Subgroups — #16830048

Order 3712.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271717	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271710	C₂	1	1	yes	maximal, characteristic
3	#14271560	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271534	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271512	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271508	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271395	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271393	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271391	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271389	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#90773	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#90784	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#90783	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#90805	—	2	1	no	—
17	C₂×SD₁₆×C₂₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
20	#11745308	C₂²	1	1	yes	characteristic
21	#11745306	C₂²	1	1	yes	characteristic
22	C₂×#90784	C₂²	1	1	yes	characteristic
23	#11745302	C₂²	1	1	yes	characteristic
24	#11745300	C₂²	1	1	yes	characteristic
25	C₂×#90783	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊D₂₉	—	2	1	no	—
27	C₂×#90773	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
30	#11745259	C₂²	1	1	yes	characteristic
31	#11745258	—	2	1	no	—
32	#11745257	C₂²	1	1	yes	characteristic
33	#11745251	—	2	1	no	—
34	#11745250	C₂²	1	1	yes	characteristic
35	#11745250	C₂²	1	1	yes	characteristic
36	#11745247	—	2	1	no	—
37	#11745246	—	2	1	no	—
38	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
39	#11745234	—	2	1	no	—
40	C₄²×D₂₉	C₂²	1	1	yes	characteristic
41	C₄×Dic₅₈	—	2	1	no	—
42	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
43	C₂₉×#106	C₂²	1	1	yes	characteristic
44	C₂₉×#103	C₂²	1	1	yes	characteristic
45	C₂₉×#102	C₂²	1	1	yes	characteristic
46	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
47	#11745183	C₂²	1	1	yes	characteristic
48	#11745180	C₂²	1	1	yes	characteristic
49	#11745169	C₂²	1	1	yes	characteristic
50	#11745165	C₂²	1	1	yes	characteristic
51	#11745164	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
53	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
54	#11745158	C₂²	1	1	yes	characteristic
55	#11745157	C₂²	1	1	yes	characteristic
56	#11745156	C₂²	1	1	yes	characteristic
57	#11745156	C₂²	1	1	yes	characteristic
58	C₄×#6597	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	—	2	1	no	—
62	C₂×C₂²⋊D₂₉	—	2	1	no	—
63	C₂×C₂²⋊D₂₉	—	2	1	no	—
64	C₂×C₂²⋊D₂₉	—	2	1	no	—
65	#90805	—	4	4	no	—
66	C₂×D₁₁₆	—	2	1	no	—
67	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₂₉	—	2	1	no	—
69	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
71	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
72	C₂×Dic₅₈	—	2	1	no	—
73	C₂×Dic₅₈	—	2	1	no	—
74	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
75	SD₁₆×C₂₉	—	2	2	no	—
76	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₉	—	2	1	no	—
79	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₂₉	—	2	1	no	—
82	C₄²×C₂₉	—	2	1	no	—
83	C₄²×C₂₉	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₂₉	—	2	1	no	—
85	#90784	—	2	2	no	—
86	#90783	—	2	2	no	—
87	#90781	—	2	1	no	—
88	#90781	—	2	1	no	—
89	#90781	—	2	1	no	—
90	#90781	—	2	1	no	—
91	C₄⋊Dic₂₉	—	2	1	no	—
92	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
94	#90779	—	2	1	no	—
95	#90779	—	2	1	no	—
96	#90779	—	2	1	no	—
97	#90779	—	2	1	no	—
98	#90779	—	2	1	no	—
99	#90779	—	2	1	no	—
100	C₄×Dic₂₉	—	2	1	no	—
101	C₄×Dic₂₉	D₄	1	1	yes	characteristic
102	C₄×Dic₂₉	D₄	1	1	yes	characteristic
103	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
104	C₂×#6597	C₂³	1	1	yes	characteristic
105	C₂×#6597	C₂³	1	1	yes	characteristic
106	#90773	—	2	2	no	—
107	C₈×D₂₉	—	2	2	no	—
108	C₈×D₂₉	—	2	2	no	—
109	C₂³×C₂₉	—	2	1	no	—
110	C₂²×D₂₉	—	2	1	no	—
