Subgroups — #16018323

Order 3328.

834 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718855	C₂	1	4	yes	maximal, complemented
2	#13125264	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117095	C₂	1	4	yes	maximal, complemented
4	#13117093	C₂	1	1	yes	maximal, characteristic, complemented
5	#13117091	C₂	1	1	yes	maximal, characteristic, complemented
6	#13117088	C₂	1	1	yes	maximal, characteristic, complemented
7	#13117085	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#135	—	2	1	no	—
11	C₂×C₁₃×#135	C₂²	1	2	yes	—
12	C₁₃×#532	—	2	1	no	—
13	C₂×#88158	—	2	1	no	—
14	C₂×#88158	C₂²	1	2	yes	—
15	#11719182	—	2	1	no	—
16	C₁₃×#495	—	2	1	no	—
17	C₁₃×#495	C₂²	1	2	yes	—
18	C₁₃×#494	C₂²	1	1	yes	characteristic
19	C₁₃×#493	C₂²	1	1	yes	characteristic
20	C₁₃×#492	—	2	1	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	#11718855	—	2	16	no	—
23	C₂×#88069	C₂²	1	4	yes	—
24	C₂×#88068	C₂²	1	4	yes	—
25	C₂×#88067	C₂²	1	4	yes	—
26	C₂×#88066	C₂²	1	4	yes	—
27	C₂²×#6090	C₂²	1	2	yes	—
28	#11718360	C₂²	1	1	yes	characteristic
29	#11718357	—	2	1	no	—
30	#11718357	C₂²	1	2	yes	—
31	#11718356	C₂²	1	1	yes	characteristic
32	#11718354	—	2	1	no	—
33	C₄×#6090	C₂²	1	1	yes	characteristic
34	C₄×#6090	C₂²	1	1	yes	characteristic
35	C₄×#6090	C₂²	1	2	yes	—
36	C₂×C₁₃×#41	—	2	2	no	—
37	C₂×#6127	—	2	2	no	—
38	C₁₃×#135	—	2	8	no	—
39	C₁₃×#135	—	4	4	no	—
40	C₂×C₁₃×Q₁₆	—	2	1	no	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×SD₁₆	—	2	2	no	—
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×#127	—	2	1	no	—
50	C₁₃×#124	—	2	2	no	—
51	C₁₃×#121	—	4	2	no	—
52	C₄×D₄×C₁₃	—	4	2	no	—
53	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
54	#88184	—	4	2	no	—
55	C₂×Dic₅₂	—	2	1	no	—
56	C₂×Dic₅₂	C₂³	1	2	yes	—
57	#88158	—	2	8	no	—
58	#88158	—	4	4	no	—
59	C₂×D₁₀₄	—	2	1	no	—
60	C₂×D₁₀₄	C₂³	1	2	yes	—
61	C₂×#6095	—	2	2	no	—
62	C₂×#6095	—	2	2	no	—
63	#88129	—	2	2	no	—
64	#88128	—	2	1	no	—
65	C₄×D₅₂	—	4	2	no	—
66	#88123	—	2	1	no	—
67	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#88069	—	2	8	no	—
72	#88068	—	2	8	no	—
73	#88067	—	2	8	no	—
74	#88066	—	2	8	no	—
75	C₂×#6090	D₄	1	8	yes	complemented
76	C₂×#6090	C₂³	1	2	yes	—
77	C₂×#6090	C₂³	1	2	yes	—
78	#33713	—	13	1	no	maximal, Sylow
79	C₁₃×#41	—	4	4	no	—
80	C₁₃×#41	—	4	4	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×D₄×C₁₃	—	2	2	no	—
83	C₂×D₄×C₁₃	—	4	2	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂²×C₄×C₁₃	—	4	2	no	—
86	C₂²×C₄×C₁₃	D₈	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₃	—	4	2	no	—
89	#6127	—	4	4	no	—
90	#6127	—	4	4	no	—
91	C₂×D₅₂	—	2	2	no	—
92	C₂×D₅₂	—	4	2	no	—
93	C₂×C₄×D₁₃	—	4	2	no	—
94	C₂×Dic₂₆	—	2	2	no	—
95	C₁₃×Q₁₆	—	2	4	no	—
96	C₁₃×Q₁₆	—	4	2	no	—
97	C₁₃×SD₁₆	—	4	4	no	—
98	C₁₃×SD₁₆	—	4	4	no	—
99	C₁₃×D₈	—	2	4	no	—
100	C₁₃×D₈	—	4	2	no	—
101	C₂×C₈×C₁₃	D₈	1	4	yes	—
102	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	4	no	—
109	C₄²×C₁₃	D₈	1	2	yes	—
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	#6103	—	4	4	no	—
113	C₄⋊Dic₁₃	—	4	2	no	—
114	Dic₅₂	—	2	4	no	—
115	Dic₅₂	—	4	2	no	—
116	D₁₀₄	—	2	4	no	—
117	D₁₀₄	—	4	2	no	—
118	#6095	—	4	4	no	—
