Subgroups — #16786830

Order 3328.

386 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131193	C₂	1	2	yes	maximal, complemented
2	#13129772	C₂	1	2	yes	maximal, complemented
3	#13129756	C₂	1	2	yes	maximal
4	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊#1055	C₄	1	1	yes	characteristic
7	C₂×C₂²⋊#1055	C₂²	1	1	yes	characteristic
8	#11718586	—	2	2	no	—
9	#11718586	—	2	2	no	—
10	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
11	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₄	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂²⋊#6092	C₂²	1	4	yes	—
15	C₄⋊#6092	—	2	2	no	—
16	C₄×#6092	—	2	2	no	—
17	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	—	2	1	no	—
21	C₂×C₄²×C₁₃	—	2	1	no	—
22	C₂×C₄²×C₁₃	—	2	1	no	—
23	C₂×C₄²×C₁₃	—	2	1	no	—
24	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂³	1	1	yes	characteristic
32	C₂×#6092	—	2	4	no	—
33	C₂²⋊#1055	—	2	2	no	—
34	C₂²⋊#1055	—	2	2	no	—
35	C₄⋊#1055	—	2	2	no	—
36	C₄⋊#1055	—	2	2	no	—
37	C₄×#1055	—	2	1	no	—
38	C₄×#1055	—	2	1	no	—
39	C₄×#1055	D₄	1	2	yes	—
40	C₄×#1055	D₄	1	2	yes	—
41	#11459	—	13	1	no	maximal, Sylow
42	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	4	1	no	—
52	C₂²×C₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	C₁₆	1	4	yes	complemented
54	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₁₃	—	2	1	no	—
61	C₄²×C₁₃	—	2	1	no	—
62	C₄²×C₁₃	—	2	1	no	—
63	C₄²×C₁₃	—	2	1	no	—
64	C₄²×C₁₃	—	2	1	no	—
65	C₄²×C₁₃	—	2	1	no	—
66	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
69	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	—	4	1	no	—
73	C₂×#1055	—	4	1	no	—
74	C₂×#1055	—	4	1	no	—
75	C₂×#1055	—	4	1	no	—
76	C₂×#1055	C₂²⋊C₄	1	2	yes	—
77	C₂×#1055	C₂×D₄	1	2	yes	—
78	#6092	—	4	4	no	—
79	C₂×#431	—	13	1	no	—
80	#2162	—	13	2	no	—
81	#1322	—	13	2	no	—
82	#1314	—	13	2	no	—
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	—	2	1	no	—
85	C₂³×C₁₃	—	4	1	no	—
86	C₂³×C₁₃	—	4	1	no	—
87	C₂³×C₁₃	C₂×C₁₆	1	2	yes	—
88	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	1	no	—
102	C₂×C₄×C₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	—	4	1	no	—
104	C₂×C₄×C₁₃	—	4	1	no	—
105	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
111	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
114	#1055	—	2	2	no	—
115	#1055	—	2	2	no	—
116	#1055	—	4	1	no	—
117	#1055	—	4	1	no	—
118	C₂²×C₄²	#261	1	1	yes	characteristic
119	#431	—	26	2	no	—
120	#431	—	26	2	no	—
121	C₂×C₄⋊C₈	—	13	1	no	—
122	C₂×C₄⋊C₈	—	13	1	no	—
123	C₂×C₂²⋊C₈	—	13	1	no	—
124	C₂×C₂²⋊C₈	—	13	1	no	—
125	C₂×C₄×C₈	—	13	1	no	—
126	C₂×C₄×C₈	—	13	1	no	—
127	C₄⋊C₁₆	—	26	2	no	—
128	C₂²⋊C₁₆	—	13	4	no	—
129	C₄×C₁₆	—	26	2	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	—	4	2	no	—
135	C₂²×C₁₃	—	4	1	no	—
136	C₂²×C₁₃	—	4	1	no	—
137	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
139	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	—	4	1	no	—
145	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
146	C₄×C₁₃	C₂²⋊C₁₆	1	2	yes	—
147	C₄×C₁₃	#349	1	2	yes	—
148	C₄×C₁₃	#410	1	1	yes	characteristic
149	C₄×C₁₃	#413	1	1	yes	characteristic
150	C₂³×C₄	#1057	1	1	yes	characteristic
151	C₂³×C₄	#1057	1	1	yes	characteristic
152	C₂³×C₄	C₂×#261	1	1	yes	characteristic
153	C₂²×C₈	—	13	1	no	—
154	C₂²×C₈	—	13	1	no	—
