Subgroups — #15774406

Order 3328.

314 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
2	#13113595	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	C₂²×#6090	—	2	1	no	—
16	#11718585	—	2	1	no	—
17	C₄⋊#6090	—	2	1	no	—
18	C₄⋊#6090	C₂²	1	2	yes	—
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	C₂×#6099	—	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	C₂×#6090	—	4	2	no	—
45	C₄⋊#1055	—	2	2	no	—
46	C₄⋊#1055	—	4	1	no	—
47	#11983	—	13	1	no	maximal, Sylow
48	C₂×Q₈×C₁₃	—	2	2	no	—
49	C₂×Q₈×C₁₃	—	4	1	no	—
50	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₁₃×Q₁₆	—	4	2	no	—
54	C₁₃×Q₁₆	—	4	1	no	—
55	C₁₃×Q₁₆	—	4	1	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	4	1	no	—
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	4	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	1	no	—
67	C₁₃×C₄⋊C₄	—	4	1	no	—
68	C₁₃×C₄⋊C₄	—	4	1	no	—
69	C₁₃×C₂²⋊C₄	—	4	2	no	—
70	C₄²×C₁₃	—	4	2	no	—
71	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃	D₈	1	1	yes	characteristic
73	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
75	#6099	—	8	2	no	—
76	C₂×#1055	—	4	2	no	—
77	#6090	—	4	2	no	—
78	#3170	D₁₃	1	1	yes	characteristic
79	C₂×C₄⋊C₁₆	—	13	1	no	—
80	#1567	—	13	1	no	—
81	#1365	—	13	2	no	—
82	#1362	—	13	2	no	—
83	C₂³×C₁₃	#102	1	1	yes	characteristic
84	Q₈×C₁₃	—	4	2	no	—
85	Q₈×C₁₃	—	4	2	no	—
86	Q₈×C₁₃	—	8	2	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	4	1	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	—	4	2	no	—
92	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
95	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
96	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₈×C₁₃	—	2	1	no	—
101	C₈×C₁₃	—	2	1	no	—
102	C₈×C₁₃	—	2	1	no	—
103	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₈×C₁₃	#137	1	1	yes	characteristic
105	#1055	—	8	2	no	—
106	#533	—	2	1	no	—
107	C₂²×C₁₆	—	26	1	no	—
108	#500	C₂×D₁₃	1	1	yes	characteristic
109	#494	C₂×D₁₃	1	1	yes	characteristic
110	#487	—	2	1	no	—
111	#467	—	2	1	no	—
112	#467	Dic₁₃	1	2	yes	—
113	C₄×Q₁₆	—	2	1	no	—
114	C₄×Q₁₆	Dic₁₃	1	2	yes	—
115	#423	—	26	1	no	—
116	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
117	C₄⋊C₁₆	—	13	2	no	—
118	C₄⋊C₁₆	—	26	1	no	—
119	#335	—	13	2	no	—
120	#335	—	26	1	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	1	no	—
123	C₂²×C₁₃	#358	1	2	yes	—
124	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	2	1	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	8	2	no	—
129	C₄×C₁₃	—	8	2	no	—
130	C₄×C₁₃	#358	1	2	yes	—
131	C₄×C₁₃	#413	1	1	yes	characteristic
132	C₄×C₁₃	#417	1	1	yes	characteristic
133	C₄×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
134	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
135	C₂×Q₁₆	—	2	1	no	—
136	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
137	C₂×M₁₆	—	52	1	no	—
138	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
140	C₄⋊Q₈	—	2	1	no	—
141	C₄⋊Q₈	—	2	1	no	—
142	#125	—	4	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	—	4	1	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	4	1	no	—
147	#117	—	4	1	no	—
148	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
149	C₂×C₁₆	—	26	2	no	—
150	C₂×C₁₆	—	52	2	no	—
151	#107	—	2	1	no	—
152	#107	C₂×Dic₁₃	1	2	yes	—
153	C₄⋊C₈	—	26	2	no	—
154	C₄⋊C₈	—	52	1	no	—
155	#103	—	2	2	no	—
156	#103	—	2	2	no	—
157	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
161	C₂×C₁₃	—	2	2	no	—
162	C₂×C₁₃	#1547	1	1	yes	characteristic
163	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
164	C₂×C₁₃	#2148	1	1	yes	characteristic
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	4	1	no	—
167	C₂²×C₄	#6104	1	1	yes	characteristic
168	C₂²×C₄	#6105	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
170	Q₁₆	—	4	2	no	—
171	Q₁₆	—	4	1	no	—
172	Q₁₆	—	4	1	no	—
173	M₁₆	—	104	2	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	52	2	no	—
178	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
180	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
181	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₂²⋊C₄	—	4	2	no	—
189	C₄²	—	4	2	no	—
190	C₄²	#6104	1	1	yes	characteristic
191	C₄²	#6105	1	1	yes	characteristic
192	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
194	C₁₆	—	52	2	no	—
195	C₁₃	#11983	1	1	yes	characteristic, complemented, Sylow
196	C₂³	#88072	1	1	yes	characteristic
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	#88068	1	2	yes	—
206	C₂×C₄	#88069	1	2	yes	—
207	C₂×C₄	#88072	1	1	yes	characteristic
208	C₂×C₄	#88072	1	1	yes	characteristic
209	C₂×C₄	#88072	1	1	yes	characteristic
210	C₂×C₄	C₂×#6104	1	1	yes	characteristic
211	C₂×C₄	C₂×#6105	1	1	yes	characteristic
212	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	104	2	no	—
217	C₈	#88203	1	1	yes	characteristic
218	C₈	#88205	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#11718156	1	2	yes	—
222	C₂²	C₂×#88072	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	#11718156	1	2	yes	—
229	C₄	C₂×#88068	1	1	yes	characteristic
230	C₄	C₂×#88069	1	1	yes	characteristic
231	C₄	#11718893	1	1	yes	characteristic
232	C₄	#11718898	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	#13113513	1	1	yes	characteristic
235	C₂	C₂×#11718156	1	1	yes	characteristic
236	C₂	#13116585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#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	#11983
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ 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
195. C₁₃	#11983	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6090	2
#13113595	2
#13112894	2
#13112885	2
C₁₃×#3170	2
#11983	13