Subgroups — #16017211

Order 3328.

806 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	C₂×#11718854	C₂	1	2	yes	maximal, complemented
2	C₂×#11718356	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116876	C₂	1	4	yes	maximal, complemented
4	#13116871	C₂	1	4	yes	maximal, complemented
5	#13116592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116551	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×#41	—	2	1	no	—
10	C₂²×D₁₀₄	—	2	1	no	—
11	C₂²×D₁₀₄	C₂²	1	2	yes	—
12	C₂×#88128	—	2	1	no	—
13	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	complemented
14	C₄×C₁₃×SD₁₆	—	2	2	no	—
15	C₄×C₁₃×D₈	C₂²	1	2	yes	complemented
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#416	—	2	1	no	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	#11718854	—	2	8	no	—
20	C₂×#88068	C₂²	1	4	yes	complemented
21	C₂×#88066	C₂²	1	4	yes	complemented
22	C₂×#88052	C₂²	1	2	yes	—
23	#11718356	—	2	2	no	—
24	#11718356	C₂²	1	4	yes	—
25	#11718083	C₂²	1	8	yes	complemented
26	C₄⋊#6090	C₂²	1	4	yes	complemented
27	C₂×C₁₃×#41	—	2	1	no	—
28	C₂²×D₅₂	—	2	2	no	—
29	C₂²×D₅₂	—	4	2	no	—
30	C₁₃×#135	—	2	2	no	—
31	C₁₃×#135	D₄	1	4	yes	complemented
32	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×SD₁₆	—	2	1	no	—
34	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	—	2	2	no	—
38	C₄×D₄×C₁₃	—	2	2	no	—
39	C₄×D₄×C₁₃	—	4	2	no	—
40	C₁₃×#117	—	2	1	no	—
41	C₁₃×#117	—	4	1	no	—
42	C₂×C₄²×C₁₃	—	4	1	no	—
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×D₁₀₄	—	2	4	no	—
45	C₂×D₁₀₄	—	2	4	no	—
46	C₂×D₁₀₄	—	2	4	no	—
47	C₂×D₁₀₄	—	2	2	no	—
48	C₂×D₁₀₄	—	4	4	no	—
49	C₂×D₁₀₄	C₂³	1	4	yes	—
50	#88128	—	2	4	no	—
51	#88128	—	4	1	no	—
52	#88128	—	4	1	no	—
53	C₁₃×#107	C₂³	1	2	yes	—
54	C₁₃×#106	—	2	1	no	—
55	C₁₃×#103	—	2	2	no	—
56	C₁₃×#102	—	2	2	no	—
57	C₄×C₈×C₁₃	D₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂³	1	2	yes	—
59	#88068	—	2	8	no	—
60	#88066	—	2	8	no	—
61	#88052	—	2	4	no	—
62	C₂×#6090	C₂³	1	4	yes	—
63	#33582	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	—	4	2	no	—
65	C₁₃×#41	—	2	4	no	—
66	C₁₃×#41	—	4	2	no	—
67	C₂×Q₈×C₁₃	—	2	1	no	—
68	C₂×D₄×C₁₃	—	2	1	no	—
69	C₂×D₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	D₈	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×D₅₂	—	2	4	no	—
75	C₂×D₅₂	—	4	4	no	—
76	C₂×D₅₂	—	4	4	no	—
77	C₂×D₅₂	—	4	4	no	—
78	C₂×D₅₂	—	4	4	no	—
79	C₂×D₅₂	—	4	2	no	—
80	C₂×D₅₂	—	8	8	no	—
81	C₁₃×Q₁₆	—	2	1	no	—
82	C₁₃×Q₁₆	C₂×D₄	1	2	yes	complemented
83	C₁₃×SD₁₆	—	2	4	no	—
84	C₁₃×D₈	—	2	1	no	—
85	C₁₃×D₈	C₂×D₄	1	2	yes	complemented
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₂²⋊C₄	—	4	1	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₄²×C₁₃	—	4	1	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	2	yes	—
101	D₁₀₄	—	2	4	no	—
102	D₁₀₄	—	2	4	no	—
103	D₁₀₄	—	4	8	no	—
104	D₁₀₄	—	4	4	no	—
105	#6090	—	2	4	no	—
106	C₂×#509	—	13	2	no	—
107	C₂×#493	—	13	1	no	—
108	C₄×#135	D₁₃	1	1	yes	characteristic, complemented
109	#2207	—	13	4	no	—
