Subgroups — #16018824

Order 3328.

626 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125415	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718360	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117092	C₂	1	2	yes	maximal, complemented
4	#13117064	C₂	1	2	yes	maximal, complemented
5	#13116882	C₂	1	4	yes	maximal, complemented
6	#13116755	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₅₂	—	2	1	no	—
9	C₂²×Dic₅₂	C₂²	1	2	yes	—
10	#11719182	—	2	1	no	—
11	C₂×#88123	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718776	—	2	1	no	—
14	#11718776	C₂²	1	2	yes	—
15	C₂×#88040	C₂²	1	8	yes	—
16	#11718360	—	2	1	no	—
17	#11718360	—	2	1	no	—
18	#11718360	C₂²	1	2	yes	—
19	#11718360	C₂²	1	2	yes	—
20	#11718356	C₂²	1	1	yes	characteristic
21	#11718355	—	2	1	no	—
22	C₄×D₁₀₄	—	2	1	no	—
23	C₄×D₁₀₄	C₂²	1	2	yes	—
24	#11718343	C₂²	1	1	yes	characteristic
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	C₂×#6127	—	4	1	no	—
30	C₂²×Dic₂₆	—	2	2	no	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	#88184	—	2	2	no	—
35	#88184	—	4	1	no	—
36	C₂×C₄⋊Dic₁₃	—	4	1	no	—
37	C₂×C₄⋊Dic₁₃	—	4	1	no	—
38	C₂×Dic₅₂	—	2	2	no	—
39	C₂×Dic₅₂	—	2	2	no	—
40	C₂×Dic₅₂	—	2	2	no	—
41	C₂×Dic₅₂	—	4	4	no	—
42	C₂×Dic₅₂	—	4	4	no	—
43	C₂×Dic₅₂	C₂³	1	4	yes	—
44	C₂×D₁₀₄	—	2	1	no	—
45	C₂×D₁₀₄	C₂³	1	2	yes	—
46	#88129	—	4	1	no	—
47	#88128	—	2	1	no	—
48	C₄×D₅₂	—	2	2	no	—
49	C₄×D₅₂	—	4	1	no	—
50	#88123	—	2	1	no	—
51	#88123	—	2	2	no	—
52	#88123	—	2	2	no	—
53	#88123	—	4	1	no	—
54	#88123	—	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	#88061	—	2	2	no	—
61	#88061	—	2	2	no	—
62	#88058	—	2	1	no	—
63	#88058	C₂³	1	2	yes	—
64	#88040	—	4	8	no	—
65	#33772	—	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	C₂×C₂²⋊D₁₃	—	4	2	no	—
70	C₂²×Dic₁₃	—	4	2	no	—
71	#6127	—	8	4	no	—
72	C₂×D₅₂	—	2	2	no	—
73	C₂×D₅₂	—	4	1	no	—
74	C₂×D₅₂	—	4	1	no	—
75	C₂×C₄×D₁₃	—	4	2	no	—
76	C₂×Dic₂₆	—	2	4	no	—
77	C₂×Dic₂₆	—	4	4	no	—
78	C₂×Dic₂₆	—	4	4	no	—
79	C₂×Dic₂₆	—	4	1	no	—
80	C₂×Dic₂₆	—	4	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₁₃	D₈	1	1	yes	characteristic
88	C₄²×C₁₃	D₈	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	#6103	—	4	2	no	—
92	#6103	—	4	2	no	—
93	C₄⋊Dic₁₃	—	2	2	no	—
94	C₄⋊Dic₁₃	—	4	2	no	—
95	C₄⋊Dic₁₃	—	4	2	no	—
96	C₄⋊Dic₁₃	—	4	2	no	—
97	C₄⋊Dic₁₃	—	4	2	no	—
98	C₄⋊Dic₁₃	—	4	1	no	—
99	Dic₅₂	—	2	8	no	—
100	Dic₅₂	—	4	4	no	—
101	Dic₅₂	—	4	4	no	—
102	Dic₅₂	—	4	4	no	—
103	D₁₀₄	—	2	4	no	—
104	D₁₀₄	—	4	2	no	—
105	C₁₃×C₁₆	—	2	4	no	—
106	#3169	—	13	1	no	—
107	C₂×#494	—	13	1	no	—
108	#2248	—	13	2	no	—
109	#2241	—	13	2	no	—
110	#2208	—	13	4	no	—
111	#2185	—	13	4	no	—
112	#2162	D₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂²×D₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
