Subgroups — #16018835

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718356	C₂	1	1	yes	maximal, characteristic
2	#13117093	C₂	1	2	yes	maximal, complemented
3	#13117061	C₂	1	2	yes	maximal, complemented
4	#13116876	C₂	1	4	yes	maximal, complemented
5	#13116764	C₂	1	4	yes	maximal, complemented
6	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#533	—	2	1	no	—
9	C₂²×D₁₀₄	—	2	1	no	—
10	C₂²×D₁₀₄	C₂²	1	2	yes	—
11	C₂×#88128	—	2	1	no	—
12	C₁₃×#500	C₂²	1	1	yes	characteristic, complemented
13	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#487	—	2	1	no	—
15	C₁₃×#467	—	2	1	no	—
16	C₁₃×#467	C₂²	1	2	yes	—
17	C₄×C₁₃×Q₁₆	—	2	1	no	—
18	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88068	C₂²	1	8	yes	complemented
21	#11718356	—	2	1	no	—
22	#11718356	—	2	1	no	—
23	#11718356	C₂²	1	2	yes	—
24	#11718356	C₂²	1	2	yes	—
25	C₂²⋊#6090	C₂²	1	2	yes	—
26	#11718083	C₂²	1	8	yes	complemented
27	C₄⋊#6090	C₂²	1	2	yes	complemented
28	C₄×#6090	C₂²	1	2	yes	complemented
29	C₂²×D₅₂	—	2	2	no	—
30	C₂²×D₅₂	—	4	1	no	—
31	C₂²×D₅₂	—	4	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	—	2	1	no	—
37	C₁₃×C₄⋊Q₈	—	2	1	no	—
38	C₁₃×#125	—	4	1	no	—
39	C₁₃×C₂²⋊Q₈	—	2	2	no	—
40	C₁₃×C₂²⋊Q₈	—	4	1	no	—
41	C₄×Q₈×C₁₃	—	2	2	no	—
42	C₄×Q₈×C₁₃	—	4	1	no	—
43	C₁₃×#117	—	4	1	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×D₁₀₄	—	2	2	no	—
46	C₂×D₁₀₄	—	2	2	no	—
47	C₂×D₁₀₄	—	2	2	no	—
48	C₂×D₁₀₄	—	4	4	no	—
49	C₂×D₁₀₄	—	4	4	no	—
50	C₂×D₁₀₄	C₂³	1	4	yes	—
51	#88128	—	2	2	no	—
52	#88128	—	2	2	no	—
53	#88128	—	4	1	no	—
54	#88128	—	4	1	no	—
55	C₁₃×#107	—	2	1	no	—
56	C₁₃×#107	C₂³	1	2	yes	—
57	C₁₃×#103	—	2	2	no	—
58	C₁₃×#103	—	2	2	no	—
59	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	#88068	—	4	8	no	—
64	C₂×#6090	C₂³	1	4	yes	—
65	#33774	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	—	4	2	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×D₅₂	—	2	4	no	—
73	C₂×D₅₂	—	4	4	no	—
74	C₂×D₅₂	—	4	4	no	—
75	C₂×D₅₂	—	4	2	no	—
76	C₂×D₅₂	—	4	2	no	—
77	C₂×D₅₂	—	4	2	no	—
78	C₂×D₅₂	—	4	2	no	—
79	C₂×D₅₂	—	4	2	no	—
80	C₂×D₅₂	—	8	4	no	—
81	C₂×D₅₂	—	8	4	no	—
82	C₁₃×Q₁₆	—	2	4	no	—
83	C₁₃×Q₁₆	—	4	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	4	1	no	—
92	C₁₃×C₄⋊C₄	—	4	1	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₂²⋊C₄	—	4	2	no	—
97	C₄²×C₁₃	—	4	2	no	—
98	C₄²×C₁₃	D₈	1	1	yes	characteristic
99	C₄²×C₁₃	D₈	1	1	yes	characteristic
100	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	D₁₀₄	—	2	8	no	—
103	D₁₀₄	—	4	4	no	—
104	D₁₀₄	—	4	4	no	—
105	D₁₀₄	—	4	4	no	—
106	#6090	—	2	4	no	—
107	#3170	D₁₃	1	1	yes	characteristic, complemented
108	C₂×#493	—	13	1	no	—
109	#2248	—	13	2	no	—
110	#2240	—	13	2	no	—
111	#2207	—	13	4	no	—