110	#2206	—	13	4	no	—
111	#2150	—	13	1	no	—
112	#2140	—	13	2	no	—
113	C₂³×C₁₃	—	4	1	no	—
114	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₁₃	—	4	4	no	—
116	C₂²×D₁₃	—	8	8	no	—
117	C₂²×D₁₃	—	8	4	no	—
118	Q₈×C₁₃	—	2	2	no	—
119	Q₈×C₁₃	—	4	1	no	—
120	D₄×C₁₃	—	2	2	no	—
121	D₄×C₁₃	—	4	4	no	—
122	D₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	8	4	no	—
128	C₂×C₄×C₁₃	#121	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	D₅₂	—	4	4	no	—
135	D₅₂	—	4	4	no	—
136	D₅₂	—	8	8	no	—
137	D₅₂	—	8	4	no	—
138	D₅₂	—	8	8	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	#121	1	2	yes	—
141	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
144	C₂²×D₈	—	13	2	no	—
145	C₂²×D₈	—	26	1	no	—
146	C₂×#127	—	26	1	no	—
147	C₄×#41	—	2	1	no	—
148	#509	—	26	8	no	—
149	C₂×SD₃₂	—	13	4	no	—
150	C₂×D₁₆	—	13	4	no	—
151	C₂×M₃₂	—	13	2	no	—
152	#493	—	13	4	no	—
153	#493	—	26	2	no	—
154	C₄×Q₁₆	C₂×D₁₃	1	2	yes	complemented
155	C₄×SD₁₆	—	2	2	no	—
156	C₄×D₈	C₂×D₁₃	1	2	yes	complemented
157	#427	C₂×D₁₃	1	1	yes	characteristic
158	#416	—	2	1	no	—
159	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
160	C₄⋊C₁₆	—	13	4	no	—
161	#357	—	13	8	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	4	2	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	#459	1	2	yes	—
166	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₁₃	—	8	8	no	—
168	C₂×D₁₃	—	8	4	no	—
169	C₂×D₁₃	—	16	8	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	4	2	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	#459	1	2	yes	—
174	C₄×C₁₃	#509	1	4	yes	complemented
175	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
176	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×#41	—	2	1	no	—
178	C₂²×D₄	—	26	2	no	—
179	C₂²×D₄	—	52	2	no	—
180	#135	—	2	2	no	—
181	#135	D₅₂	1	4	yes	complemented
182	C₂×Q₁₆	C₂²×D₁₃	1	1	yes	characteristic
183	C₂×SD₁₆	—	2	1	no	—
184	C₂×D₈	—	13	4	no	—
185	C₂×D₈	—	26	4	no	—
186	C₂×D₈	—	26	4	no	—
187	C₂×D₈	—	26	2	no	—
188	C₂×D₈	—	26	4	no	—
189	C₂×D₈	—	52	4	no	—
190	C₂×D₈	C₂²×D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
193	#127	—	26	4	no	—
194	#127	—	52	1	no	—
195	#127	—	52	1	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	4	2	no	—
199	#117	—	2	1	no	—
200	#117	—	4	1	no	—
201	C₂×C₄²	—	4	1	no	—
202	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
203	SD₃₂	—	26	8	no	—
204	D₁₆	—	26	8	no	—
205	M₃₂	—	26	4	no	—
206	C₂×C₁₆	—	13	4	no	—
207	#107	C₂²×D₁₃	1	2	yes	—
208	#106	—	2	1	no	—
209	#103	—	2	2	no	—
210	#102	—	2	2	no	—
211	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
212	C₄×C₈	C₂²×D₁₃	1	2	yes	—
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	—	8	2	no	—
215	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#509	1	2	yes	—
217	D₁₃	—	16	4	no	—
218	C₂⁴	—	52	2	no	—
219	#41	—	2	4	no	—
220	#41	—	4	2	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	26	4	no	—
225	C₂×D₄	—	52	4	no	—
226	C₂×D₄	—	52	4	no	—
227	C₂×D₄	—	52	4	no	—
