Subgroups — #16789631

Order 3328.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131168	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718262	C₂	1	2	yes	maximal
3	C₄×#88106	C₂	1	1	yes	maximal, characteristic
4	#13129878	C₂	1	2	yes	maximal
5	#13114706	C₂	1	1	yes	maximal, characteristic
6	C₂×#88106	C₂²	1	1	yes	characteristic
7	C₂×#88106	C₂²	1	2	yes	—
8	C₂×#88106	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×#88055	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718262	—	2	4	no	—
14	#11718231	C₄	1	2	yes	complemented
15	C₄⋊#6092	C₄	1	2	yes	complemented
16	C₄⋊#6092	C₄	1	2	yes	complemented
17	C₄×#6092	C₄	1	2	yes	complemented
18	#11718142	C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	—	2	1	no	—
22	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
23	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂³	1	1	yes	characteristic
25	#88106	—	2	2	no	—
26	#88106	—	2	1	no	—
27	#88106	—	2	1	no	—
28	#88106	—	2	2	no	—
29	#88106	D₄	1	2	yes	—
30	#88106	Q₈	1	2	yes	—
31	C₂×#6092	C₂×C₄	1	2	yes	—
32	C₂×#6092	C₂×C₄	1	4	yes	—
33	C₂×#6092	C₂×C₄	1	2	yes	—
34	C₁₃×C₄⋊C₈	—	2	1	no	—
35	C₁₃×C₄⋊C₈	—	2	1	no	—
36	#88071	—	2	1	no	—
37	#88055	—	2	2	no	—
38	#88055	—	2	2	no	—
39	C₄×#1055	C₂×C₄	1	1	yes	characteristic
40	C₄×#1055	C₂×C₄	1	1	yes	characteristic
41	C₄×#1055	C₂×C₄	1	1	yes	characteristic
42	C₄×#1055	C₂×C₄	1	1	yes	characteristic
43	#12666	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	C₄²	1	1	yes	characteristic
52	C₄²×C₁₃	C₄²	1	1	yes	characteristic
53	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
54	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
55	C₄⋊Dic₁₃	—	4	2	no	—
56	C₄×Dic₁₃	—	2	2	no	—
57	C₄×Dic₁₃	—	2	2	no	—
58	C₂×#1055	—	2	1	no	—
59	C₂×#1055	—	2	1	no	—
60	C₂×#1055	C₄²	1	1	yes	characteristic
61	C₂×#1055	C₄²	1	1	yes	characteristic
62	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
63	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
64	C₂×#1055	C₂×D₄	1	1	yes	characteristic
65	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
66	C₂×#1055	#41	1	1	yes	characteristic
67	C₂×#1055	#41	1	1	yes	characteristic
68	#6092	—	2	2	no	—
69	#6092	—	2	4	no	—
70	#6092	C₄⋊C₄	1	4	yes	complemented
71	#2150	—	13	1	no	—
72	C₂×#347	—	13	2	no	—
73	C₄×M₃₂	—	13	1	no	—
74	#1770	—	13	1	no	—
75	#1375	—	13	2	no	—
76	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	4	1	no	—
79	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#97	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
89	C₂×Dic₁₃	—	4	2	no	—
90	C₂×Dic₁₃	—	4	2	no	—
91	C₂×Dic₁₃	—	4	2	no	—
92	C₈×C₁₃	—	4	2	no	—
93	#1055	—	2	1	no	—
94	#1055	—	2	1	no	—
95	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
96	#1055	#117	1	1	yes	characteristic
97	#1055	C₄×D₄	1	1	yes	characteristic
98	#1055	C₄×Q₈	1	1	yes	characteristic
99	C₂×M₃₂	—	13	1	no	—
100	C₂×M₃₂	—	13	1	no	—
101	C₂×M₃₂	—	13	2	no	—
102	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
103	C₂×#97	—	13	2	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₄×C₄⋊C₄	—	13	1	no	—
106	C₄⋊C₁₆	—	13	2	no	—
107	C₄⋊C₁₆	—	13	2	no	—
108	#347	—	26	4	no	—
109	#346	—	13	2	no	—
110	C₄×C₁₆	—	13	2	no	—
111	#336	—	13	2	no	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
115	C₂²×C₁₃	#436	1	1	yes	characteristic
116	C₂²×C₁₃	#446	1	1	yes	characteristic
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	#347	1	2	yes	complemented