119	#6095	—	4	4	no	—
120	#6090	C₂×D₄	1	4	yes	—
121	#6090	C₂×D₄	1	4	yes	—
122	C₂×#508	—	13	4	no	—
123	#3147	—	13	1	no	—
124	#3147	D₁₃	1	1	yes	characteristic, complemented
125	#2249	—	13	4	no	—
126	#2248	—	13	1	no	—
127	#2248	—	13	1	no	—
128	#2247	—	13	1	no	—
129	#2246	—	13	1	no	—
130	C₂×C₄×C₁₆	—	13	1	no	—
131	C₂³×C₁₃	—	4	2	no	—
132	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂²×D₁₃	—	4	2	no	—
134	Q₈×C₁₃	—	4	2	no	—
135	Q₈×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	no	—
138	D₄×C₁₃	—	8	4	no	—
139	D₄×C₁₃	—	8	4	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	8	4	no	—
143	C₂×C₄×C₁₃	#127	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₁₃	—	8	4	no	—
150	C₂×Dic₁₃	—	4	2	no	—
151	D₅₂	—	4	2	no	—
152	D₅₂	—	4	2	no	—
153	D₅₂	—	8	4	no	—
154	C₄×D₁₃	—	8	4	no	—
155	Dic₂₆	—	4	2	no	—
156	Dic₂₆	—	4	2	no	—
157	C₈×C₁₃	#127	1	2	yes	—
158	C₈×C₁₃	C₂×D₈	1	2	yes	—
159	C₈×C₁₃	C₂×D₈	1	2	yes	—
160	C₈×C₁₃	C₂²×D₄	1	2	yes	—
161	C₂×#135	—	2	1	no	—
162	C₂×#135	—	13	2	no	—
163	C₂×#135	—	26	1	no	—
164	C₂×#135	C₂×D₁₃	1	2	yes	—
165	#532	—	2	1	no	—
166	#532	—	26	1	no	—
167	#508	—	26	16	no	—
168	C₂×Q₃₂	—	13	4	no	—
169	C₂×SD₃₂	—	13	4	no	—
170	C₂×SD₃₂	—	13	4	no	—
171	C₂×D₁₆	—	13	4	no	—
172	C₂²×C₁₆	—	13	2	no	—
173	#495	—	2	1	no	—
174	#495	—	13	2	no	—
175	#495	—	26	1	no	—
176	#495	C₂×D₁₃	1	2	yes	—
177	#494	—	13	1	no	—
178	#494	C₂×D₁₃	1	1	yes	characteristic
179	#493	—	13	1	no	—
180	#493	C₂×D₁₃	1	1	yes	characteristic
181	#492	—	2	1	no	—
182	#492	—	26	1	no	—
183	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
184	C₄×C₁₆	—	13	1	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₄×C₁₆	—	13	2	no	—
187	C₂²×C₁₃	—	2	2	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	4	no	—
190	C₂²×C₁₃	#493	1	2	yes	—
191	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
192	C₂×D₁₃	—	8	2	no	—
193	C₂×D₁₃	—	8	4	no	—
194	C₄×C₁₃	—	2	2	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	#493	1	2	yes	—
197	C₄×C₁₃	#508	1	4	yes	complemented
198	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
199	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
200	Dic₁₃	—	8	2	no	—
201	C₂×#41	—	2	2	no	—
202	C₂×#41	—	26	2	no	—
203	#135	—	2	8	no	—
204	#135	—	4	4	no	—
205	#135	—	26	8	no	—
206	#135	—	52	4	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	—	13	2	no	—
209	C₂×Q₁₆	—	26	1	no	—
210	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×SD₁₆	—	26	2	no	—
214	C₂×SD₁₆	—	26	2	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	13	2	no	—
217	C₂×D₈	—	26	1	no	—
218	C₂×D₈	C₂²×D₁₃	1	2	yes	—
219	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	26	1	no	—
223	#127	—	2	1	no	—
224	#127	—	26	1	no	—
225	#124	—	2	2	no	—
226	#124	—	26	2	no	—
227	#121	—	4	2	no	—
228	#121	—	52	2	no	—
229	C₄×D₄	—	4	2	no	—
230	C₄×D₄	—	52	2	no	—
231	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
232	Q₃₂	—	26	8	no	—
233	SD₃₂	—	26	8	no	—
234	SD₃₂	—	26	8	no	—
235	D₁₆	—	26	8	no	—
236	C₂×C₁₆	—	13	8	no	—
237	C₂×C₁₆	—	13	2	no	—
238	C₂×C₁₆	—	13	2	no	—
239	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
240	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