228	C₂×D₄	—	52	2	no	—
229	C₂×D₄	—	52	4	no	—
230	C₂×D₄	—	104	8	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	#6104	1	2	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
235	Q₁₆	—	2	1	no	—
236	Q₁₆	C₂×D₅₂	1	2	yes	complemented
237	SD₁₆	—	2	4	no	—
238	D₈	—	2	1	no	—
239	D₈	—	26	4	no	—
240	D₈	—	26	4	no	—
241	D₈	—	52	8	no	—
242	D₈	—	52	4	no	—
243	D₈	C₂×D₅₂	1	2	yes	complemented
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂×D₅₂	1	2	yes	—
246	C₂×C₈	#6127	1	2	yes	—
247	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₂²⋊C₄	—	4	1	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	—	4	2	no	—
256	C₄²	#6104	1	1	yes	characteristic
257	C₄²	#6104	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
259	C₁₆	—	26	4	no	—
260	C₁₃	#33582	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	4	1	no	—
262	C₂³	—	52	4	no	—
263	C₂³	—	104	8	no	—
264	C₂³	—	104	4	no	—
265	C₂³	C₂×#6104	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	—	4	1	no	—
268	D₄	—	2	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	1	no	—
271	D₄	—	52	4	no	—
272	D₄	—	52	4	no	—
273	D₄	—	104	8	no	—
274	D₄	—	104	4	no	—
275	D₄	—	104	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	#88184	1	2	yes	—
282	C₂×C₄	C₂×#6104	1	2	yes	—
283	C₂×C₄	C₂×#6104	1	1	yes	characteristic
284	C₂×C₄	C₂×#6104	1	2	yes	—
285	C₂×C₄	#88203	1	2	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	#88184	1	2	yes	—
289	C₈	C₂²×D₅₂	1	1	yes	characteristic
290	C₈	C₂×#6127	1	1	yes	characteristic
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	104	8	no	—
295	C₂²	—	104	4	no	—
296	C₂²	—	208	8	no	—
297	C₂²	#11718676	1	2	yes	—
298	C₂²	C₂²×#6104	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	2	no	—
302	C₄	#11718676	1	2	yes	—
303	C₄	#11718854	1	4	yes	complemented
304	C₄	C₂×#88184	1	1	yes	characteristic
305	C₄	C₂×#88203	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	8	2	no	—
308	C₂	—	208	4	no	—
309	C₂	C₂×#11718676	1	1	yes	characteristic
310	C₂	C₂×#11718854	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	#33582
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#11718854	C₂	semidirect
2. C₂×#11718356	C₂	semidirect
3. #13116876	C₂	semidirect
4. #13116871	C₂	semidirect
5. #13116592	C₂	semidirect
6. #13116551	C₂	semidirect
7. C₄×C₁₃×#135	C₂	semidirect
13. C₄×C₁₃×Q₁₆	C₂²	semidirect
15. C₄×C₁₃×D₈	C₂²	semidirect
20. C₂×#88068	C₂²	semidirect
21. C₂×#88066	C₂²	semidirect
25. #11718083	C₂²	semidirect
26. C₄⋊#6090	C₂²	semidirect
31. C₁₃×#135	D₄	semidirect
82. C₁₃×Q₁₆	C₂×D₄	semidirect
85. C₁₃×D₈	C₂×D₄	semidirect
108. C₄×#135	D₁₃	semidirect
154. C₄×Q₁₆	C₂×D₁₃	semidirect
156. C₄×D₈	C₂×D₁₃	semidirect
174. C₄×C₁₃	#509	semidirect
181. #135	D₅₂	semidirect
236. Q₁₆	C₂×D₅₂	semidirect
243. D₈	C₂×D₅₂	semidirect
260. C₁₃	#33582	semidirect
303. C₄	#11718854	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718854	2
C₂×#11718356	2
#13116876	2
#13116871	2
#13116592	2
#13116551	2
C₄×C₁₃×#135	2
#33582	13