121	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₁₃	#405	1	1	yes	characteristic
123	C₄×C₁₃	#436	1	1	yes	characteristic
124	C₄×C₁₃	#446	1	1	yes	characteristic
125	Dic₁₃	—	8	2	no	—
126	C₂²×C₈	—	13	1	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	C₂×#261	1	2	yes	—
129	C₂×C₄⋊C₄	—	26	1	no	—
130	C₂×C₄²	—	13	2	no	—
131	C₂×C₄²	C₂×#261	1	1	yes	characteristic
132	M₃₂	—	13	2	no	—
133	M₃₂	—	13	2	no	—
134	M₃₂	—	26	2	no	—
135	M₃₂	—	26	1	no	—
136	M₃₂	—	26	1	no	—
137	M₃₂	—	26	2	no	—
138	C₂×C₁₆	—	13	2	no	—
139	C₂×C₁₆	—	13	2	no	—
140	C₂×C₁₆	—	13	4	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	2	1	no	—
143	#97	—	26	2	no	—
144	#97	—	26	2	no	—
145	C₄×C₈	—	13	1	no	—
146	C₄×C₈	—	13	1	no	—
147	C₄×C₈	—	13	1	no	—
148	C₄×C₈	—	13	1	no	—
149	#95	—	26	1	no	—
150	C₂×C₁₃	—	2	1	no	—
151	C₂×C₁₃	—	2	1	no	—
152	C₂×C₁₃	#1770	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#347	1	1	yes	characteristic
154	C₂×C₁₃	#2110	1	1	yes	characteristic
155	C₂²×C₄	—	26	2	no	—
156	C₂²×C₄	#6118	1	1	yes	characteristic
157	C₂²×C₄	#6118	1	1	yes	characteristic
158	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	13	1	no	—
163	C₂×C₈	—	13	1	no	—
164	C₂×C₈	—	13	1	no	—
165	C₂×C₈	—	13	1	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	1	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	26	1	no	—
171	C₂×C₈	—	26	1	no	—
172	C₄⋊C₄	—	52	2	no	—
173	C₄²	—	26	2	no	—
174	C₄²	—	26	2	no	—
175	C₄²	C₄×#261	1	1	yes	characteristic
176	C₄²	C₄×#261	1	1	yes	characteristic
177	C₄²	#6122	1	1	yes	characteristic
178	C₄²	#6122	1	1	yes	characteristic
179	C₁₆	—	13	4	no	—
180	C₁₆	—	26	4	no	—
181	C₁₆	—	26	2	no	—
182	C₁₃	#12666	1	1	yes	characteristic, complemented, Sylow
183	C₂³	C₂×#6118	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	52	2	no	—
187	C₂×C₄	—	52	2	no	—
188	C₂×C₄	—	52	2	no	—
189	C₂×C₄	#88110	1	1	yes	characteristic
190	C₂×C₄	#88110	1	1	yes	characteristic
191	C₂×C₄	#88110	1	1	yes	characteristic
192	C₂×C₄	#88110	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
194	C₂×C₄	#88238	1	1	yes	characteristic
195	C₂×C₄	D₄×#261	1	1	yes	characteristic
196	C₂×C₄	#88240	1	1	yes	characteristic
197	C₂×C₄	Q₈×#261	1	1	yes	characteristic
198	C₂×C₄	C₂×#6122	1	1	yes	characteristic
199	C₈	—	4	2	no	—
200	C₈	—	13	1	no	—
201	C₈	—	13	1	no	—
202	C₈	—	13	1	no	—
203	C₈	—	13	1	no	—
204	C₈	—	26	1	no	—
205	C₈	—	26	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#11719076	1	1	yes	characteristic
209	C₂²	#11719080	1	1	yes	characteristic
210	C₂²	C₂×#88110	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	104	2	no	—
215	C₄	#11718262	1	2	yes	complemented
216	C₄	#11719072	1	1	yes	characteristic
217	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
218	C₄	#11719145	1	1	yes	characteristic
219	C₄	#11719158	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#13130632	1	1	yes	characteristic
223	C₂	C₂×#11718262	1	1	yes	characteristic
224	C₂	#13131059	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#12666
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. #11718231	C₄	semidirect
15. C₄⋊#6092	C₄	semidirect
16. C₄⋊#6092	C₄	semidirect
17. C₄×#6092	C₄	semidirect
70. #6092	C₄⋊C₄	semidirect
120. C₄×C₁₃	#347	semidirect
182. C₁₃	#12666	semidirect
215. C₄	#11718262	semidirect

Maximal subgroups

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

subgroup	index
#13131168	2
C₂×#11718262	2
C₄×#88106	2
#13129878	2
#13114706	2
#12666	13