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	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂²⋊D₁₃	—	8	4	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	8	4	no	—
129	D₅₂	—	4	2	no	—
130	D₅₂	—	4	2	no	—
131	D₅₂	—	8	2	no	—
132	D₅₂	—	8	2	no	—
133	C₄×D₁₃	—	8	4	no	—
134	Dic₂₆	—	4	4	no	—
135	Dic₂₆	—	4	4	no	—
136	Dic₂₆	—	8	4	no	—
137	Dic₂₆	—	8	2	no	—
138	C₈×C₁₃	—	2	1	no	—
139	C₈×C₁₃	—	2	1	no	—
140	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
141	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
142	C₂²×Q₁₆	—	13	2	no	—
143	C₂²×Q₁₆	—	26	1	no	—
144	#532	—	26	1	no	—
145	C₂×C₄⋊Q₈	—	26	1	no	—
146	C₂×SD₃₂	—	13	8	no	—
147	#500	—	13	1	no	—
148	#494	—	13	2	no	—
149	#494	—	13	2	no	—
150	#494	—	26	1	no	—
151	#494	—	26	1	no	—
152	#493	—	13	1	no	—
153	#486	—	26	1	no	—
154	#466	—	13	2	no	—
155	#466	—	26	1	no	—
156	C₄×D₈	—	13	2	no	—
157	C₄×D₈	—	26	1	no	—
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
160	#358	—	13	8	no	—
161	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
162	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
163	C₂²×C₁₃	—	2	1	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
166	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₁₃	—	8	2	no	—
168	C₂×D₁₃	—	8	2	no	—
169	C₂×D₁₃	—	8	2	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
173	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
174	C₄×C₁₃	#510	1	2	yes	—
175	C₄×C₁₃	#535	1	1	yes	characteristic
176	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	Dic₁₃	—	8	2	no	—
178	Dic₁₃	—	8	4	no	—
179	C₂×#41	—	52	1	no	—
180	C₂²×Q₈	—	26	2	no	—
181	C₂×Q₁₆	—	13	4	no	—
182	C₂×Q₁₆	—	26	2	no	—
183	C₂×Q₁₆	—	26	2	no	—
184	C₂×Q₁₆	—	26	2	no	—
185	C₂×Q₁₆	—	52	4	no	—
186	C₂×Q₁₆	—	52	4	no	—
187	C₂×D₈	—	13	2	no	—
188	C₂×D₈	—	26	1	no	—
189	C₂²×C₈	D₅₂	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
191	C₄⋊Q₈	—	26	1	no	—
192	C₄⋊Q₈	—	26	2	no	—
193	C₄⋊Q₈	—	26	2	no	—
194	C₄⋊Q₈	—	52	1	no	—
195	C₄⋊Q₈	—	52	1	no	—
196	#127	—	26	1	no	—
197	#124	—	52	1	no	—
198	#121	—	26	2	no	—
199	#121	—	52	1	no	—
200	C₄×D₄	—	26	2	no	—
201	C₄×D₄	—	52	1	no	—
202	C₂×C₄⋊C₄	—	52	1	no	—
203	C₂×C₄⋊C₄	—	52	1	no	—
204	C₂×C₄²	D₅₂	1	1	yes	characteristic
205	SD₃₂	—	52	8	no	—
206	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
207	#107	—	13	2	no	—
208	#107	—	26	1	no	—
209	#102	—	26	2	no	—
210	#102	—	26	2	no	—
211	C₄×C₈	D₅₂	1	1	yes	characteristic
212	C₄×C₈	D₅₂	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	#3205	1	1	yes	characteristic
217	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
218	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
219	D₁₃	—	16	2	no	—
220	#41	—	104	4	no	—
221	C₂×Q₈	—	26	4	no	—
222	C₂×Q₈	—	52	4	no	—
223	C₂×Q₈	—	52	4	no	—
224	C₂×Q₈	—	52	1	no	—
225	C₂×Q₈	—	52	2	no	—
226	C₂×D₄	—	26	2	no	—
227	C₂×D₄	—	52	2	no	—
228	C₂×D₄	—	52	1	no	—
229	C₂×D₄	—	52	1	no	—