112	#2187	—	13	4	no	—
113	#2162	—	13	1	no	—
114	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₁₃	—	4	4	no	—
116	C₂²×D₁₃	—	8	4	no	—
117	C₂²×D₁₃	—	8	4	no	—
118	C₂²×D₁₃	—	8	4	no	—
119	Q₈×C₁₃	—	4	2	no	—
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
128	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	D₅₂	—	4	4	no	—
135	D₅₂	—	4	4	no	—
136	D₅₂	—	8	4	no	—
137	D₅₂	—	8	4	no	—
138	D₅₂	—	8	4	no	—
139	D₅₂	—	8	4	no	—
140	D₅₂	—	8	4	no	—
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₈×C₁₃	H(5,𝔽₂)	1	2	yes	—
145	C₂²×D₈	—	13	2	no	—
146	C₂²×D₈	—	26	1	no	—
147	#533	—	2	1	no	—
148	C₂×#127	—	26	1	no	—
149	C₂×SD₃₂	—	13	8	no	—
150	#500	C₂×D₁₃	1	1	yes	characteristic, complemented
151	#494	C₂×D₁₃	1	1	yes	characteristic, complemented
152	#493	—	13	2	no	—
153	#493	—	13	2	no	—
154	#493	—	26	1	no	—
155	#493	—	26	1	no	—
156	#487	—	2	1	no	—
157	#467	—	2	1	no	—
158	#467	C₂×D₁₃	1	2	yes	—
159	C₄×Q₁₆	—	2	1	no	—
160	C₄×Q₁₆	C₂×D₁₃	1	2	yes	complemented
161	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
162	C₄⋊C₁₆	—	13	2	no	—
163	#357	—	13	8	no	—
164	C₂²⋊C₁₆	—	13	2	no	—
165	C₄×C₁₆	—	13	2	no	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
169	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₁₃	—	8	4	no	—
171	C₂×D₁₃	—	8	4	no	—
172	C₂×D₁₃	—	8	4	no	—
173	C₂×D₁₃	—	16	8	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	8	2	no	—
177	C₄×C₁₃	—	8	2	no	—
178	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
179	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
180	C₄×C₁₃	#509	1	2	yes	—
181	C₄×C₁₃	#535	1	1	yes	characteristic
182	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂²×D₄	—	26	2	no	—
184	C₂²×D₄	—	52	1	no	—
185	C₂²×D₄	—	52	1	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
188	C₂×D₈	—	13	4	no	—
189	C₂×D₈	—	26	2	no	—
190	C₂×D₈	—	26	2	no	—
191	C₂×D₈	—	26	2	no	—
192	C₂×D₈	—	52	4	no	—
193	C₂×D₈	—	52	4	no	—
194	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	—	2	1	no	—
198	#127	—	26	2	no	—
199	#127	—	26	2	no	—
200	#127	—	52	1	no	—
201	#127	—	52	1	no	—
202	#125	—	4	1	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	4	1	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	4	1	no	—
207	#117	—	4	1	no	—
208	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
209	SD₃₂	—	52	8	no	—
210	C₂×C₁₆	—	13	4	no	—
211	#107	—	2	1	no	—
212	#107	C₂²×D₁₃	1	2	yes	—
213	#103	—	2	2	no	—
214	#103	—	2	2	no	—
215	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	#3205	1	1	yes	characteristic
221	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
223	D₁₃	—	16	4	no	—
224	C₂⁴	—	52	2	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×D₄	—	26	4	no	—
228	C₂×D₄	—	52	4	no	—
229	C₂×D₄	—	52	4	no	—
