Subgroups — #16784416

Order 3328.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130933	C₂	1	2	yes	maximal
2	C₂×#11718254	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718251	C₂	1	1	yes	maximal, characteristic
4	#13129672	C₂	1	2	yes	maximal
5	#106×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#1057	—	2	2	no	—
7	C₂×C₄⋊#1057	C₂²	1	2	yes	—
8	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
9	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#106	C₄	1	1	yes	characteristic
11	C₂×#88057	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×Dic₁₃	C₂²	1	2	yes	—
14	#11718290	—	2	2	no	—
15	#11718254	—	2	2	no	—
16	#11718251	—	2	2	no	—
17	#11718121	C₄	1	2	yes	—
18	C₂²×#1057	—	2	1	no	—
19	C₂²×#1057	—	2	2	no	—
20	C₂²×#1057	D₄	1	1	yes	characteristic
21	C₂²×#1057	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₁₃	—	2	2	no	—
26	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₄⋊#1057	—	2	2	no	—
29	C₄⋊#1057	—	2	2	no	—
30	C₄⋊#1057	—	4	4	no	—
31	C₄×#1057	—	2	1	no	—
32	C₄×#1057	—	2	1	no	—
33	C₄×#1057	—	2	1	no	—
34	C₄×#1057	Q₈	1	2	yes	—
35	C₁₃×#106	C₈	1	4	yes	complemented
36	#88071	—	2	2	no	—
37	#88057	—	2	2	no	—
38	C₈×Dic₁₃	D₄	1	2	yes	—
39	C₈×Dic₁₃	Q₈	1	2	yes	—
40	#9596	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	#41	1	1	yes	characteristic
46	C₂×#1057	—	2	1	no	—
47	C₂×#1057	—	2	1	no	—
48	C₂×#1057	—	2	2	no	—
49	C₂×#1057	—	2	2	no	—
50	C₂×#1057	—	2	2	no	—
51	C₂×#1057	—	4	2	no	—
52	C₂×#1057	—	4	2	no	—
53	C₂×#1057	—	4	2	no	—
54	C₂×#1057	—	4	2	no	—
55	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
60	C₄⋊Dic₁₃	—	2	2	no	—
61	C₄⋊Dic₁₃	—	2	2	no	—
62	C₄×Dic₁₃	—	4	2	no	—
63	C₄×Dic₁₃	—	4	2	no	—
64	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
65	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
67	C₄×Dic₁₃	#41	1	1	yes	characteristic
68	C₂×#1055	—	2	1	no	—
69	C₂×#1055	—	2	1	no	—
70	C₂×#1055	C₄⋊C₄	1	2	yes	—
71	#1939	—	13	2	no	—
72	C₄×#106	—	13	1	no	—
73	C₂×#335	—	13	1	no	—
74	C₂×#322	—	13	1	no	—
75	#1279	—	13	2	no	—
76	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	4	2	no	—
79	C₂×C₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	—	4	2	no	—
88	C₂×Dic₁₃	—	4	2	no	—
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	#121	1	1	yes	characteristic
91	C₂×Dic₁₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×Dic₁₃	#125	1	1	yes	characteristic
93	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×Dic₁₃	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×Dic₁₃	#137	1	2	yes	—
96	#1057	—	4	2	no	—
97	#1057	—	4	1	no	—
98	#1057	—	4	1	no	—
99	#1057	—	8	2	no	—
100	C₈×C₁₃	C₄⋊C₈	1	4	yes	complemented
101	#1055	—	2	2	no	—
102	C₂×#106	—	13	1	no	—
103	C₂×#106	#261	1	1	yes	characteristic
104	C₂×C₄⋊C₈	—	13	2	no	—
105	C₂×C₄⋊C₈	—	26	2	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₄×C₄⋊C₄	—	13	2	no	—
110	#340	—	13	2	no	—
111	#336	—	26	2	no	—
112	#335	—	26	2	no	—
113	#322	—	26	2	no	—
114	C₂²×C₁₃	#389	1	1	yes	characteristic
115	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
116	C₂²×C₁₃	#436	1	1	yes	characteristic
117	C₂²×C₁₃	#440	1	2	yes	—
118	C₂²×C₁₃	#444	1	2	yes	—
119	C₄×C₁₃	—	4	4	no	—
120	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₄×C₁₃	#424	1	1	yes	characteristic
122	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
123	C₄×C₁₃	#446	1	1	yes	characteristic
124	Dic₁₃	—	2	1	no	—
125	Dic₁₃	—	2	1	no	—
126	Dic₁₃	—	8	2	no	—
