Subgroups — #15768709

Order 3328.

950 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	2	yes	maximal
2	C₂×#11718116	C₂	1	4	yes	maximal
3	C₂×C₁₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
5	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	4	yes	—
7	C₁₃×#376	—	2	4	no	—
8	C₂×C₁₃×#95	C₄	1	4	yes	—
9	C₂×C₄⋊#1055	—	2	4	no	—
10	C₂×C₄⋊#1055	—	2	4	no	—
11	C₂×C₄⋊#1055	C₄	1	8	yes	complemented
12	#11718116	—	2	8	no	—
13	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
14	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	—	2	4	no	—
17	C₂×C₄²×C₁₃	—	2	4	no	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₁₃	D₄	1	4	yes	—
20	C₂×C₄²×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	D₄	1	4	yes	—
22	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
23	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
24	C₂²×#1055	—	2	4	no	—
25	C₂²×#1055	—	2	4	no	—
26	C₂²×#1055	—	2	8	no	—
27	C₂²×#1055	—	2	8	no	—
28	C₂²×#1055	C₂×C₄	1	8	yes	—
29	C₁₃×#95	—	2	8	no	—
30	C₄⋊#1055	—	2	16	no	—
31	#10833	—	13	1	no	maximal, Sylow
32	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	—	2	8	no	—
34	C₂²×C₄×C₁₃	—	2	4	no	—
35	C₂²×C₄×C₁₃	—	2	4	no	—
36	C₂²×C₄×C₁₃	—	2	16	no	—
37	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
38	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
47	C₂²×C₄×C₁₃	#41	1	4	yes	—
48	C₄²×C₁₃	—	2	8	no	—
49	C₄²×C₁₃	—	2	8	no	—
50	C₄²×C₁₃	C₂²⋊C₄	1	8	yes	—
51	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
52	C₂×#1055	—	2	8	no	—
53	C₂×#1055	—	2	16	no	—
54	C₂×#1055	—	4	8	no	—
55	C₂×#1055	—	4	4	no	—
56	C₂×#1055	—	4	4	no	—
57	C₂²×C₄⋊C₈	—	13	2	no	—
58	C₂×#376	D₁₃	1	1	yes	characteristic
59	C₂×#336	—	13	4	no	—
60	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
61	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
64	C₂³×C₁₃	#117	1	2	yes	—
65	C₂³×C₁₃	#131	1	8	yes	—
66	C₂×C₄×C₁₃	—	2	8	no	—
67	C₂×C₄×C₁₃	—	2	8	no	—
68	C₂×C₄×C₁₃	—	2	8	no	—
69	C₂×C₄×C₁₃	—	2	8	no	—
70	C₂×C₄×C₁₃	—	2	16	no	—
71	C₂×C₄×C₁₃	—	4	8	no	—
72	C₂×C₄×C₁₃	—	4	8	no	—
73	C₂×C₄×C₁₃	#95	1	8	yes	—
74	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
78	C₂×C₄×C₁₃	#121	1	4	yes	—
79	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
80	C₂×C₄×C₁₃	#124	1	2	yes	—
81	C₂×C₄×C₁₃	#125	1	2	yes	—
82	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
84	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
85	#1055	—	4	8	no	—
86	C₂³×C₈	—	13	4	no	—
87	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
88	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
89	C₂×C₄⋊C₈	—	13	8	no	—
90	C₂×C₄⋊C₈	—	26	4	no	—
91	C₂×C₄⋊C₈	—	26	4	no	—
92	#376	—	2	4	no	—
93	C₂×#95	Dic₁₃	1	4	yes	—
94	#336	—	26	8	no	—
95	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
96	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
97	C₂²×C₁₃	#380	1	2	yes	—
98	C₂²×C₁₃	#382	1	1	yes	characteristic
99	C₂²×C₁₃	#384	1	1	yes	characteristic
100	C₂²×C₁₃	#405	1	4	yes	—
101	C₂²×C₁₃	#408	1	4	yes	—
102	C₂²×C₁₃	#424	1	8	yes	—
103	C₂²×C₁₃	#424	1	4	yes	—
104	C₂²×C₁₃	#433	1	8	yes	—
105	C₄×C₁₃	—	2	8	no	—
106	C₄×C₁₃	—	4	8	no	—
107	C₄×C₁₃	#389	1	4	yes	—
108	C₄×C₁₃	#390	1	2	yes	—
109	C₄×C₁₃	#391	1	2	yes	—
110	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
111	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