243	C₂×C₁₃	—	2	2	no	—
244	C₂×C₁₃	—	16	2	no	—
245	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
246	C₂×C₁₃	C₂×#508	1	2	yes	—
247	D₁₃	—	16	2	no	—
248	#41	—	4	4	no	—
249	#41	—	4	4	no	—
250	#41	—	52	4	no	—
251	#41	—	52	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	26	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	26	2	no	—
258	C₂×D₄	—	52	2	no	—
259	C₂×D₄	—	52	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	52	2	no	—
262	C₂²×C₄	#6104	1	2	yes	—
263	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	Q₁₆	—	2	4	no	—
265	Q₁₆	—	4	2	no	—
266	Q₁₆	—	26	4	no	—
267	Q₁₆	—	52	2	no	—
268	SD₁₆	—	4	4	no	—
269	SD₁₆	—	4	4	no	—
270	SD₁₆	—	52	4	no	—
271	SD₁₆	—	52	4	no	—
272	D₈	—	2	4	no	—
273	D₈	—	4	2	no	—
274	D₈	—	26	4	no	—
275	D₈	—	52	2	no	—
276	C₂×C₈	#6104	1	4	yes	—
277	C₂×C₈	D₄×D₁₃	1	4	yes	—
278	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
279	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₂²⋊C₄	—	4	4	no	—
285	C₂²⋊C₄	—	52	4	no	—
286	C₄²	#6104	1	2	yes	—
287	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
289	C₁₆	—	13	4	no	—
290	C₁₆	—	13	4	no	—
291	C₁₃	#33713	1	1	yes	characteristic, complemented, Sylow
292	C₂³	—	4	2	no	—
293	C₂³	—	52	2	no	—
294	C₂³	C₂×#6104	1	1	yes	characteristic
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	52	2	no	—
298	Q₈	—	52	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	52	2	no	—
304	D₄	—	52	2	no	—
305	D₄	—	104	4	no	—
306	D₄	—	104	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	52	2	no	—
311	C₂×C₄	—	104	4	no	—
312	C₂×C₄	C₂×#6104	1	2	yes	—
313	C₂×C₄	C₂×#6104	1	2	yes	—
314	C₂×C₄	C₂×#6104	1	2	yes	—
315	C₂×C₄	C₂×#6104	1	1	yes	characteristic
316	C₂×C₄	#88194	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
318	C₈	C₂×#6104	1	2	yes	—
319	C₈	C₂×#6104	1	2	yes	—
320	C₈	#88194	1	2	yes	—
321	C₈	C₂×D₄×D₁₃	1	2	yes	—
322	C₂²	—	2	2	no	—
323	C₂²	—	8	4	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	104	4	no	—
326	C₂²	—	104	2	no	—
327	C₂²	#11718734	1	2	yes	—
328	C₂²	C₂²×#6104	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	104	2	no	—
332	C₄	#11718734	1	2	yes	—
333	C₄	#11718855	1	4	yes	complemented
334	C₄	C₂²×#6104	1	1	yes	characteristic
335	C₄	C₂×#88194	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	16	2	no	—
338	C₂	—	208	2	no	—
339	C₂	C₂×#11718734	1	1	yes	characteristic
340	C₂	C₂×#11718855	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3147	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#3147	D₁₃
hypocenter	C₁₃	#33713
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#11718855	C₂	semidirect
2. #13125264	C₂	semidirect
3. #13117095	C₂	semidirect
4. #13117093	C₂	semidirect
5. #13117091	C₂	semidirect
6. #13117088	C₂	semidirect
7. #13117085	C₂	semidirect
8. C₂×C₄×#6090	C₂	semidirect
9. C₁₃×#3147	C₂	semidirect
75. C₂×#6090	D₄	semidirect
124. #3147	D₁₃	semidirect
197. C₄×C₁₃	#508	semidirect
291. C₁₃	#33713	semidirect
333. C₄	#11718855	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718855	2
#13125264	2
#13117095	2
#13117093	2
#13117091	2
#13117088	2
#13117085	2
C₂×C₄×#6090	2
C₁₃×#3147	2
#33713	13