Subgroups — #15772890

Order 3328.

306 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116639	C₂	1	1	yes	maximal, characteristic
2	#13116639	C₂	1	1	yes	maximal, characteristic
3	#13112818	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₁₃×#446	—	2	1	no	—
7	C₁₃×#446	—	2	1	no	—
8	C₁₃×#446	C₄	1	2	yes	—
9	C₁₃×#446	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#97	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²⋊#6090	—	2	1	no	—
15	C₂²⋊#6090	—	2	1	no	—
16	C₄⋊#6090	—	2	1	no	—
17	C₄⋊#6090	C₂²	1	2	yes	—
18	#11718053	—	2	1	no	—
19	#11718053	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	—	2	1	no	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	—	2	1	no	—
25	C₄×Q₈×C₁₃	—	2	1	no	—
26	C₄×Q₈×C₁₃	C₈	1	2	yes	—
27	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	—	2	1	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	—	2	2	no	—
33	C₁₃×C₄⋊C₈	—	2	2	no	—
34	C₁₃×C₄⋊C₈	—	2	1	no	—
35	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₁₃×#97	—	2	2	no	—
37	C₁₃×#97	—	2	2	no	—
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×#6090	—	2	2	no	—
43	C₂×#6090	—	2	2	no	—
44	#11644	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	2	1	no	—
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	—	4	2	no	—
57	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	—	2	2	no	—
67	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
70	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	#6090	—	4	2	no	—
72	#6090	—	4	2	no	—
73	C₂×#446	D₁₃	1	1	yes	characteristic
74	#2163	—	13	1	no	—
75	#2163	—	13	1	no	—
76	#1338	—	13	4	no	—
77	C₂³×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
78	Q₈×C₁₃	—	4	2	no	—
79	Q₈×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	4	1	no	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	—	4	2	no	—
98	C₈×C₁₃	#137	1	2	yes	—
99	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
100	#446	—	2	1	no	—
101	#446	—	2	1	no	—
102	#446	Dic₁₃	1	2	yes	—
103	#446	C₂×D₁₃	1	2	yes	—
104	C₂×C₄⋊C₈	—	2	1	no	—
105	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic
106	C₂×#97	—	2	1	no	—
107	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
108	C₄⋊C₁₆	—	13	2	no	—
109	C₄⋊C₁₆	—	26	1	no	—
110	C₂²⋊C₁₆	—	26	1	no	—
111	C₂²⋊C₁₆	—	26	1	no	—
112	#346	—	13	2	no	—
113	#346	—	26	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	—	2	1	no	—
116	C₂²×C₁₃	#349	1	2	yes	—
117	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	#419	1	1	yes	characteristic
126	C₄×C₁₃	#421	1	1	yes	characteristic
127	C₂²×Q₈	#1055	1	1	yes	characteristic
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
131	C₄×Q₈	—	2	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×Q₈	#1055	1	2	yes	—
134	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	#1055	1	1	yes	characteristic
137	C₂×C₄²	—	2	1	no	—
138	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
139	C₂×C₁₆	—	26	2	no	—
140	C₂×C₁₆	—	26	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	1	no	—
144	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
145	#97	—	2	2	no	—
146	#97	—	2	2	no	—
147	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
150	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
151	C₂×C₁₃	—	2	2	no	—
152	C₂×C₁₃	#1479	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#349	1	1	yes	characteristic
154	C₂×C₁₃	#2117	1	1	yes	characteristic
155	C₂×Q₈	—	2	1	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	C₂×#1055	1	2	yes	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
161	C₂²×C₄	#6099	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	C₂×#1055	1	2	yes	—
175	C₄²	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂×#1055	1	1	yes	characteristic
178	C₄²	#6099	1	1	yes	characteristic
179	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
180	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
181	C₁₆	—	52	2	no	—
182	C₁₆	—	52	2	no	—
183	C₁₃	#11644	1	1	yes	characteristic, complemented, Sylow
184	C₂³	C₂²⋊#1055	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
196	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
197	C₂×C₄	C₂²⋊#1055	1	1	yes	characteristic
198	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
199	C₂×C₄	C₂×#6099	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	#88196	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#11718145	1	2	yes	—
209	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	#11718886	1	1	yes	characteristic
218	C₄	#11718897	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#13113231	1	1	yes	characteristic
221	C₂	C₂×#11718145	1	1	yes	characteristic
222	C₂	#13116464	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₁₃	C₂²⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	#11644
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #13112818	C₂	semidirect
183. C₁₃	#11644	semidirect

Maximal subgroups

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

subgroup	index
#13116639	2
#13112818	2
C₂×C₁₃×#446	2
#11644	13