112	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
113	C₂²×C₈	—	13	8	no	—
114	C₂²×C₈	—	26	8	no	—
115	C₂²×C₈	—	26	4	no	—
116	C₂²×C₈	—	26	8	no	—
117	C₂²×C₈	—	26	4	no	—
118	C₂×C₄²	—	2	4	no	—
119	C₂×C₄²	—	2	4	no	—
120	C₂×C₄²	Dic₂₆	1	4	yes	—
121	C₂×C₄²	Dic₂₆	1	2	yes	—
122	C₂×C₄²	C₄×D₁₃	1	4	yes	—
123	C₂×C₄²	D₅₂	1	4	yes	—
124	C₂×C₄²	D₅₂	1	2	yes	—
125	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
126	C₄⋊C₈	—	26	16	no	—
127	#95	—	2	8	no	—
128	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
129	C₂×C₁₃	#1729	1	2	yes	—
130	C₂×C₁₃	#1768	1	4	yes	—
131	C₂×C₁₃	#1798	1	4	yes	—
132	C₂×C₁₃	#1832	1	2	yes	—
133	C₂×C₁₃	#1843	1	2	yes	—
134	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	16	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
140	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
141	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
142	C₂²×C₄	C₂²⋊Dic₁₃	1	4	yes	—
143	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
144	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
146	C₂²×C₄	C₂×D₅₂	1	2	yes	—
147	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
148	C₂²×C₄	#6127	1	4	yes	—
149	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
150	C₂×C₈	—	26	16	no	—
151	C₂×C₈	—	26	8	no	—
152	C₂×C₈	—	52	4	no	—
153	C₂×C₈	—	52	4	no	—
154	C₂×C₈	—	52	8	no	—
155	C₄²	—	2	8	no	—
156	C₄²	—	2	8	no	—
157	C₄²	#6101	1	8	yes	—
158	C₄²	#6103	1	8	yes	—
159	C₁₃	#10833	1	1	yes	characteristic, complemented, Sylow
160	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
161	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
162	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
163	C₂³	#88180	1	2	yes	—
164	C₂³	#88202	1	8	yes	—
165	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	16	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	#88071	1	8	yes	—
174	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
175	C₂×C₄	#88123	1	2	yes	—
176	C₂×C₄	#88123	1	1	yes	characteristic
177	C₂×C₄	#88124	1	2	yes	—
178	C₂×C₄	C₄×D₅₂	1	4	yes	—
179	C₂×C₄	#88128	1	1	yes	characteristic
180	C₂×C₄	#88129	1	2	yes	—
181	C₂×C₄	C₂×#6101	1	2	yes	—
182	C₂×C₄	#88178	1	4	yes	—
183	C₂×C₄	C₂×#6103	1	2	yes	—
184	C₂×C₄	#88184	1	4	yes	—
185	C₈	—	52	8	no	—
186	C₂²	C₄×C₄⋊Dic₁₃	1	2	yes	—
187	C₂²	#11718596	1	1	yes	characteristic
188	C₂²	#11718597	1	1	yes	characteristic
189	C₂²	#11718658	1	4	yes	—
190	C₂²	#11718660	1	8	yes	—
191	C₂²	#11718660	1	4	yes	—
192	C₂²	#11718661	1	8	yes	—
193	C₂²	C₂×#88071	1	1	yes	characteristic
194	C₂²	#11718873	1	2	yes	—
195	C₂²	#11718895	1	4	yes	—
196	C₄	—	2	8	no	—
197	C₄	—	4	8	no	—
198	C₄	#11718589	1	4	yes	—
199	C₄	#11718590	1	2	yes	—
200	C₄	#11718600	1	2	yes	—
201	C₂	#13113066	1	1	yes	characteristic
202	C₂	#13114533	1	2	yes	—
203	C₂	#13114693	1	4	yes	—
204	C₂	#13114820	1	4	yes	—
205	C₂	#13114956	1	2	yes	—
206	C₂	#13114990	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₃
commutator	C₂³×C₁₃	C₂×C₄²
Fitting	C₂×C₁₃×#376	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#376	D₁₃
hypocenter	C₁₃	#10833
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
11. C₂×C₄⋊#1055	C₄	semidirect
159. C₁₃	#10833	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#1055	2
C₂×#11718116	2
C₂×C₁₃×#376	2
#10833	13