Subgroups — #16018847

Order 3328.

582 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718360	C₂	1	1	yes	maximal, characteristic
3	#13117089	C₂	1	2	yes	maximal, complemented
4	#13117064	C₂	1	2	yes	maximal, complemented
5	#13116887	C₂	1	4	yes	maximal, complemented
6	#13116772	C₂	1	4	yes	maximal
7	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₅₂	—	2	1	no	—
9	C₂²×Dic₅₂	C₂²	1	2	yes	—
10	#11719175	—	2	1	no	—
11	C₂×#88123	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718777	—	2	1	no	—
14	#11718777	C₂²	1	2	yes	—
15	C₂×Dic₁₀₄	C₂²	1	8	yes	—
16	#11718360	—	2	1	no	—
17	#11718360	—	2	1	no	—
18	#11718360	C₂²	1	1	yes	characteristic
19	#11718360	C₂²	1	2	yes	—
20	#11718360	C₂²	1	2	yes	—
21	C₄×Dic₅₂	—	2	1	no	—
22	C₄×Dic₅₂	C₂²	1	2	yes	—
23	#11718343	C₂²	1	1	yes	characteristic
24	#11718342	—	2	1	no	—
25	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
28	#11718095	C₂²	1	8	yes	—
29	C₂²×Dic₂₆	—	2	2	no	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	#88180	—	4	1	no	—
34	C₂×C₄⋊Dic₁₃	—	4	1	no	—
35	C₂×C₄⋊Dic₁₃	—	4	1	no	—
36	#88178	—	2	2	no	—
37	#88178	—	4	1	no	—
38	C₂×Dic₅₂	—	2	2	no	—
39	C₂×Dic₅₂	—	2	2	no	—
40	C₂×Dic₅₂	—	2	2	no	—
41	C₂×Dic₅₂	—	2	1	no	—
42	C₂×Dic₅₂	—	4	4	no	—
43	C₂×Dic₅₂	—	4	4	no	—
44	C₂×Dic₅₂	C₂³	1	4	yes	—
45	C₂×Dic₅₂	C₂³	1	2	yes	—
46	#88124	—	4	1	no	—
47	#88123	—	2	1	no	—
48	#88123	—	2	2	no	—
49	#88123	—	2	2	no	—
50	#88123	—	2	1	no	—
51	#88123	—	4	1	no	—
52	#88123	—	4	1	no	—
53	C₄×Dic₂₆	—	2	2	no	—
54	C₄×Dic₂₆	—	4	1	no	—
55	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	#88058	—	2	1	no	—
61	#88058	C₂³	1	2	yes	—
62	#88056	—	2	2	no	—
63	#88056	—	2	2	no	—
64	Dic₁₀₄	—	4	8	no	—
65	#33775	—	13	1	no	maximal, Sylow
66	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	4	2	no	—
70	C₂×Dic₂₆	—	2	2	no	—
71	C₂×Dic₂₆	—	2	4	no	—
72	C₂×Dic₂₆	—	4	4	no	—
73	C₂×Dic₂₆	—	4	4	no	—
74	C₂×Dic₂₆	—	4	1	no	—
75	C₂×Dic₂₆	—	4	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₄²×C₁₃	D₈	1	1	yes	characteristic
83	C₄²×C₁₃	D₈	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²⋊Dic₁₃	—	4	2	no	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	4	1	no	—
89	C₄⋊Dic₁₃	—	4	2	no	—
90	C₄⋊Dic₁₃	—	4	2	no	—
91	C₄⋊Dic₁₃	—	4	2	no	—
92	C₄⋊Dic₁₃	—	4	2	no	—
93	C₄⋊Dic₁₃	—	4	1	no	—
94	C₄⋊Dic₁₃	—	4	1	no	—
95	#6101	—	4	2	no	—
96	#6101	—	4	2	no	—
97	C₄×Dic₁₃	—	4	2	no	—
98	Dic₅₂	—	2	8	no	—
99	Dic₅₂	—	2	4	no	—
100	Dic₅₂	—	4	4	no	—
101	Dic₅₂	—	4	4	no	—
102	Dic₅₂	—	4	2	no	—
103	Dic₅₂	—	4	4	no	—
104	C₁₃×C₁₆	—	2	4	no	—
105	#3170	—	13	1	no	—
106	C₂×#494	—	13	1	no	—
107	#2247	—	13	2	no	—
108	#2241	—	13	2	no	—
109	#2209	—	13	4	no	—
110	#2188	—	13	4	no	—
111	#2162	D₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	Q₃₂	1	4	yes	complemented
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	4	4	no	—
126	C₂×Dic₁₃	—	8	4	no	—
127	Dic₂₆	—	4	4	no	—
128	Dic₂₆	—	4	2	no	—
129	Dic₂₆	—	4	4	no	—
130	Dic₂₆	—	4	2	no	—
131	Dic₂₆	—	8	4	no	—
132	Dic₂₆	—	8	2	no	—
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
136	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
137	C₂²×Q₁₆	—	13	2	no	—
138	C₂²×Q₁₆	—	26	1	no	—
139	#533	—	26	1	no	—
140	C₂×C₄⋊Q₈	—	26	1	no	—
141	C₂×Q₃₂	—	13	8	no	—
142	#500	—	13	1	no	—
143	#494	—	13	2	no	—
144	#494	—	13	2	no	—
145	#494	—	13	1	no	—
146	#494	—	26	1	no	—
147	#494	—	26	1	no	—
148	#487	—	26	1	no	—
149	#467	—	13	2	no	—
150	#467	—	26	1	no	—
151	C₄×Q₁₆	—	13	2	no	—
152	C₄×Q₁₆	—	26	1	no	—
153	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
154	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
155	#358	—	13	8	no	—
156	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
157	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
161	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
165	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
166	C₄×C₁₃	#510	1	2	yes	—
167	C₄×C₁₃	#535	1	1	yes	characteristic
168	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	Dic₁₃	—	8	2	no	—
170	Dic₁₃	—	8	4	no	—
