Subgroups — #16807055

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131197	C₂	1	4	yes	maximal, complemented
2	#13131193	C₂	1	4	yes	maximal, complemented
3	#13131168	C₂	1	2	yes	maximal, complemented
4	#13131063	C₂	1	2	yes	maximal
5	C₄×#88106	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×#1055	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₂³×#1055	C₄	1	1	yes	characteristic
9	C₂³×#1055	C₂²	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂×C₄×#1055	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₄	1	2	yes	—
13	C₂×C₄×#1055	C₄	1	2	yes	—
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	2	yes	—
18	C₂×C₄×#1055	C₂²	1	2	yes	—
19	C₂²⋊#6092	C₂²	1	8	yes	—
20	#11718231	C₂²	1	4	yes	complemented
21	C₄⋊#6092	C₂²	1	8	yes	complemented
22	C₄×#6092	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₁₃	C₈	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₈	1	2	yes	—
26	C₂×C₄²×C₁₃	C₈	1	2	yes	—
27	C₂×C₄²×C₁₃	C₈	1	2	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×#1055	—	2	4	no	—
33	C₂²×#1055	—	2	4	no	—
34	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	2	yes	—
36	C₂²×#1055	C₂×C₄	1	2	yes	—
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	2	yes	—
39	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
40	C₂²×#1055	C₂³	1	2	yes	—
41	C₂²×#1055	C₂³	1	1	yes	characteristic
42	#88106	—	2	8	no	—
43	C₂×#6092	C₂³	1	8	yes	—
44	C₄×#1055	C₂×C₄	1	2	yes	—
45	C₄×#1055	C₂×C₄	1	2	yes	—
46	C₄×#1055	C₂×C₄	1	2	yes	—
47	C₄×#1055	C₂×C₄	1	2	yes	—
48	C₄×#1055	C₂×C₄	1	1	yes	characteristic
49	C₄×#1055	C₂×C₄	1	2	yes	—
50	C₄×#1055	C₂×C₄	1	1	yes	characteristic
51	C₄×#1055	C₂³	1	1	yes	characteristic
52	C₄×#1055	C₂³	1	2	yes	—
53	C₄×#1055	C₂³	1	1	yes	characteristic
54	#33418	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	8	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×C₈	1	2	yes	—
70	C₄²×C₁₃	C₂×C₈	1	2	yes	—
71	C₄²×C₁₃	C₂×C₈	1	2	yes	—
72	C₄²×C₁₃	C₂×C₈	1	2	yes	—
73	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₁₃	C₂×C₈	1	2	yes	—
75	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	4	no	—
85	C₂×#1055	M₁₆	1	4	yes	—
86	C₂×#1055	C₂²×C₄	1	2	yes	—
87	C₂×#1055	C₂²×C₄	1	2	yes	—
88	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
89	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
90	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
91	C₂×#1055	#41	1	4	yes	—
92	C₂×#1055	C₂⁴	1	1	yes	characteristic
93	#6092	—	2	8	no	—
94	C₂²×C₄×C₈	—	13	1	no	—
95	#2163	—	13	4	no	—
96	#2162	—	13	4	no	—
97	#2150	—	13	2	no	—
98	#2112	—	13	2	no	—
99	C₄×M₃₂	—	13	2	no	—
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	4	no	—
102	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	M₃₂	1	8	yes	complemented
114	C₂×C₄×C₁₃	#117	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	#1055	—	2	2	no	—
124	#1055	—	2	2	no	—
125	#1055	#117	1	2	yes	—
126	#1055	C₂×M₁₆	1	2	yes	—
127	#1055	C₂×M₁₆	1	2	yes	—
128	#1055	C₂×#41	1	2	yes	—
129	C₂³×C₈	—	13	1	no	—
130	C₂³×C₈	—	13	1	no	—
131	C₂²×C₄²	#261	1	1	yes	characteristic
132	C₂×M₃₂	—	13	4	no	—
133	C₂×C₄×C₈	—	13	1	no	—
134	C₂×C₄×C₈	—	13	1	no	—
135	C₂×C₄×C₈	—	13	2	no	—
136	C₂×C₄×C₈	—	13	2	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	C₂×C₄×C₈	—	13	2	no	—
139	C₂×C₄×C₈	—	13	2	no	—
140	C₂×C₄×C₈	—	13	1	no	—
141	C₄⋊C₁₆	—	13	8	no	—
142	C₂²⋊C₁₆	—	13	8	no	—
143	#346	—	13	4	no	—
144	C₄×C₁₆	—	13	4	no	—
145	C₂²×C₁₃	—	2	8	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	#431	1	2	yes	—
150	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
151	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	#431	1	2	yes	—
156	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
157	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
158	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
