Subgroups — #16782405

Order 3328.

462 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130769	C₂	1	2	yes	maximal
2	C₂×#11718255	C₂	1	2	yes	maximal
3	C₂×#11718253	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#88112	—	2	4	no	—
6	C₂×C₄⋊#1057	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₄	1	1	yes	characteristic
14	#11718255	—	2	4	no	—
15	#11718253	—	2	4	no	—
16	C₂²×#1057	—	2	4	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
22	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
23	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	#88112	—	4	8	no	—
27	C₄⋊#1057	—	2	4	no	—
28	C₄⋊#1057	—	2	4	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₈×Dic₁₃	—	2	2	no	—
31	C₈×Dic₁₃	C₈	1	4	yes	complemented
32	C₈×Dic₁₃	D₄	1	2	yes	—
33	C₈×Dic₁₃	D₄	1	2	yes	—
34	C₈×Dic₁₃	Q₈	1	2	yes	—
35	C₈×Dic₁₃	Q₈	1	2	yes	—
36	C₄×#1055	—	2	2	no	—
37	#8805	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
41	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×#1057	—	4	4	no	—
44	C₂×#1057	—	4	8	no	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
51	C₄²×C₁₃	—	2	1	no	—
52	C₄²×C₁₃	—	2	1	no	—
53	C₄×Dic₁₃	—	2	4	no	—
54	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
55	C₄×Dic₁₃	M₁₆	1	2	yes	—
56	C₄×Dic₁₃	SD₁₆	1	8	yes	—
57	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×#1055	—	2	1	no	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	C₄⋊C₄	1	2	yes	—
64	C₂×#1055	C₄⋊C₄	1	2	yes	—
65	C₂×#1055	C₂×C₈	1	2	yes	—
66	C₂×#1055	M₁₆	1	2	yes	—
67	#1842	—	13	2	no	—
68	C₄²×C₈	—	13	1	no	—
69	C₂×#334	—	13	2	no	—
70	C₂×#334	—	13	2	no	—
71	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	#106	1	4	yes	—
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₁₃	—	2	2	no	—
82	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
83	C₂×Dic₁₃	#106	1	4	yes	—
84	C₂×Dic₁₃	#108	1	4	yes	—
85	C₂×Dic₁₃	#127	1	1	yes	characteristic
86	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×Dic₁₃	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×Dic₁₃	C₂×SD₁₆	1	4	yes	—
90	#1057	—	8	4	no	—
91	C₈×C₁₃	—	2	2	no	—
92	C₈×C₁₃	C₄⋊C₈	1	4	yes	complemented
93	#1055	—	2	2	no	—
94	#1055	C₄⋊C₈	1	4	yes	complemented
95	C₂×C₄⋊C₈	—	13	4	no	—
96	C₂×C₄⋊C₈	—	26	4	no	—
97	C₂×C₄×C₈	—	13	1	no	—
98	C₂×C₄×C₈	—	13	1	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₈	—	13	1	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	#261	1	1	yes	characteristic
103	C₄³	—	13	1	no	—
104	#334	—	26	4	no	—
105	#334	—	26	4	no	—
106	C₂²×C₁₃	#384	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₁₃	#423	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×#106	1	2	yes	—
110	C₂²×C₁₃	C₂×#108	1	2	yes	—
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₁₃	#423	1	1	yes	characteristic
115	C₄×C₁₃	#423	1	1	yes	characteristic
116	Dic₁₃	—	2	1	no	—
117	Dic₁₃	—	2	1	no	—
118	Dic₁₃	#334	1	8	yes	complemented
119	Dic₁₃	#492	1	2	yes	—
120	Dic₁₃	#498	1	2	yes	—
121	C₂²×C₈	—	13	1	no	—
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	—	26	4	no	—
124	C₂²×C₈	C₂×#261	1	1	yes	characteristic
125	C₂²×C₈	C₂×#261	1	1	yes	characteristic
