Subgroups — #15761812

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115773	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#1055	C₂	1	2	yes	maximal
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₄×C₁₃×Q₁₆	—	2	2	no	—
9	C₄×C₁₃×Q₁₆	—	2	2	no	—
10	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
11	C₂×C₁₃×#103	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	Q₈⋊#1055	—	2	4	no	—
17	#11718116	—	2	4	no	—
18	#11718050	—	2	2	no	—
19	#11718046	—	2	2	no	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×Q₁₆	—	2	4	no	—
22	C₂×C₁₃×Q₁₆	—	2	2	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₃	—	2	2	no	—
26	C₄×Q₈×C₁₃	—	2	2	no	—
27	C₄×Q₈×C₁₃	—	2	2	no	—
28	C₄×Q₈×C₁₃	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×#1055	—	2	1	no	—
34	C₂²×#1055	—	2	2	no	—
35	C₂²×#1055	D₄	1	2	yes	—
36	C₁₃×#107	—	2	2	no	—
37	C₁₃×#103	—	2	4	no	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	C₄⋊#1055	—	2	2	no	—
41	C₄⋊#1055	—	2	2	no	—
42	C₄×#1055	—	2	1	no	—
43	C₄×#1055	—	2	1	no	—
44	C₄×#1055	—	2	1	no	—
45	C₄×#1055	D₄	1	2	yes	—
46	#9662	—	13	1	no	maximal, Sylow
47	C₂×Q₈×C₁₃	—	2	2	no	—
48	C₂×Q₈×C₁₃	—	2	2	no	—
49	C₂×Q₈×C₁₃	—	2	2	no	—
50	C₂×Q₈×C₁₃	—	2	2	no	—
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
58	C₁₃×Q₁₆	—	2	8	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₄²×C₁₃	—	4	2	no	—
68	C₄²×C₁₃	—	4	2	no	—
69	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₃	#41	1	1	yes	characteristic
73	C₂×#1055	—	2	2	no	—
74	C₂×#1055	—	2	2	no	—
75	C₂×#1055	—	2	2	no	—
76	C₂×#1055	—	4	2	no	—
77	C₂×#1055	—	4	2	no	—
78	C₂×#1055	—	4	4	no	—
79	C₂×#1055	Q₁₆	1	4	yes	complemented
80	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
81	#1984	—	13	2	no	—
82	C₂×#335	—	13	1	no	—
83	C₂×Q₈⋊C₈	—	13	2	no	—
84	C₂×#322	—	13	1	no	—
85	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₁₃	—	2	4	no	—
87	Q₈×C₁₃	—	2	4	no	—
88	Q₈×C₁₃	—	4	2	no	—
89	Q₈×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₁₃	#131	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₈×C₁₃	—	2	2	no	—
107	C₈×C₁₃	—	2	2	no	—
108	#1055	—	2	4	no	—
109	#1055	—	4	2	no	—
110	#1055	—	8	2	no	—
111	C₂²×Q₁₆	Dic₁₃	1	1	yes	characteristic
112	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
113	C₄×Q₁₆	—	2	2	no	—
114	C₄×Q₁₆	—	2	2	no	—
115	C₂×#107	Dic₁₃	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	13	2	no	—
117	C₂×#103	Dic₁₃	1	2	yes	—
118	C₂×C₄×C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	1	no	—
120	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
121	#336	—	26	4	no	—
122	#335	—	26	2	no	—
123	Q₈⋊C₈	—	26	4	no	—
124	#322	—	26	2	no	—
125	C₂²×C₁₃	#390	1	1	yes	characteristic
126	C₂²×C₁₃	#435	1	2	yes	—
127	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
128	C₂²×C₁₃	#444	1	2	yes	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	—	4	4	no	—
132	C₄×C₁₃	—	8	2	no	—
133	C₄×C₁₃	#407	1	1	yes	characteristic
134	C₄×C₁₃	#408	1	1	yes	characteristic
135	C₄×C₁₃	#435	1	2	yes	—
136	C₄×C₁₃	#467	1	2	yes	—
137	C₄×C₁₃	#494	1	1	yes	characteristic
138	C₄×C₁₃	#495	1	1	yes	characteristic
139	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
140	C₂×Q₁₆	—	2	4	no	—
141	C₂×Q₁₆	—	2	2	no	—
142	C₂²×C₈	—	13	2	no	—
143	C₂²×C₈	—	26	1	no	—
144	C₂²×C₈	—	26	2	no	—
145	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
155	#107	—	2	2	no	—
156	C₄⋊C₈	—	26	2	no	—
157	C₄⋊C₈	—	26	2	no	—
158	#103	—	2	4	no	—
159	C₄×C₈	—	13	2	no	—
160	C₄×C₈	—	26	1	no	—
161	C₄×C₈	—	26	1	no	—
162	C₄×C₈	—	26	1	no	—
163	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
164	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
165	C₂×C₁₃	#1584	1	4	yes	—
166	C₂×C₁₃	#1955	1	1	yes	characteristic
167	C₂×C₁₃	#1969	1	1	yes	characteristic
168	C₂×C₁₃	#1975	1	1	yes	characteristic
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	#6099	1	4	yes	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₄	#6129	1	1	yes	characteristic
179	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
180	Q₁₆	—	2	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	13	4	no	—
184	C₂×C₈	—	26	2	no	—
185	C₂×C₈	—	26	2	no	—
186	C₂×C₈	—	26	2	no	—
187	C₂×C₈	—	52	2	no	—
188	C₂×C₈	—	52	4	no	—
189	C₂×C₈	—	52	2	no	—
190	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
191	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	D₄×D₁₃	1	1	yes	characteristic
199	C₄²	#6129	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
202	C₁₃	#9662	1	1	yes	characteristic, complemented, Sylow
203	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
215	C₂×C₄	#88173	1	2	yes	—
216	C₂×C₄	C₂×#6099	1	2	yes	—
217	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
218	C₂×C₄	#88190	1	1	yes	characteristic
219	C₂×C₄	#88192	1	1	yes	characteristic
220	C₂×C₄	#88192	1	1	yes	characteristic
221	C₂×C₄	#88194	1	1	yes	characteristic
222	C₂×C₄	#88202	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	26	4	no	—
227	C₈	—	52	2	no	—
228	C₈	—	104	2	no	—
229	C₂²	#11718672	1	2	yes	—
230	C₂²	Q₁₆×Dic₁₃	1	2	yes	—
231	C₂²	#11718858	1	2	yes	—
232	C₂²	#11718882	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	#11718672	1	2	yes	—
238	C₄	#11718842	1	1	yes	characteristic
239	C₄	#11718849	1	2	yes	—
240	C₄	#11718852	1	1	yes	characteristic
241	C₄	#11718868	1	1	yes	characteristic
242	C₄	#11718895	1	1	yes	characteristic
243	C₂	#13113687	1	4	yes	—
244	C₂	#13115635	1	1	yes	characteristic
245	C₂	#13115698	1	1	yes	characteristic
246	C₂	#13115730	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#9662
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
79. C₂×#1055	Q₁₆	semidirect
202. C₁₃	#9662	semidirect

Maximal subgroups

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

subgroup	index
#13115773	2
C₂×#11718050	2
C₂×Q₈⋊#1055	2
C₂×#11718046	2
C₂×C₄×C₁₃×Q₁₆	2
#9662	13