Subgroups — #16789048

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 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	1	yes	maximal, characteristic
2	C₂³×#6092	C₂	1	1	yes	maximal, characteristic
3	#13129874	C₂	1	4	yes	maximal, complemented
4	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
6	C₂²×#6099	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂×#88106	—	2	2	no	—
9	C₂×#88106	—	2	2	no	—
10	C₂×#88106	C₄	1	4	yes	complemented
11	C₂²×#6092	—	2	2	no	—
12	C₂²×#6092	—	2	2	no	—
13	C₂²×#6092	C₄	1	4	yes	complemented
14	C₂²×#6092	C₂²	1	2	yes	—
15	C₂²×#6092	C₂²	1	2	yes	—
16	#11718658	—	2	2	no	—
17	#11718658	—	2	2	no	—
18	C₂×#88043	C₄	1	2	yes	—
19	C₂×C₄×#1055	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₁₃×#117	—	2	2	no	—
22	C₁₃×#117	—	2	2	no	—
23	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
24	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
26	C₂×#6099	—	2	2	no	—
27	C₂×#6099	—	2	2	no	—
28	C₂×#6099	C₈	1	4	yes	—
29	C₂²×#1055	C₂×C₄	1	2	yes	—
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂×C₄	1	2	yes	—
32	C₂²×#1055	C₂×C₄	1	2	yes	—
33	C₂²×#1055	D₄	1	2	yes	—
34	C₂²×#1055	D₄	1	2	yes	—
35	C₂²×#1055	D₄	1	2	yes	—
36	C₂²×#1055	Q₈	1	2	yes	—
37	C₂²×#1055	C₂³	1	1	yes	characteristic
38	#88106	—	2	8	no	—
39	C₂×#6092	—	2	8	no	—
40	C₂×#6092	—	2	2	no	—
41	C₂×#6092	—	2	2	no	—
42	C₂×#6092	—	2	2	no	—
43	C₂×#6092	—	2	2	no	—
44	C₂×#6092	C₂×C₄	1	4	yes	—
45	C₂×#6092	C₂×C₄	1	4	yes	—
46	#88043	—	2	4	no	—
47	C₄×#1055	—	2	2	no	—
48	C₄×#1055	—	2	2	no	—
49	#12545	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
58	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	#6099	—	2	8	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	2	no	—
67	C₂×#1055	C₄²	1	2	yes	—
68	C₂×#1055	C₂²⋊C₄	1	2	yes	—
69	C₂×#1055	C₂²⋊C₄	1	4	yes	—
70	C₂×#1055	C₂²⋊C₄	1	4	yes	—
71	C₂×#1055	C₄⋊C₄	1	2	yes	—
72	C₂×#1055	C₄⋊C₄	1	4	yes	—
73	C₂×#1055	C₄⋊C₄	1	4	yes	—
74	C₂×#1055	C₂×C₈	1	4	yes	—
75	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
76	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
77	C₂×#1055	C₂×D₄	1	1	yes	characteristic
78	C₂×#1055	C₂×D₄	1	1	yes	characteristic
79	C₂×#1055	C₂×D₄	1	1	yes	characteristic
80	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
81	#6092	—	2	4	no	—
82	#6092	—	2	4	no	—
83	C₂²×M₃₂	—	13	1	no	—
84	C₂³×C₁₆	—	13	1	no	—
85	C₂×#405	—	13	1	no	—
86	#1374	—	13	4	no	—
87	C₂³×C₁₃	—	2	4	no	—
88	C₂³×C₁₃	C₄×C₈	1	2	yes	—
89	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
90	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
91	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	8	no	—
94	C₂×C₄×C₁₃	#95	1	4	yes	—
95	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
96	C₂×C₄×C₁₃	#97	1	4	yes	—
97	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
101	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	#1055	—	2	4	no	—
108	#1055	#95	1	4	yes	—
109	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	#1055	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
112	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×M₁₆	—	13	1	no	—
114	C₂³×C₈	—	13	1	no	—
115	C₂×#117	#261	1	1	yes	characteristic
116	C₂×M₃₂	—	13	4	no	—
117	C₂×M₃₂	—	26	2	no	—
118	C₂×M₃₂	—	26	2	no	—
119	C₂²×C₁₆	—	13	2	no	—
120	C₂²×C₁₆	—	13	4	no	—
121	C₂²×C₁₆	—	13	2	no	—
122	C₂²×C₁₆	—	26	2	no	—
123	C₂²×C₁₆	—	26	2	no	—
124	#405	—	26	2	no	—
125	#405	—	26	2	no	—
126	C₂×#97	—	13	2	no	—
127	C₂×C₄×C₈	—	13	2	no	—
128	C₂²×C₁₃	—	2	4	no	—
129	C₂²×C₁₃	—	2	4	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	#336	1	4	yes	—
132	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₂²×C₁₃	#504	1	4	yes	—
136	C₄×C₁₃	—	4	4	no	—
137	C₄×C₁₃	#336	1	4	yes	—