127	Dic₁₃	#469	1	1	yes	characteristic
128	Dic₁₃	#470	1	1	yes	characteristic
129	Dic₁₃	#501	1	2	yes	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	13	1	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₂²×C₈	—	26	1	no	—
134	C₂²×C₈	—	26	2	no	—
135	C₂²×C₈	C₂×#261	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	13	2	no	—
137	C₂×C₄⋊C₄	C₂×#261	1	2	yes	—
138	C₂×C₄²	—	13	1	no	—
139	C₂×C₄²	—	26	2	no	—
140	#106	—	26	2	no	—
141	#106	#1057	1	4	yes	complemented
142	C₄⋊C₈	—	26	2	no	—
143	C₄⋊C₈	—	26	2	no	—
144	C₄⋊C₈	—	52	4	no	—
145	C₄×C₈	—	13	2	no	—
146	C₄×C₈	—	13	2	no	—
147	C₄×C₈	—	13	2	no	—
148	C₄×C₈	—	26	1	no	—
149	C₄×C₈	—	26	1	no	—
150	C₄×C₈	—	26	1	no	—
151	#95	—	26	2	no	—
152	C₂×C₁₃	SD₁₆⋊C₈	1	2	yes	—
153	C₂×C₁₃	#1592	1	2	yes	—
154	C₂×C₁₃	#1939	1	1	yes	characteristic
155	C₂×C₁₃	#1944	1	1	yes	characteristic
156	C₂×C₁₃	#1946	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂²×C₄	—	26	2	no	—
161	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	26	1	no	—
164	C₂×C₈	—	26	1	no	—
165	C₂×C₈	—	26	2	no	—
166	C₂×C₈	—	26	1	no	—
167	C₂×C₈	—	26	1	no	—
168	C₂×C₈	—	26	2	no	—
169	C₂×C₈	—	26	2	no	—
170	C₂×C₈	—	52	2	no	—
171	C₂×C₈	—	52	2	no	—
172	C₂×C₈	—	52	2	no	—
173	C₂×C₈	—	52	2	no	—
174	C₂×C₈	#6118	1	2	yes	—
175	C₂×C₈	C₄⋊#261	1	2	yes	—
176	C₂×C₈	C₂×#1057	1	2	yes	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	26	2	no	—
179	C₄⋊C₄	—	26	2	no	—
180	C₄⋊C₄	C₂×#1057	1	4	yes	—
181	C₄²	—	13	1	no	—
182	C₄²	—	13	1	no	—
183	C₄²	—	13	1	no	—
184	C₄²	—	13	1	no	—
185	C₄²	—	52	2	no	—
186	C₄²	—	52	2	no	—
187	C₁₃	#9596	1	1	yes	characteristic, complemented, Sylow
188	C₂³	D₄×#261	1	1	yes	characteristic
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	2	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	13	1	no	—
198	C₂×C₄	—	26	2	no	—
199	C₂×C₄	—	52	2	no	—
200	C₂×C₄	—	52	2	no	—
201	C₂×C₄	—	52	2	no	—
202	C₂×C₄	C₂×#6118	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
204	C₂×C₄	#88238	1	1	yes	characteristic
205	C₂×C₄	#88240	1	1	yes	characteristic
206	C₂×C₄	Q₈×#261	1	1	yes	characteristic
207	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
208	C₈	—	26	2	no	—
209	C₈	—	52	1	no	—
210	C₈	—	52	1	no	—
211	C₈	—	52	2	no	—
212	C₈	—	104	2	no	—
213	C₈	C₄⋊#1057	1	4	yes	complemented
214	C₂²	#11719076	1	1	yes	characteristic
215	C₂²	#11719083	1	1	yes	characteristic
216	C₂²	#11719106	1	2	yes	—
217	C₂²	#11719108	1	2	yes	—
218	C₂²	D₄×#1057	1	1	yes	characteristic
219	C₄	—	4	4	no	—
220	C₄	—	13	1	no	—
221	C₄	—	13	1	no	—
222	C₄	—	13	2	no	—
223	C₄	—	26	1	no	—
224	C₄	—	26	1	no	—
225	C₄	—	104	2	no	—
226	C₄	#11719079	1	1	yes	characteristic
227	C₄	#11719080	1	1	yes	characteristic
228	C₄	C₂×C₄⋊#1057	1	1	yes	characteristic
229	C₄	Q₈×#1057	1	1	yes	characteristic
230	C₂	SD₁₆⋊#1057	1	2	yes	—
231	C₂	#13130478	1	2	yes	—
232	C₂	#13130933	1	1	yes	characteristic
233	C₂	#13130942	1	1	yes	characteristic
234	C₂	#13130946	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#106	C₄
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×#106	#261
hypocenter	C₁₃	#9596
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
35. C₁₃×#106	C₈	semidirect
100. C₈×C₁₃	C₄⋊C₈	semidirect
141. #106	#1057	semidirect
187. C₁₃	#9596	semidirect
213. C₈	C₄⋊#1057	semidirect

Maximal subgroups

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

subgroup	index
#13130933	2
C₂×#11718254	2
C₂×#11718251	2
#13129672	2
#106×Dic₁₃	2
#9596	13