Subgroups — #15779334

Order 3328.

382 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	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718103	C₂	1	2	yes	maximal
3	#13116591	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
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×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88052	C₂²	1	2	yes	—
13	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₁₃×#364	—	2	2	no	—
16	#11718103	—	2	4	no	—
17	#11718093	—	2	4	no	—
18	#11718055	—	2	2	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	—	2	2	no	—
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₁₃×M₃₂	—	2	2	no	—
28	C₁₃×M₃₂	—	2	2	no	—
29	C₂×C₁₃×C₁₆	—	2	2	no	—
30	C₄×C₈×C₁₃	D₄	1	2	yes	—
31	C₄×C₈×C₁₃	Q₈	1	2	yes	—
32	C₈×Dic₁₃	—	2	2	no	—
33	C₈×Dic₁₃	C₈	1	2	yes	—
34	C₈×Dic₁₃	C₈	1	2	yes	—
35	C₈×Dic₁₃	D₄	1	2	yes	—
36	C₈×Dic₁₃	D₄	1	2	yes	—
37	C₈×Dic₁₃	Q₈	1	2	yes	—
38	C₈×Dic₁₃	Q₈	1	2	yes	—
39	#88052	—	2	2	no	—
40	#88052	—	2	2	no	—
41	C₂×#6090	—	2	2	no	—
42	C₄×#1055	—	2	2	no	—
43	#12900	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	—	2	1	no	—
47	C₂²×Dic₁₃	M₁₆	1	2	yes	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
51	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₁₃	—	2	1	no	—
58	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
59	C₄×Dic₁₃	—	2	8	no	—
60	C₄×Dic₁₃	—	2	1	no	—
61	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
62	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
63	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
64	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	—	2	1	no	—
67	C₂×#1055	C₄⋊C₄	1	2	yes	—
68	C₂×#1055	C₄⋊C₄	1	2	yes	—
69	C₂×#1055	C₂×C₈	1	2	yes	—
70	C₂×#1055	M₁₆	1	2	yes	—
71	C₁₃×C₁₆	—	4	2	no	—
72	#6090	—	4	2	no	—
73	C₂×#364	—	13	2	no	—
74	C₂×#364	—	13	1	no	—
75	C₂×#364	D₁₃	1	1	yes	characteristic
76	#2150	—	13	2	no	—
77	C₄²×C₈	—	13	1	no	—
78	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	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×Dic₁₃	—	2	4	no	—
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	2	4	no	—
90	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
91	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
92	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
93	C₈×C₁₃	#127	1	1	yes	characteristic
94	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
95	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
97	#1055	—	2	2	no	—
98	#1055	C₄⋊C₈	1	4	yes	—
99	C₂×M₃₂	—	13	2	no	—
100	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	—	13	1	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
107	C₄³	—	13	1	no	—
108	#364	—	2	2	no	—
109	#364	—	26	4	no	—
110	#364	—	26	2	no	—
111	C₄⋊C₁₆	—	26	4	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₁₃	#423	1	1	yes	characteristic
115	C₂²×C₁₃	#423	1	1	yes	characteristic
116	C₄×C₁₃	—	2	4	no	—
117	C₄×C₁₃	#384	1	1	yes	characteristic
118	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₄×C₁₃	#423	1	1	yes	characteristic
121	Dic₁₃	—	2	4	no	—
122	Dic₁₃	—	2	2	no	—
123	Dic₁₃	#364	1	4	yes	complemented
124	C₂²×C₈	—	13	1	no	—
125	C₂²×C₈	—	13	1	no	—
126	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
128	C₂×C₄²	—	13	1	no	—
129	C₂×C₄²	—	13	1	no	—
130	C₂×C₄²	—	26	2	no	—
131	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	2	2	no	—
134	M₃₂	—	26	2	no	—
135	M₃₂	—	26	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	26	2	no	—
138	C₄×C₈	—	13	2	no	—
139	C₄×C₈	—	13	2	no	—
140	C₄×C₈	—	13	2	no	—
141	C₄×C₈	—	13	2	no	—
142	C₄×C₈	—	13	2	no	—
143	C₄×C₈	—	13	2	no	—
144	C₄×C₈	—	26	2	no	—
145	C₄×C₈	—	26	2	no	—
146	C₄×C₈	Dic₂₆	1	2	yes	—
147	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
148	C₂×C₁₃	—	2	1	no	—
149	C₂×C₁₃	—	2	1	no	—
150	C₂×C₁₃	#1842	1	1	yes	characteristic
151	C₂×C₁₃	C₂×#364	1	2	yes	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	13	2	no	—
154	C₂²×C₄	—	26	1	no	—
155	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	13	2	no	—
158	C₂×C₈	—	13	2	no	—
159	C₂×C₈	—	13	2	no	—
160	C₂×C₈	—	26	1	no	—
161	C₂×C₈	—	26	1	no	—
162	C₂×C₈	C₈×D₁₃	1	2	yes	—
163	C₂×C₈	#6094	1	2	yes	—
164	C₂×C₈	#6101	1	2	yes	—
165	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
166	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
167	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
168	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
171	C₄²	—	2	1	no	—
172	C₄²	—	13	2	no	—
173	C₄²	—	13	1	no	—
174	C₄²	—	13	1	no	—
175	C₄²	—	13	2	no	—
176	C₄²	—	26	8	no	—
177	C₄²	—	26	1	no	—
178	C₄²	#6101	1	2	yes	—
179	C₁₆	—	4	2	no	—
180	C₁₆	—	52	2	no	—
181	C₁₃	#12900	1	1	yes	characteristic, complemented, Sylow
182	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	13	2	no	—
186	C₂×C₄	—	13	2	no	—
187	C₂×C₄	—	26	4	no	—
188	C₂×C₄	—	26	4	no	—
189	C₂×C₄	—	26	2	no	—
190	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
191	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
192	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
193	C₂×C₄	C₂×#6094	1	1	yes	characteristic
194	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
195	C₂×C₄	C₂×#6101	1	1	yes	characteristic
196	C₈	—	13	4	no	—
197	C₈	—	26	2	no	—
198	C₈	#88054	1	4	yes	—
199	C₈	#88142	1	1	yes	characteristic
200	C₈	#88142	1	1	yes	characteristic
201	C₈	#88194	1	1	yes	characteristic
202	C₈	#88198	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
205	C₂²	#11718498	1	1	yes	characteristic
206	C₂²	#11718779	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	13	4	no	—
209	C₄	—	26	4	no	—
210	C₄	—	26	2	no	—
211	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
212	C₄	#11718498	1	1	yes	characteristic
213	C₄	#11718637	1	1	yes	characteristic
214	C₄	C₂×#88054	1	1	yes	characteristic
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	#13114987	1	1	yes	characteristic
218	C₂	D₁₃×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#364	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#364	D₁₃
hypocenter	C₁₃	#12900
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
123. Dic₁₃	#364	semidirect
181. C₁₃	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718055	2
C₂×#11718103	2
#13116591	2
C₄×C₈×Dic₁₃	2
C₂×C₁₃×#364	2
#12900	13