155	C₂²×C₈	—	13	1	no	—
156	C₂²×C₈	—	13	1	no	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	#1057	1	1	yes	characteristic
162	C₂×C₄²	#1057	1	1	yes	characteristic
163	C₂×C₄²	C₂×#261	1	1	yes	characteristic
164	C₂×C₄²	C₂×#261	1	1	yes	characteristic
165	C₂×C₁₆	—	26	4	no	—
166	C₄⋊C₈	—	26	2	no	—
167	C₄⋊C₈	—	26	2	no	—
168	C₂²⋊C₈	—	26	2	no	—
169	C₂²⋊C₈	—	26	2	no	—
170	C₄×C₈	—	13	2	no	—
171	C₄×C₈	—	13	2	no	—
172	C₄×C₈	—	26	1	no	—
173	C₄×C₈	—	26	1	no	—
174	C₂×C₁₃	—	2	1	no	—
175	C₂×C₁₃	—	2	1	no	—
176	C₂×C₁₃	—	4	2	no	—
177	C₂×C₁₃	#1458	1	1	yes	characteristic
178	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
179	C₂×C₁₃	C₂×#349	1	1	yes	characteristic
180	C₂⁴	C₂×#1057	1	1	yes	characteristic
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	—	4	1	no	—
191	C₂²×C₄	#6092	1	4	yes	complemented
192	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
193	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
194	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
195	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
196	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
197	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
198	C₂×C₈	—	13	2	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	—	26	1	no	—
201	C₂×C₈	—	26	1	no	—
202	C₂×C₈	—	52	1	no	—
203	C₂×C₈	—	52	1	no	—
204	C₂×C₈	—	52	1	no	—
205	C₂×C₈	—	52	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	#6122	1	1	yes	characteristic
213	C₄²	#6122	1	1	yes	characteristic
214	C₄²	C₂²⋊#261	1	1	yes	characteristic
215	C₄²	C₂²⋊#261	1	1	yes	characteristic
216	C₁₆	—	52	4	no	—
217	C₁₃	#11459	1	1	yes	characteristic, complemented, Sylow
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	4	1	no	—
221	C₂³	—	4	1	no	—
222	C₂³	C₂×#6092	1	2	yes	—
223	C₂³	C₂²×#1057	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	C₂×#6092	1	2	yes	—
241	C₂×C₄	C₂×#6092	1	2	yes	—
242	C₂×C₄	#88106	1	2	yes	—
243	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
246	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
247	C₂×C₄	C₂×#6122	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
249	C₈	—	26	2	no	—
250	C₈	—	26	2	no	—
251	C₈	—	52	1	no	—
252	C₈	—	52	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	C₂²×#6092	1	1	yes	characteristic
261	C₂²	C₂×#88106	1	1	yes	characteristic
262	C₂²	C₂×C₂²⋊#1057	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	C₂²⋊#6092	1	2	yes	—
269	C₄	C₂²⋊#6092	1	2	yes	—
270	C₄	#11718284	1	2	yes	—
271	C₄	#11719116	1	1	yes	characteristic
272	C₄	#11719132	1	1	yes	characteristic
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	4	2	no	—
276	C₂	#13130067	1	1	yes	characteristic
277	C₂	C₂×C₂²⋊#6092	1	1	yes	characteristic
278	C₂	C₂×#11718284	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²⋊#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₄²×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	#11459
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #13131193	C₂	semidirect
2. #13129772	C₂	semidirect
53. C₂²×C₄×C₁₃	C₁₆	semidirect
191. C₂²×C₄	#6092	semidirect
217. C₁₃	#11459	semidirect

Maximal subgroups

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

subgroup	index
#13131193	2
#13129772	2
#13129756	2
C₂×#11718586	2
#11459	13