126	C₂×C₄²	—	13	2	no	—
127	C₂×C₄²	—	13	1	no	—
128	C₂×C₄²	—	13	1	no	—
129	C₂×C₄²	—	13	2	no	—
130	C₂×C₄²	C₂×#261	1	1	yes	characteristic
131	C₄⋊C₈	—	26	4	no	—
132	C₄⋊C₈	—	26	4	no	—
133	C₄⋊C₈	—	52	8	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	13	4	no	—
136	C₄×C₈	—	13	2	no	—
137	C₄×C₈	—	13	2	no	—
138	C₄×C₈	—	13	2	no	—
139	C₄×C₈	—	13	2	no	—
140	C₄×C₈	—	26	2	no	—
141	C₄×C₈	—	26	2	no	—
142	C₂×C₁₃	C₂×#334	1	2	yes	—
143	C₂×C₁₃	#1566	1	2	yes	—
144	C₂×C₁₃	#1842	1	1	yes	characteristic
145	C₂×C₁₃	#1844	1	1	yes	characteristic
146	C₂×C₁₃	#1848	1	1	yes	characteristic
147	C₂²×C₄	—	13	1	no	—
148	C₂²×C₄	—	13	2	no	—
149	C₂²×C₄	—	13	1	no	—
150	C₂²×C₄	C₄⋊#261	1	2	yes	—
151	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	13	2	no	—
155	C₂×C₈	—	13	2	no	—
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	13	2	no	—
158	C₂×C₈	—	26	1	no	—
159	C₂×C₈	—	26	1	no	—
160	C₂×C₈	—	52	4	no	—
161	C₂×C₈	—	52	8	no	—
162	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
163	C₂×C₈	C₄⋊#261	1	2	yes	—
164	C₂×C₈	C₄⋊#261	1	2	yes	—
165	C₂×C₈	#6122	1	2	yes	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	13	8	no	—
169	C₄²	—	13	2	no	—
170	C₄²	—	13	2	no	—
171	C₄²	—	13	1	no	—
172	C₄²	—	13	1	no	—
173	C₄²	—	13	1	no	—
174	C₄²	—	13	1	no	—
175	C₄²	—	26	4	no	—
176	C₁₃	#8805	1	1	yes	characteristic, complemented, Sylow
177	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	13	4	no	—
181	C₂×C₄	—	13	1	no	—
182	C₂×C₄	—	13	1	no	—
183	C₂×C₄	—	13	4	no	—
184	C₂×C₄	—	13	4	no	—
185	C₂×C₄	—	13	4	no	—
186	C₂×C₄	—	13	1	no	—
187	C₂×C₄	—	13	1	no	—
188	C₂×C₄	—	26	2	no	—
189	C₂×C₄	#88101	1	4	yes	—
190	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
191	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
192	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
195	C₂×C₄	C₂×#6122	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	—	13	4	no	—
198	C₈	—	26	2	no	—
199	C₈	—	104	4	no	—
200	C₈	#88112	1	4	yes	complemented
201	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
202	C₂²	#11719059	1	1	yes	characteristic
203	C₂²	C₂×#88101	1	2	yes	—
204	C₂²	D₁₃⋊#108	1	2	yes	—
205	C₂²	#11719129	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	—	13	2	no	—
208	C₄	—	13	2	no	—
209	C₄	—	13	8	no	—
210	C₄	—	26	1	no	—
211	C₄	—	26	1	no	—
212	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
213	C₄	#11719053	1	1	yes	characteristic
214	C₄	C₂×#88112	1	1	yes	characteristic
215	C₄	#11719129	1	1	yes	characteristic
216	C₂	D₁₃⋊#334	1	2	yes	—
217	C₂	#13130390	1	2	yes	—
218	C₂	#13130769	1	1	yes	characteristic
219	C₂	#13130771	1	1	yes	characteristic
220	C₂	#13130776	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#8805
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
31. C₈×Dic₁₃	C₈	semidirect
92. C₈×C₁₃	C₄⋊C₈	semidirect
94. #1055	C₄⋊C₈	semidirect
118. Dic₁₃	#334	semidirect
176. C₁₃	#8805	semidirect
200. C₈	#88112	semidirect

Maximal subgroups

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

subgroup	index
#13130769	2
C₂×#11718255	2
C₂×#11718253	2
C₄×C₈×Dic₁₃	2
#8805	13