159	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
160	C₂³×C₄	#1057	1	2	yes	—
161	C₂³×C₄	C₂×#261	1	1	yes	characteristic
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	—	13	2	no	—
166	C₂²×C₈	—	13	2	no	—
167	C₂²×C₈	—	13	2	no	—
168	C₂²×C₈	—	13	1	no	—
169	C₂²×C₈	—	13	1	no	—
170	C₂²×C₈	—	26	4	no	—
171	C₂²×C₈	—	26	4	no	—
172	C₂×C₄²	#1057	1	2	yes	—
173	C₂×C₄²	#1057	1	2	yes	—
174	C₂×C₄²	#1057	1	2	yes	—
175	C₂×C₄²	C₂×#261	1	2	yes	—
176	C₂×C₄²	C₂×#261	1	1	yes	characteristic
177	C₂×C₄²	C₂×#261	1	1	yes	characteristic
178	C₂×C₄²	C₂×#261	1	2	yes	—
179	M₃₂	—	26	8	no	—
180	C₂×C₁₆	—	13	8	no	—
181	C₄×C₈	—	13	1	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	1	no	—
184	C₄×C₈	—	13	2	no	—
185	C₄×C₈	—	13	2	no	—
186	C₄×C₈	—	13	2	no	—
187	C₄×C₈	—	13	2	no	—
188	C₄×C₈	—	13	1	no	—
189	C₄×C₈	—	13	1	no	—
190	C₄×C₈	—	13	2	no	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	—	2	4	no	—
193	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
194	C₂×C₁₃	C₂²×M₃₂	1	2	yes	—
195	C₂⁴	C₂×#1057	1	1	yes	characteristic
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
200	C₂²×C₄	C₂×#1057	1	2	yes	—
201	C₂²×C₄	C₂×#1057	1	2	yes	—
202	C₂²×C₄	C₂×#1057	1	2	yes	—
203	C₂²×C₄	C₂×#1057	1	2	yes	—
204	C₂²×C₄	C₂×#1057	1	2	yes	—
205	C₂²×C₄	C₂×#1057	1	2	yes	—
206	C₂²×C₄	C₂×#1057	1	2	yes	—
207	C₂²×C₄	C₂²×#261	1	2	yes	—
208	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
209	C₂×C₈	—	13	4	no	—
210	C₂×C₈	—	13	4	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	2	no	—
216	C₂×C₈	—	13	2	no	—
217	C₂×C₈	—	26	4	no	—
218	C₂×C₈	—	26	4	no	—
219	C₂×C₈	—	26	4	no	—
220	C₂×C₈	—	26	4	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	—	26	2	no	—
223	C₄²	C₂×#1057	1	2	yes	—
224	C₄²	C₂×#1057	1	2	yes	—
225	C₄²	C₂×#1057	1	2	yes	—
226	C₄²	C₂×#1057	1	2	yes	—
227	C₄²	C₂×#1057	1	1	yes	characteristic
228	C₄²	C₂×#1057	1	2	yes	—
229	C₄²	C₂×#1057	1	1	yes	characteristic
230	C₄²	C₂²×#261	1	1	yes	characteristic
231	C₄²	C₂²×#261	1	2	yes	—
232	C₄²	C₂²×#261	1	1	yes	characteristic
233	C₁₆	—	26	8	no	—
234	C₁₃	#33418	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	2	4	no	—
236	C₂³	—	2	2	no	—
237	C₂³	C₂²×#1057	1	2	yes	—
238	C₂³	C₂²×#1057	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	#88106	1	8	yes	complemented
249	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
250	C₂×C₄	#88237	1	2	yes	—
251	C₂×C₄	C₂²×#1057	1	2	yes	—
252	C₂×C₄	C₂²×#1057	1	2	yes	—
253	C₂×C₄	C₂²×#1057	1	2	yes	—
254	C₂×C₄	C₂²×#1057	1	2	yes	—
255	C₂×C₄	C₂²×#1057	1	2	yes	—
256	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
257	C₂×C₄	C₂³×#261	1	1	yes	characteristic
258	C₈	—	13	2	no	—
259	C₈	—	13	2	no	—
260	C₈	—	13	2	no	—
261	C₈	—	13	2	no	—
262	C₈	—	26	2	no	—
263	C₈	—	26	2	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	C₂×#88106	1	4	yes	—
269	C₂²	#11719119	1	2	yes	—
270	C₂²	C₂³×#1057	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	C₂×#88106	1	4	yes	—
275	C₄	C₂×#88106	1	4	yes	—
276	C₄	#11719119	1	2	yes	—
277	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
278	C₄	C₂×#88237	1	1	yes	characteristic
279	C₂	—	2	4	no	—
280	C₂	—	2	2	no	—
281	C₂	C₂×#11719119	1	1	yes	characteristic
282	C₂	C₂²×#88106	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₄²×C₁₃	C₄
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×C₄²	#261
hypocenter	C₁₃	#33418
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #13131197	C₂	semidirect
2. #13131193	C₂	semidirect
3. #13131168	C₂	semidirect
5. C₄×#88106	C₂	semidirect
20. #11718231	C₂²	semidirect
21. C₄⋊#6092	C₂²	semidirect
22. C₄×#6092	C₂²	semidirect
113. C₂×C₄×C₁₃	M₃₂	semidirect
234. C₁₃	#33418	semidirect
248. C₂×C₄	#88106	semidirect

Maximal subgroups

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

subgroup	index
#13131197	2
#13131193	2
#13131168	2
#13131063	2
C₄×#88106	2
C₂²×C₄×#1055	2
#33418	13