138	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
139	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
140	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₂³×C₄	C₂×#261	1	1	yes	characteristic
143	C₂×M₁₆	—	13	4	no	—
144	C₂×M₁₆	—	26	2	no	—
145	C₂×M₁₆	—	26	2	no	—
146	C₂²×C₈	—	13	2	no	—
147	C₂²×C₈	—	13	2	no	—
148	C₂²×C₈	—	13	2	no	—
149	C₂²×C₈	—	13	2	no	—
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	—	13	2	no	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	13	1	no	—
154	C₂²×C₈	—	13	2	no	—
155	#117	—	2	2	no	—
156	#117	—	2	2	no	—
157	C₂×C₄⋊C₄	#1057	1	2	yes	—
158	C₂×C₂²⋊C₄	#1057	1	2	yes	—
159	C₂×C₄²	C₂×#261	1	2	yes	—
160	M₃₂	—	26	8	no	—
161	C₂×C₁₆	—	13	4	no	—
162	C₂×C₁₆	—	13	4	no	—
163	C₂×C₁₆	—	26	8	no	—
164	C₂×C₁₆	—	26	2	no	—
165	C₂×C₁₆	—	26	2	no	—
166	C₂×C₁₆	—	26	2	no	—
167	C₂×C₁₆	—	26	2	no	—
168	#97	—	26	4	no	—
169	C₄×C₈	—	26	2	no	—
170	C₄×C₈	—	26	2	no	—
171	C₂×C₁₃	—	2	4	no	—
172	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
173	C₂×C₁₃	#2108	1	2	yes	—
174	C₂×C₁₃	#2113	1	2	yes	—
175	C₂×C₁₃	#2151	1	2	yes	—
176	C₂⁴	C₂×#1057	1	1	yes	characteristic
177	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
178	C₂²×C₄	#6118	1	2	yes	—
179	C₂²×C₄	C₄×#261	1	2	yes	—
180	C₂²×C₄	C₄⋊#261	1	2	yes	—
181	C₂²×C₄	C₂×#1057	1	4	yes	—
182	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
183	C₂²×C₄	#6122	1	2	yes	—
184	C₂²×C₄	C₂²⋊#261	1	2	yes	—
185	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
186	M₁₆	—	26	8	no	—
187	C₂×C₈	—	13	4	no	—
188	C₂×C₈	—	13	4	no	—
189	C₂×C₈	—	13	4	no	—
190	C₂×C₈	—	13	4	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	13	1	no	—
193	C₂×C₈	—	13	2	no	—
194	C₂×C₈	—	13	1	no	—
195	C₂×C₈	—	13	2	no	—
196	C₂×C₈	—	13	1	no	—
197	C₂×C₈	—	13	1	no	—
198	C₂×C₈	—	13	1	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	—	13	4	no	—
201	C₂×C₈	—	26	2	no	—
202	C₂×C₈	—	26	2	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₁₆	—	26	4	no	—
208	C₁₆	—	26	4	no	—
209	C₁₃	#12545	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	2	4	no	—
211	C₂³	C₄×#1057	1	2	yes	—
212	C₂³	C₄⋊#1057	1	2	yes	—
213	C₂³	C₂²⋊#1057	1	2	yes	—
214	C₂³	C₂²×#1057	1	1	yes	characteristic
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	C₄×#1057	1	2	yes	—
218	C₂×C₄	C₄⋊#1057	1	2	yes	—
219	C₂×C₄	#88110	1	4	yes	—
220	C₂×C₄	#88111	1	4	yes	—
221	C₂×C₄	#88112	1	4	yes	—
222	C₂×C₄	(C₂×#261)⋊C₄	1	4	yes	—
223	C₂×C₄	C₂²⋊#1057	1	2	yes	—
224	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₂×#6118	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
228	C₂×C₄	C₂×#6122	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
230	C₈	—	13	1	no	—
231	C₈	—	13	1	no	—
232	C₈	—	13	4	no	—
233	C₈	—	13	1	no	—
234	C₈	—	13	1	no	—
235	C₈	—	26	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#11718290	1	4	yes	—
240	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
241	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
242	C₂²	#11719162	1	4	yes	—
243	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
244	C₄	—	4	4	no	—
245	C₄	#11718290	1	4	yes	—
246	C₄	C₂×#88110	1	1	yes	characteristic
247	C₄	C₂×#88111	1	1	yes	characteristic
248	C₄	C₂×#88112	1	1	yes	characteristic
249	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	C₂×#11718290	1	1	yes	characteristic
252	C₂	#13131045	1	2	yes	—
253	C₂	#13131066	1	2	yes	—
254	C₂	#13131171	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×#117	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#12545
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #13129874	C₂	semidirect
10. C₂×#88106	C₄	semidirect
13. C₂²×#6092	C₄	semidirect
209. C₁₃	#12545	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
C₂³×#6092	2
#13129874	2
C₂×#11718658	2
#12545	13