230	C₂²×C₄	—	52	2	no	—
231	C₂²×C₄	—	52	2	no	—
232	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
233	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
235	Q₁₆	—	26	8	no	—
236	Q₁₆	—	52	4	no	—
237	Q₁₆	—	52	4	no	—
238	Q₁₆	—	52	4	no	—
239	D₈	—	26	4	no	—
240	D₈	—	52	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
244	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
245	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
246	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
247	C₄⋊C₄	—	26	2	no	—
248	C₄⋊C₄	—	52	2	no	—
249	C₄⋊C₄	—	52	2	no	—
250	C₄⋊C₄	—	52	2	no	—
251	C₄⋊C₄	—	52	2	no	—
252	C₄⋊C₄	—	52	1	no	—
253	C₂²⋊C₄	—	52	2	no	—
254	C₂²⋊C₄	—	52	2	no	—
255	C₄²	D₁₀₄	1	1	yes	characteristic
256	C₄²	D₁₀₄	1	1	yes	characteristic
257	C₄²	C₂×D₅₂	1	1	yes	characteristic
258	C₄²	C₂×D₅₂	1	1	yes	characteristic
259	C₁₆	—	2	4	no	—
260	C₁₃	#33772	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	52	2	no	—
262	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
263	Q₈	—	52	4	no	—
264	Q₈	—	52	4	no	—
265	Q₈	—	104	4	no	—
266	Q₈	—	104	2	no	—
267	D₄	—	52	2	no	—
268	D₄	—	52	2	no	—
269	D₄	—	104	4	no	—
270	D₄	—	104	2	no	—
271	D₄	—	104	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	52	2	no	—
276	C₂×C₄	—	52	4	no	—
277	C₂×C₄	—	104	4	no	—
278	C₂×C₄	—	104	4	no	—
279	C₂×C₄	#88040	1	4	yes	complemented
280	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
281	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
282	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
283	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
284	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	#88162	1	2	yes	—
289	C₈	#88256	1	2	yes	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	104	2	no	—
293	C₂²	—	104	2	no	—
294	C₂²	—	104	2	no	—
295	C₂²	C₂×#88040	1	2	yes	—
296	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	104	2	no	—
300	C₄	—	104	4	no	—
301	C₄	C₂×#88040	1	2	yes	—
302	C₄	C₂×#88040	1	2	yes	—
303	C₄	#11718570	1	2	yes	—
304	C₄	#11719230	1	1	yes	characteristic
305	C₄	C₂×#88162	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	208	2	no	—
308	C₂	#13125756	1	1	yes	characteristic
309	C₂	C₂²×#88040	1	1	yes	characteristic
310	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	#13125415	C₂
hypercenter	#2162	D₁₃
hypocenter	C₁₃	#33772
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
2. C₂×#11718360	C₂	semidirect
3. #13117092	C₂	semidirect
4. #13117064	C₂	semidirect
5. #13116882	C₂	semidirect
6. #13116755	C₂	semidirect
7. C₁₃×#2162	C₂	semidirect
112. #2162	D₁₃	semidirect
118. C₂×C₄×C₁₃	SD₃₂	semidirect
260. C₁₃	#33772	semidirect
279. C₂×C₄	#88040	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
C₂×#11718360	2
#13117092	2
#13117064	2
#13116882	2
#13116755	2
C₁₃×#2162	2
#33772	13