Subgroups — #15774254

Order 3328.

310 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116635	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	#13112894	C₂	1	2	yes	maximal, complemented
4	#13112885	C₂	1	2	yes	maximal, complemented
5	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	—	2	1	no	—
7	C₁₃×#500	C₂²	1	1	yes	characteristic
8	C₁₃×#494	C₂²	1	1	yes	characteristic
9	C₁₃×#487	—	2	1	no	—
10	C₁₃×#467	—	2	1	no	—
11	C₁₃×#467	C₄	1	2	yes	—
12	C₄×C₁₃×Q₁₆	—	2	1	no	—
13	C₄×C₁₃×Q₁₆	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	—	2	1	no	—
16	C₂²⋊#6090	—	2	1	no	—
17	C₄⋊#6090	C₂²	1	2	yes	—
18	C₄×#6090	—	2	1	no	—
19	#11718050	—	2	1	no	—
20	#11718050	C₂²	1	2	yes	—
21	C₂×C₁₃×Q₁₆	—	2	1	no	—
22	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₁₃×C₄⋊Q₈	—	2	1	no	—
26	C₁₃×C₄⋊Q₈	—	2	1	no	—
27	C₁₃×#125	—	4	1	no	—
28	C₁₃×C₂²⋊Q₈	—	2	2	no	—
29	C₁₃×C₂²⋊Q₈	—	4	1	no	—
30	C₄×Q₈×C₁₃	—	2	2	no	—
31	C₄×Q₈×C₁₃	—	4	1	no	—
32	C₁₃×#117	—	4	1	no	—
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×#1055	—	4	1	no	—
35	C₁₃×#107	—	2	1	no	—
36	C₁₃×#107	C₂×C₄	1	2	yes	—
37	C₁₃×#103	—	2	2	no	—
38	C₁₃×#103	—	2	2	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6090	—	2	2	no	—
44	C₄⋊#1055	—	2	2	no	—
45	C₄⋊#1055	—	4	1	no	—
46	#11949	—	13	1	no	maximal, Sylow
47	C₂×Q₈×C₁₃	—	2	2	no	—
48	C₂×Q₈×C₁₃	—	4	1	no	—
49	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₁₃×Q₁₆	—	4	2	no	—
53	C₁₃×Q₁₆	—	4	2	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	4	2	no	—
62	C₁₃×C₄⋊C₄	—	4	2	no	—
63	C₁₃×C₄⋊C₄	—	4	1	no	—
64	C₁₃×C₄⋊C₄	—	4	1	no	—
65	C₁₃×C₄⋊C₄	—	4	1	no	—
66	C₁₃×C₂²⋊C₄	—	4	2	no	—
67	C₄²×C₁₃	—	4	2	no	—
68	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₁₃	D₈	1	1	yes	characteristic
70	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×#1055	—	4	2	no	—
73	C₂×#1055	—	8	2	no	—
74	#6090	—	4	2	no	—
75	#3170	D₁₃	1	1	yes	characteristic
76	#2162	—	13	1	no	—
77	C₂×#335	—	13	1	no	—
78	#1365	—	13	2	no	—
79	#1362	—	13	2	no	—
80	C₂³×C₁₃	#102	1	1	yes	characteristic
81	Q₈×C₁₃	—	4	2	no	—
82	Q₈×C₁₃	—	4	2	no	—
83	Q₈×C₁₃	—	8	2	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	4	1	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
93	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	#135	1	2	yes	—
101	#1055	—	8	2	no	—
102	#533	—	2	1	no	—
103	#500	C₂×D₁₃	1	1	yes	characteristic
104	#494	C₂×D₁₃	1	1	yes	characteristic
105	#487	—	2	1	no	—
106	#467	—	2	1	no	—
107	#467	Dic₁₃	1	2	yes	—
108	C₄×Q₁₆	—	2	1	no	—
109	C₄×Q₁₆	Dic₁₃	1	2	yes	—
110	C₂×C₄⋊C₈	—	26	1	no	—
111	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
112	C₄⋊C₁₆	—	13	2	no	—
113	C₂²⋊C₁₆	—	26	1	no	—
114	C₄×C₁₆	—	26	1	no	—
115	#335	—	13	2	no	—
116	#335	—	26	1	no	—
117	C₂²×C₁₃	—	2	1	no	—
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	#358	1	2	yes	—
120	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	—	8	2	no	—
125	C₄×C₁₃	—	8	2	no	—
126	C₄×C₁₃	#358	1	2	yes	—
127	C₄×C₁₃	#413	1	1	yes	characteristic
128	C₄×C₁₃	#416	1	1	yes	characteristic
129	C₄×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
130	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
131	C₂×Q₁₆	—	2	1	no	—
132	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
133	C₂²×C₈	—	52	1	no	—
134	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	1	no	—
137	C₄⋊Q₈	—	2	1	no	—
138	#125	—	4	1	no	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₂²⋊Q₈	—	4	1	no	—
141	C₄×Q₈	—	2	2	no	—
142	C₄×Q₈	—	4	1	no	—
143	#117	—	4	1	no	—
144	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
145	C₂×C₁₆	—	26	2	no	—
146	#107	—	2	1	no	—
147	#107	C₂×Dic₁₃	1	2	yes	—
148	C₄⋊C₈	—	26	2	no	—
149	C₄⋊C₈	—	52	1	no	—
150	#103	—	2	2	no	—
151	#103	—	2	2	no	—
152	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₂×C₁₃	—	2	2	no	—
157	C₂×C₁₃	#1540	1	1	yes	characteristic
158	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
159	C₂×C₁₃	#2148	1	1	yes	characteristic
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	4	1	no	—
162	C₂²×C₄	#6104	1	1	yes	characteristic
163	C₂²×C₄	#6105	1	1	yes	characteristic
164	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
165	Q₁₆	—	4	2	no	—
166	Q₁₆	—	4	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	52	2	no	—
170	C₂×C₈	—	104	2	no	—
171	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	—	4	2	no	—
183	C₄²	#6104	1	1	yes	characteristic
184	C₄²	#6105	1	1	yes	characteristic
185	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
186	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
187	C₁₆	—	52	2	no	—
188	C₁₃	#11949	1	1	yes	characteristic, complemented, Sylow
189	C₂³	#88072	1	1	yes	characteristic
190	Q₈	—	4	2	no	—
191	Q₈	—	4	2	no	—
192	Q₈	—	8	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	#88068	1	2	yes	—
199	C₂×C₄	#88069	1	2	yes	—
200	C₂×C₄	#88072	1	1	yes	characteristic
201	C₂×C₄	#88072	1	1	yes	characteristic
202	C₂×C₄	#88072	1	1	yes	characteristic
203	C₂×C₄	C₂×#6104	1	1	yes	characteristic
204	C₂×C₄	C₂×#6105	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	104	2	no	—
210	C₈	#88204	1	2	yes	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#11718156	1	2	yes	—
214	C₂²	C₂×#88072	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	#11718156	1	2	yes	—
221	C₄	C₂×#88068	1	1	yes	characteristic
222	C₄	C₂×#88069	1	1	yes	characteristic
223	C₄	#11718894	1	1	yes	characteristic
224	C₄	#11718898	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	#13113485	1	1	yes	characteristic
227	C₂	C₂×#11718156	1	1	yes	characteristic
228	C₂	#13116585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	#102
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#11949
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #13112894	C₂	semidirect
4. #13112885	C₂	semidirect
188. C₁₃	#11949	semidirect

Maximal subgroups

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

subgroup	index
#13116635	2
C₂×#11718050	2
#13112894	2
#13112885	2
C₁₃×#3170	2
#11949	13