230	C₂×D₄	—	52	2	no	—
231	C₂×D₄	—	52	2	no	—
232	C₂×D₄	—	52	2	no	—
233	C₂×D₄	—	52	2	no	—
234	C₂×D₄	—	52	2	no	—
235	C₂×D₄	—	104	4	no	—
236	C₂×D₄	—	104	4	no	—
237	C₂²×C₄	#6104	1	1	yes	characteristic
238	C₂²×C₄	#6104	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
240	Q₁₆	—	2	4	no	—
241	Q₁₆	—	4	2	no	—
242	D₈	—	26	8	no	—
243	D₈	—	52	4	no	—
244	D₈	—	52	4	no	—
245	D₈	—	52	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	#6104	1	1	yes	characteristic
261	C₄²	#6104	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₁₆	—	26	4	no	—
265	C₁₃	#33774	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	52	4	no	—
267	C₂³	—	104	4	no	—
268	C₂³	—	104	4	no	—
269	C₂³	—	104	4	no	—
270	C₂³	C₂×#6104	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	8	2	no	—
274	D₄	—	52	4	no	—
275	D₄	—	52	4	no	—
276	D₄	—	104	4	no	—
277	D₄	—	104	4	no	—
278	D₄	—	104	4	no	—
279	D₄	—	104	4	no	—
280	D₄	—	104	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	#88068	1	4	yes	complemented
287	C₂×C₄	C₂×#6104	1	1	yes	characteristic
288	C₂×C₄	C₂×#6104	1	1	yes	characteristic
289	C₂×C₄	C₂×#6104	1	1	yes	characteristic
290	C₂×C₄	C₂×#6104	1	1	yes	characteristic
291	C₂×C₄	C₂×#6104	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#88203	1	2	yes	—
296	C₈	#88256	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	104	4	no	—
300	C₂²	—	104	4	no	—
301	C₂²	—	104	4	no	—
302	C₂²	—	208	8	no	—
303	C₂²	C₂×#88068	1	2	yes	—
304	C₂²	C₂²×#6104	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₂×#88068	1	2	yes	—
310	C₄	C₂×#88068	1	2	yes	—
311	C₄	#11718854	1	2	yes	—
312	C₄	C₂×#88203	1	1	yes	characteristic
313	C₄	#11719535	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	208	4	no	—
316	C₂	#13125751	1	1	yes	characteristic
317	C₂	C₂²×#88068	1	1	yes	characteristic
318	C₂	C₂×#11718854	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718356	C₂
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#33774
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13117093	C₂	semidirect
3. #13117061	C₂	semidirect
4. #13116876	C₂	semidirect
5. #13116764	C₂	semidirect
6. #13116635	C₂	semidirect
7. C₁₃×#3170	C₂	semidirect
12. C₁₃×#500	C₂²	semidirect
13. C₁₃×#494	C₂²	semidirect
18. C₄×C₁₃×Q₁₆	C₂²	semidirect
20. C₂×#88068	C₂²	semidirect
26. #11718083	C₂²	semidirect
27. C₄⋊#6090	C₂²	semidirect
28. C₄×#6090	C₂²	semidirect
107. #3170	D₁₃	semidirect
127. C₂×C₄×C₁₃	SD₃₂	semidirect
150. #500	C₂×D₁₃	semidirect
151. #494	C₂×D₁₃	semidirect
160. C₄×Q₁₆	C₂×D₁₃	semidirect
265. C₁₃	#33774	semidirect
286. C₂×C₄	#88068	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718356	2
#13117093	2
#13117061	2
#13116876	2
#13116764	2
#13116635	2
C₁₃×#3170	2
#33774	13