Subgroups — #16789059

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	#13129877	C₂	1	4	yes	maximal, complemented
3	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
5	C₂²×#6099	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂×#88106	—	2	4	no	—
8	C₂×#88106	—	2	4	no	—
9	C₂×#88106	C₄	1	4	yes	complemented
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂²×#6092	C₄	1	4	yes	complemented
12	#11718658	—	2	4	no	—
13	C₂×#88043	C₂²	1	2	yes	—
14	C₂×C₄×#1055	C₄	1	2	yes	—
15	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₁₃×#117	—	2	2	no	—
17	C₁₃×#117	C₈	1	4	yes	—
18	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
19	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
20	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×#6099	—	2	2	no	—
22	C₂×#6099	—	2	2	no	—
23	C₂×#6099	C₈	1	4	yes	—
24	C₂²×#1055	C₂×C₄	1	2	yes	—
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	2	yes	—
28	C₂²×#1055	D₄	1	2	yes	—
29	C₂²×#1055	D₄	1	4	yes	—
30	C₂²×#1055	Q₈	1	2	yes	—
31	C₂²×#1055	C₂³	1	1	yes	characteristic
32	#88106	—	2	8	no	—
33	#88106	—	2	4	no	—
34	#88106	—	2	4	no	—
35	C₂×#6092	—	2	8	no	—
36	C₂×#6092	C₂×C₄	1	8	yes	—
37	#88043	—	2	4	no	—
38	C₄×#1055	—	2	4	no	—
39	#12547	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
48	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₁₃×C₄⋊C₄	—	2	4	no	—
51	C₁₃×C₂²⋊C₄	—	2	4	no	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄²×C₁₃	C₂×C₈	1	4	yes	—
54	#6099	—	2	8	no	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	C₄²	1	1	yes	characteristic
58	C₂×#1055	C₄²	1	1	yes	characteristic
59	C₂×#1055	C₂²⋊C₄	1	2	yes	—
60	C₂×#1055	C₂²⋊C₄	1	4	yes	—
61	C₂×#1055	C₂²⋊C₄	1	4	yes	—
62	C₂×#1055	C₄⋊C₄	1	2	yes	—
63	C₂×#1055	C₄⋊C₄	1	4	yes	—
64	C₂×#1055	C₄⋊C₄	1	4	yes	—
65	C₂×#1055	C₂×C₈	1	4	yes	—
66	C₂×#1055	C₂²×C₄	1	2	yes	—
67	C₂×#1055	C₂×D₄	1	1	yes	characteristic
68	C₂×#1055	C₂×D₄	1	2	yes	—
69	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
70	#6092	—	2	8	no	—
71	C₂²×M₃₂	—	13	2	no	—
72	C₂×#405	—	13	1	no	—
73	#1375	—	13	4	no	—
74	C₂³×C₁₃	—	4	2	no	—
75	C₂³×C₁₃	#97	1	2	yes	—
76	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
77	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
78	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	#95	1	4	yes	—
82	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
83	C₂×C₄×C₁₃	#97	1	2	yes	—
84	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
86	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
88	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
94	#1055	—	2	4	no	—
95	#1055	#95	1	2	yes	—
96	#1055	#95	1	2	yes	—
97	#1055	C₂×C₂²⋊C₄	1	2	yes	—
98	#1055	C₂×C₄⋊C₄	1	2	yes	—
99	C₂²×M₁₆	—	13	1	no	—
100	C₂³×C₈	—	13	1	no	—
101	C₂×#117	#261	1	1	yes	characteristic
102	C₂×M₃₂	—	13	4	no	—
103	C₂×M₃₂	—	13	4	no	—
104	C₂×M₃₂	—	26	4	no	—
105	C₂×M₃₂	—	26	4	no	—
106	C₂²×C₁₆	—	13	4	no	—
107	#405	—	26	4	no	—
108	C₂×#97	—	13	2	no	—
109	C₂×C₄×C₈	—	13	2	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	4	no	—
112	C₂²×C₁₃	—	4	2	no	—
113	C₂²×C₁₃	—	4	2	no	—
114	C₂²×C₁₃	#336	1	4	yes	—
115	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₁₃	—	4	4	no	—
119	C₄×C₁₃	#336	1	4	yes	—
120	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
