Subgroups — #15754234

Order 3328.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113584	C₂	1	1	yes	maximal, characteristic
2	#13113123	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#1055	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
5	#13112674	C₂	1	1	yes	maximal, characteristic
6	#13112674	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#108	C₄	1	2	yes	—
9	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718586	C₂²	1	1	yes	characteristic
13	#11718586	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₁₃×#341	C₄	1	2	yes	—
17	#11718050	C₄	1	2	yes	—
18	#11718049	C₄	1	2	yes	—
19	#11718047	—	2	1	no	—
20	#11718046	—	2	1	no	—
21	#11718046	—	2	1	no	—
22	#11718046	C₄	1	2	yes	—
23	#11718046	C₄	1	2	yes	—
24	C₈×#1055	—	2	1	no	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×#1055	—	2	1	no	—
31	C₂²×#1055	Q₈	1	2	yes	—
32	C₁₃×#108	—	2	4	no	—
33	C₁₃×#97	—	2	1	no	—
34	C₁₃×#97	—	2	1	no	—
35	C₄×C₈×C₁₃	—	2	1	no	—
36	C₄×C₈×C₁₃	—	2	1	no	—
37	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²⋊#1055	—	2	1	no	—
42	C₂²⋊#1055	—	2	1	no	—
43	C₄⋊#1055	C₂×C₄	1	2	yes	—
44	C₄⋊#1055	C₂×C₄	1	2	yes	—
45	C₄×#1055	—	2	1	no	—
46	C₄×#1055	—	2	1	no	—
47	C₄×#1055	—	2	1	no	—
48	C₄×#1055	C₂×C₄	1	2	yes	—
49	C₄×#1055	C₂×C₄	1	2	yes	—
50	C₄×#1055	D₄	1	1	yes	characteristic
51	C₄×#1055	D₄	1	1	yes	characteristic
52	#8296	—	13	1	no	maximal, Sylow
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
56	C₁₃×M₁₆	—	2	2	no	—
57	C₁₃×M₁₆	—	2	2	no	—
58	C₁₃×M₁₆	—	2	2	no	—
59	C₁₃×M₁₆	M₁₆	1	4	yes	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
68	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	#41	1	1	yes	characteristic
72	C₂×#1055	—	2	4	no	—
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	C₄⋊C₄	1	2	yes	—
78	C₂×#1055	C₄⋊C₄	1	2	yes	—
79	C₄×#108	D₁₃	1	1	yes	characteristic
80	#1564	—	13	1	no	—
81	#1451	—	13	1	no	—
82	C₈×M₁₆	—	13	1	no	—
83	C₂×#322	—	13	1	no	—
84	#1278	—	13	1	no	—
85	#1278	—	13	1	no	—
86	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₁₃	#131	1	2	yes	—
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	—	2	1	no	—
101	C₈×C₁₃	—	2	4	no	—
102	C₈×C₁₃	—	4	2	no	—
103	#1055	—	2	4	no	—
104	#1055	—	2	2	no	—
105	#1055	—	2	2	no	—
106	#1055	—	4	2	no	—
107	#1055	—	4	2	no	—
108	#431	—	13	1	no	—
109	#431	—	13	1	no	—
110	C₂×#108	Dic₁₃	1	2	yes	—
111	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
112	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
113	C₂×C₄×C₈	—	13	1	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	#341	Dic₁₃	1	2	yes	—
117	#335	—	13	2	no	—
118	#334	—	13	2	no	—
119	#322	—	13	2	no	—
120	#322	—	13	2	no	—
121	#322	—	26	1	no	—
122	#322	—	26	1	no	—
123	#322	—	26	1	no	—
124	C₈²	—	26	1	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₁₃	#446	1	2	yes	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
132	C₄×C₁₃	#405	1	1	yes	characteristic
133	C₄×C₁₃	#430	1	2	yes	—
134	C₄×C₁₃	#435	1	2	yes	—
135	C₄×C₁₃	#443	1	2	yes	—
136	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
137	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
138	C₂²×C₈	—	13	2	no	—
139	C₂²×C₈	—	26	1	no	—
140	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
141	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
142	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
143	#108	—	2	4	no	—
144	C₄⋊C₈	—	13	2	no	—
145	C₄⋊C₈	—	13	2	no	—
146	C₂²⋊C₈	—	26	1	no	—
147	C₂²⋊C₈	—	26	1	no	—
148	#97	—	2	1	no	—
149	#97	—	2	1	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	13	2	no	—
153	C₄×C₈	—	13	2	no	—
154	C₄×C₈	—	13	1	no	—
155	C₄×C₈	—	13	1	no	—
156	C₄×C₈	—	26	1	no	—
157	C₄×C₈	—	26	1	no	—
158	C₄×C₈	—	26	1	no	—
159	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
160	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
161	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
162	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
163	C₂×C₁₃	—	2	2	no	—
164	C₂×C₁₃	#1771	1	1	yes	characteristic
165	C₂×C₁₃	#1772	1	1	yes	characteristic
166	C₂×C₁₃	#1778	1	1	yes	characteristic
167	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
168	C₂²×C₄	#6131	1	1	yes	characteristic
169	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
170	M₁₆	—	2	2	no	—
171	M₁₆	—	2	2	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	#6099	1	4	yes	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	13	2	no	—
180	C₂×C₈	—	26	4	no	—
181	C₂×C₈	—	26	2	no	—
182	C₂×C₈	—	26	2	no	—
183	C₂×C₈	—	26	2	no	—
184	C₂×C₈	—	52	2	no	—
185	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
186	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
187	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
188	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
189	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
190	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
191	C₄²	D₄×D₁₃	1	1	yes	characteristic
192	C₄²	#6129	1	1	yes	characteristic
193	C₁₃	#8296	1	1	yes	characteristic, complemented, Sylow
194	C₂³	Q₈×Dic₁₃	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	#88141	1	1	yes	characteristic
198	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
199	C₂×C₄	#88146	1	1	yes	characteristic
200	C₂×C₄	#88147	1	1	yes	characteristic
201	C₂×C₄	C₂×#6099	1	2	yes	—
202	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
203	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
204	C₂×C₄	#88202	1	2	yes	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	4	no	—
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	4	2	no	—
211	C₈	—	26	4	no	—
212	C₈	—	26	2	no	—
213	C₈	—	26	2	no	—
214	C₈	—	52	2	no	—
215	C₈	—	52	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
219	C₂²	#11718685	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	#11718554	1	2	yes	—
223	C₄	#11718556	1	2	yes	—
224	C₄	C₄×#6099	1	1	yes	characteristic
225	C₄	#11718658	1	1	yes	characteristic
226	C₄	#11718672	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	#13114711	1	1	yes	characteristic
229	C₂	#13114718	1	1	yes	characteristic
230	C₂	#13114744	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₄×C₄⋊C₄
prime radical	C₂³×C₁₃	C₄×Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#8296
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
193. C₁₃	#8296	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
#13113123	2
M₁₆×#1055	2
C₂×#11718046	2
#13112674	2
C₄×C₁₃×#108	2
#8296	13