	—
92	C₂×C₄⋊C₈	—	13	2	no	—
93	C₂×C₄⋊C₈	—	26	2	no	—
94	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
95	C₂×C₄×C₈	—	2	1	no	—
96	#384	—	26	2	no	—
97	#384	—	26	1	no	—
98	#384	—	26	1	no	—
99	C₄³	C₂×D₁₃	1	1	yes	characteristic
100	#336	—	2	1	no	—
101	#333	—	26	8	no	—
102	C₂²×C₁₃	#390	1	1	yes	characteristic
103	C₂²×C₁₃	C₂×#100	1	2	yes	—
104	C₂²×C₁₃	C₂×#103	1	2	yes	—
105	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
106	C₂²×C₁₃	#419	1	1	yes	characteristic
107	C₄×C₁₃	—	2	4	no	—
108	C₄×C₁₃	—	2	2	no	—
109	C₄×C₁₃	#333	1	8	yes	complemented
110	C₄×C₁₃	#461	1	2	yes	—
111	C₄×C₁₃	#462	1	2	yes	—
112	C₄×C₁₃	#487	1	2	yes	—
113	C₄×C₁₃	#492	1	1	yes	characteristic
114	C₄×C₁₃	#494	1	1	yes	characteristic
115	Dic₁₃	—	8	4	no	—
116	C₂²×Q₈	—	13	4	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	26	2	no	—
119	C₄⋊Q₈	—	26	4	no	—
120	C₄⋊Q₈	—	26	4	no	—
121	C₄⋊Q₈	—	52	8	no	—
122	C₂×C₄⋊C₄	—	26	4	no	—
123	C₂×C₄⋊C₄	—	26	4	no	—
124	C₂×C₄⋊C₄	—	26	2	no	—
125	C₂×C₄⋊C₄	—	26	2	no	—
126	C₂×C₄²	C₄×D₁₃	1	2	yes	—
127	C₂×C₄²	D₅₂	1	2	yes	—
128	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	26	4	no	—
133	C₄⋊C₈	—	52	4	no	—
134	C₄⋊C₈	D₅₂	1	4	yes	—
135	C₄×C₈	—	4	2	no	—
136	C₂×C₁₃	C₂×#333	1	2	yes	—
137	C₂×C₁₃	#1548	1	2	yes	—
138	C₂×C₁₃	#1956	1	1	yes	characteristic
139	C₂×C₁₃	#1965	1	1	yes	characteristic
140	C₂×C₁₃	#1981	1	1	yes	characteristic
141	C₂×Q₈	—	26	8	no	—
142	C₂×Q₈	—	52	8	no	—
143	C₂²×C₄	—	26	4	no	—
144	C₂²×C₄	#6103	1	2	yes	—
145	C₂²×C₄	#6103	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	52	2	no	—
153	C₂×C₈	—	52	4	no	—
154	C₄⋊C₄	—	52	8	no	—
155	C₄⋊C₄	—	52	2	no	—
156	C₄⋊C₄	—	52	2	no	—
157	C₄⋊C₄	—	52	4	no	—
158	C₄⋊C₄	—	52	8	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	#6095	1	4	yes	—
162	C₄²	Dic₅₂	1	4	yes	—
163	C₄²	#6105	1	2	yes	—
164	C₄²	#6107	1	2	yes	—
165	C₄²	C₂×D₅₂	1	2	yes	—
166	C₄²	#6127	1	2	yes	—
167	C₁₃	#9691	1	1	yes	characteristic, complemented, Sylow
168	C₂³	C₂×#6103	1	1	yes	characteristic
169	Q₈	—	104	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	52	8	no	—
174	C₂×C₄	—	52	4	no	—
175	C₂×C₄	#88050	1	4	yes	—
176	C₂×C₄	#88056	1	4	yes	—
177	C₂×C₄	#88056	1	4	yes	—
178	C₂×C₄	#88063	1	4	yes	—
179	C₂×C₄	C₄×D₅₂	1	2	yes	—
180	C₂×C₄	#88128	1	1	yes	characteristic
181	C₂×C₄	#88129	1	1	yes	characteristic
182	C₂×C₄	C₂×#6095	1	2	yes	—
183	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
184	C₂×C₄	#88184	1	2	yes	—
185	C₂×C₄	C₂×#6105	1	1	yes	characteristic
186	C₂×C₄	C₂×#6107	1	1	yes	characteristic
187	C₂×C₄	#88205	1	2	yes	—
188	C₈	—	4	4	no	—
189	C₈	—	104	2	no	—
190	C₂²	#11718600	1	1	yes	characteristic
191	C₂²	C₂×#88050	1	1	yes	characteristic
192	C₂²	#11718665	1	1	yes	characteristic
193	C₂²	C₂×#88056	1	2	yes	—
194	C₂²	C₂×#88063	1	2	yes	—
195	C₄	—	2	2	no	—
196	C₄	—	2	4	no	—
197	C₄	—	104	4	no	—
198	C₄	#11718073	1	8	yes	complemented
199	C₄	#11718342	1	2	yes	—
200	C₄	#11718354	1	1	yes	characteristic
201	C₄	#11718360	1	1	yes	characteristic
202	C₄	#11718680	1	2	yes	—
203	C₄	#11718692	1	2	yes	—
204	C₂	C₂×#11718073	1	2	yes	—
205	C₂	#13113516	1	2	yes	—
206	C₂	#13115638	1	1	yes	characteristic
207	C₂	#13115676	1	1	yes	characteristic
208	C₂	#13115759	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#9691
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
109. C₄×C₁₃	#333	semidirect
167. C₁₃	#9691	semidirect
198. C₄	#11718073	semidirect

Maximal subgroups

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

subgroup	index
#13123234	2
#13114988	2
C₂×#11718073	2
C₄×C₁₃×C₄⋊C₈	2
#9691	13