171	Dic₁₃	—	8	2	no	—
172	C₂²×Q₈	—	26	2	no	—
173	C₂×Q₁₆	—	13	4	no	—
174	C₂×Q₁₆	—	13	2	no	—
175	C₂×Q₁₆	—	26	2	no	—
176	C₂×Q₁₆	—	26	2	no	—
177	C₂×Q₁₆	—	26	2	no	—
178	C₂×Q₁₆	—	26	1	no	—
179	C₂×Q₁₆	—	52	4	no	—
180	C₂×Q₁₆	—	52	4	no	—
181	C₂²×C₈	D₅₂	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	26	1	no	—
184	C₄⋊Q₈	—	26	2	no	—
185	C₄⋊Q₈	—	26	2	no	—
186	C₄⋊Q₈	—	26	1	no	—
187	C₄⋊Q₈	—	52	1	no	—
188	C₄⋊Q₈	—	52	1	no	—
189	#125	—	52	1	no	—
190	C₂²⋊Q₈	—	26	2	no	—
191	C₂²⋊Q₈	—	52	1	no	—
192	C₄×Q₈	—	26	2	no	—
193	C₄×Q₈	—	52	1	no	—
194	#117	—	52	1	no	—
195	C₂×C₄⋊C₄	—	52	1	no	—
196	C₂×C₄⋊C₄	—	52	1	no	—
197	C₂×C₄²	D₅₂	1	1	yes	characteristic
198	Q₃₂	—	52	8	no	—
199	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
200	#107	—	13	2	no	—
201	#107	—	26	1	no	—
202	#103	—	26	2	no	—
203	#103	—	26	2	no	—
204	C₄×C₈	D₅₂	1	1	yes	characteristic
205	C₄×C₈	D₅₂	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
208	C₂×C₁₃	—	2	2	no	—
209	C₂×C₁₃	#3205	1	1	yes	characteristic
210	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
211	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
212	C₂×Q₈	—	26	2	no	—
213	C₂×Q₈	—	26	4	no	—
214	C₂×Q₈	—	52	4	no	—
215	C₂×Q₈	—	52	4	no	—
216	C₂×Q₈	—	52	1	no	—
217	C₂×Q₈	—	52	2	no	—
218	C₂²×C₄	—	52	2	no	—
219	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
220	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
222	Q₁₆	—	26	4	no	—
223	Q₁₆	—	26	8	no	—
224	Q₁₆	—	52	4	no	—
225	Q₁₆	—	52	4	no	—
226	Q₁₆	—	52	4	no	—
227	Q₁₆	—	52	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
231	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
232	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
233	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
234	C₄⋊C₄	—	26	2	no	—
235	C₄⋊C₄	—	52	2	no	—
236	C₄⋊C₄	—	52	2	no	—
237	C₄⋊C₄	—	52	2	no	—
238	C₄⋊C₄	—	52	2	no	—
239	C₄⋊C₄	—	52	1	no	—
240	C₄⋊C₄	—	52	1	no	—
241	C₄⋊C₄	—	52	2	no	—
242	C₄⋊C₄	—	52	2	no	—
243	C₄⋊C₄	—	52	1	no	—
244	C₂²⋊C₄	—	52	2	no	—
245	C₄²	—	52	2	no	—
246	C₄²	D₁₀₄	1	1	yes	characteristic
247	C₄²	D₁₀₄	1	1	yes	characteristic
248	C₄²	C₂×D₅₂	1	1	yes	characteristic
249	C₄²	C₂×D₅₂	1	1	yes	characteristic
250	C₁₆	—	2	4	no	—
251	C₁₃	#33775	1	1	yes	characteristic, complemented, Sylow
252	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
253	Q₈	—	52	4	no	—
254	Q₈	—	52	2	no	—
255	Q₈	—	52	4	no	—
256	Q₈	—	52	2	no	—
257	Q₈	—	104	4	no	—
258	Q₈	—	104	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	52	2	no	—
263	C₂×C₄	—	52	2	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	—	104	4	no	—
266	C₂×C₄	Dic₁₀₄	1	4	yes	complemented
267	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
268	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
269	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
270	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
271	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
273	C₈	—	2	1	no	—
274	C₈	—	2	1	no	—
275	C₈	#88162	1	2	yes	—
276	C₈	#88256	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	C₂×Dic₁₀₄	1	2	yes	—
280	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	104	2	no	—
284	C₄	—	104	4	no	—
285	C₄	—	104	2	no	—
286	C₄	C₂×Dic₁₀₄	1	2	yes	—
287	C₄	C₂×Dic₁₀₄	1	2	yes	—
288	C₄	#11718570	1	2	yes	—
289	C₄	#11719230	1	1	yes	characteristic
290	C₄	C₂×#88162	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	#13125756	1	1	yes	characteristic
293	C₂	C₂²×Dic₁₀₄	1	1	yes	characteristic
294	C₂	C₂×#11718570	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2162	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#2162	D₁₃
hypocenter	C₁₃	#33775
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #13117089	C₂	semidirect
4. #13117064	C₂	semidirect
5. #13116887	C₂	semidirect
116. C₂×C₄×C₁₃	Q₃₂	semidirect
251. C₁₃	#33775	semidirect
266. C₂×C₄	Dic₁₀₄	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
C₂×#11718360	2
#13117089	2
#13117064	2
#13116887	2
#13116772	2
C₁₃×#2162	2
#33775	13