121	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
122	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
123	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₂³×C₄	C₂×#261	1	1	yes	characteristic
125	C₂×M₁₆	—	13	4	no	—
126	C₂×M₁₆	—	26	2	no	—
127	C₂×M₁₆	—	26	2	no	—
128	C₂²×C₈	—	13	2	no	—
129	C₂²×C₈	—	13	2	no	—
130	C₂²×C₈	—	13	2	no	—
131	C₂²×C₈	—	13	2	no	—
132	C₂²×C₈	—	13	4	no	—
133	C₂²×C₈	—	13	1	no	—
134	C₂²×C₈	—	13	1	no	—
135	C₂²×C₈	—	13	2	no	—
136	#117	—	2	2	no	—
137	#117	#1057	1	4	yes	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₂²⋊C₄	—	2	1	no	—
140	C₂×C₄²	C₂×#261	1	2	yes	—
141	M₃₂	—	26	8	no	—
142	M₃₂	—	26	4	no	—
143	M₃₂	—	26	4	no	—
144	C₂×C₁₆	—	13	8	no	—
145	C₂×C₁₆	—	26	8	no	—
146	#97	—	26	4	no	—
147	C₄×C₈	—	26	4	no	—
148	C₂×C₁₃	—	2	1	no	—
149	C₂×C₁₃	—	2	4	no	—
150	C₂×C₁₃	—	2	1	no	—
151	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
152	C₂×C₁₃	#2110	1	2	yes	—
153	C₂⁴	#6122	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
156	C₂²×C₄	#6118	1	2	yes	—
157	C₂²×C₄	C₄×#261	1	2	yes	—
158	C₂²×C₄	C₄⋊#261	1	2	yes	—
159	C₂²×C₄	C₂×#1057	1	2	yes	—
160	C₂²×C₄	#6122	1	1	yes	characteristic
161	C₂²×C₄	C₂²⋊#261	1	2	yes	—
162	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
163	M₁₆	—	26	8	no	—
164	C₂×C₈	—	13	4	no	—
165	C₂×C₈	—	13	4	no	—
166	C₂×C₈	—	13	4	no	—
167	C₂×C₈	—	13	4	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	13	1	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	4	no	—
177	C₂×C₈	—	26	2	no	—
178	C₂×C₈	—	26	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₄²	—	2	2	no	—
182	C₄²	C₂×#1057	1	4	yes	—
183	C₁₆	—	26	8	no	—
184	C₁₃	#12547	1	1	yes	characteristic, complemented, Sylow
185	C₂³	—	4	2	no	—
186	C₂³	#88110	1	2	yes	—
187	C₂³	#88112	1	2	yes	—
188	C₂³	C₂²⋊#1057	1	2	yes	—
189	C₂³	C₂×#6122	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₄×#1057	1	4	yes	—
193	C₂×C₄	C₄⋊#1057	1	4	yes	—
194	C₂×C₄	#88110	1	2	yes	—
195	C₂×C₄	#88111	1	4	yes	—
196	C₂×C₄	#88112	1	2	yes	—
197	C₂×C₄	(C₂×#261)⋊C₄	1	4	yes	—
198	C₂×C₄	C₂²⋊#1057	1	2	yes	—
199	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×#6118	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
203	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
204	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
205	C₈	—	13	2	no	—
206	C₈	—	13	2	no	—
207	C₈	—	13	2	no	—
208	C₈	—	13	2	no	—
209	C₈	—	26	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	#11718290	1	4	yes	—
215	C₂²	C₂×#88110	1	1	yes	characteristic
216	C₂²	C₂×#88112	1	1	yes	characteristic
217	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
218	C₄	—	4	4	no	—
219	C₄	#11718290	1	4	yes	—
220	C₄	C₂×C₄×#1057	1	1	yes	characteristic
221	C₄	C₂×C₄⋊#1057	1	1	yes	characteristic
222	C₄	C₂×#88111	1	1	yes	characteristic
223	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	4	no	—
226	C₂	—	2	1	no	—
227	C₂	C₂×#11718290	1	1	yes	characteristic
228	C₂	#13131057	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₁₃×#117	C₄
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#12547
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #13129877	C₂	semidirect
9. C₂×#88106	C₄	semidirect
11. C₂²×#6092	C₄	semidirect
184. C₁₃	#12547	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
#13129877	2
C₂×#11718658	2
#12547	13