111	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₂₉	—	2	1	no	—
113	Q₈×C₂₉	—	2	1	no	—
114	D₄×C₂₉	—	2	1	no	—
115	D₄×C₂₉	—	2	1	no	—
116	C₂×C₄×C₂₉	—	2	1	no	—
117	C₂×C₄×C₂₉	—	2	1	no	—
118	C₂×C₄×C₂₉	—	2	1	no	—
119	C₂×C₄×C₂₉	—	4	2	no	—
120	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₂₉	—	4	2	no	—
124	C₂²⋊D₂₉	—	4	2	no	—
125	C₂²⋊D₂₉	—	4	2	no	—
126	C₂×Dic₂₉	—	2	1	no	—
127	C₂×Dic₂₉	—	2	1	no	—
128	C₂×Dic₂₉	—	2	1	no	—
129	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
132	D₁₁₆	—	2	1	no	—
133	D₁₁₆	—	2	1	no	—
134	D₁₁₆	—	4	2	no	—
135	C₄×D₂₉	—	2	1	no	—
136	C₄×D₂₉	—	2	1	no	—
137	C₄×D₂₉	—	2	2	no	—
138	C₄×D₂₉	—	4	2	no	—
139	C₄×D₂₉	SD₁₆	1	4	yes	complemented
140	Dic₅₈	—	2	1	no	—
141	Dic₅₈	—	2	1	no	—
142	Dic₅₈	—	4	2	no	—
143	Dic₅₈	—	4	2	no	—
144	C₈×C₂₉	—	2	1	no	—
145	C₈×C₂₉	#41	1	2	yes	complemented
146	#6597	—	2	1	no	—
147	#6597	#41	1	2	yes	complemented
148	#3167	—	29	1	no	maximal, Sylow
149	C₂²×C₂₉	—	4	1	no	—
150	C₂²×C₂₉	—	4	1	no	—
151	C₂²×C₂₉	—	4	1	no	—
152	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₂₉	—	2	1	no	—
154	C₂×D₂₉	—	2	1	no	—
155	C₂×D₂₉	—	4	1	no	—
156	C₂×D₂₉	—	4	1	no	—
157	C₂×D₂₉	—	4	1	no	—
158	C₂×D₂₉	C₂×SD₁₆	1	2	yes	—
159	C₄×C₂₉	—	2	1	no	—
160	C₄×C₂₉	—	4	1	no	—
161	C₄×C₂₉	—	4	1	no	—
162	C₄×C₂₉	—	4	1	no	—
163	C₄×C₂₉	C₂×SD₁₆	1	2	yes	—
164	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
166	Dic₂₉	—	2	1	no	—
167	Dic₂₉	—	4	1	no	—
168	Dic₂₉	—	4	1	no	—
169	Dic₂₉	—	4	1	no	—
170	Dic₂₉	C₂×SD₁₆	1	2	yes	—
171	Dic₂₉	#135	1	2	yes	—
172	#533	—	29	1	no	—
173	#532	—	29	1	no	—
174	#498	—	29	1	no	—
175	#492	—	29	1	no	—
176	#487	—	29	1	no	—
177	#486	—	29	1	no	—
178	#465	—	29	1	no	—
179	#465	—	29	1	no	—
180	#465	—	29	1	no	—
181	#465	—	29	1	no	—
182	C₄×SD₁₆	—	29	1	no	—
183	C₄×SD₁₆	—	29	1	no	—
184	C₄×SD₁₆	—	29	1	no	—
185	C₄×SD₁₆	D₂₉	1	1	yes	characteristic, complemented
186	C₂×C₄×C₈	—	29	1	no	—
187	C₂×C₂₉	—	8	1	no	—
188	C₂×C₂₉	#532	1	1	yes	characteristic
189	C₂×C₂₉	C₂²×SD₁₆	1	1	yes	characteristic
190	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
191	D₂₉	—	2	2	no	—
192	D₂₉	—	8	1	no	—
193	C₂×#41	—	58	1	no	—
194	C₂×SD₁₆	—	29	1	no	—
195	C₂×SD₁₆	—	29	1	no	—
196	C₂×SD₁₆	—	29	1	no	—
197	C₂×SD₁₆	C₂×D₂₉	1	1	yes	characteristic
198	C₂²×C₈	—	29	1	no	—
199	C₂²×C₈	—	29	1	no	—
200	C₄⋊Q₈	—	29	1	no	—
201	C₄⋊Q₈	—	29	1	no	—
202	C₄⋊Q₈	—	29	1	no	—
203	#127	—	29	1	no	—
204	#125	—	58	1	no	—
205	#124	—	58	1	no	—
206	C₂²⋊Q₈	—	29	1	no	—
207	C₂²⋊Q₈	—	29	1	no	—
208	C₂²⋊Q₈	—	58	1	no	—
209	#121	—	29	1	no	—
210	#121	—	29	1	no	—
211	#121	—	58	1	no	—
212	C₄×Q₈	—	29	1	no	—
213	C₄×Q₈	—	58	1	no	—
214	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
215	C₄×D₄	—	29	1	no	—
216	C₄×D₄	—	58	1	no	—
217	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
218	#117	—	58	1	no	—
219	C₂×C₄²	—	29	1	no	—
220	#106	—	29	1	no	—
221	#106	—	29	1	no	—
222	#106	—	29	1	no	—
223	#106	C₂×D₂₉	1	1	yes	characteristic
224	#103	—	29	1	no	—
225	#103	—	29	1	no	—
226	#103	—	29	1	no	—
227	#103	C₂×D₂₉	1	1	yes	characteristic
228	#102	—	29	1	no	—
229	#102	—	29	1	no	—
230	#102	—	29	1	no	—
231	#102	C₂×D₂₉	1	1	yes	characteristic
232	C₄×C₈	—	29	1	no	—
233	C₄×C₈	—	29	1	no	—
234	C₄×C₈	—	29	1	no	—
235	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
236	C₂₉	#3167	1	1	yes	characteristic, complemented, Sylow
237	#41	—	116	4	no	—
238	C₂×Q₈	—	29	1	no	—
239	C₂×Q₈	—	58	1	no	—
240	C₂×Q₈	—	58	1	no	—
241	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
242	C₂×D₄	—	29	1	no	—
243	C₂×D₄	—	58	1	no	—
244	C₂×D₄	—	58	1	no	—
245	C₂×D₄	—	58	1	no	—
246	C₂×D₄	—	58	1	no	—
247	C₂×D₄	C₂²×D₂₉	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	29	1	no	—
250	C₂²×C₄	—	29	1	no	—
251	C₂²×C₄	—	29	1	no	—
252	C₂²×C₄	—	58	1	no	—
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	58	2	no	—
255	SD₁₆	—	58	2	no	—
256	SD₁₆	—	58	2	no	—
257	C₂×C₈	—	29	1	no	—
258	C₂×C₈	—	29	1	no	—
259	C₂×C₈	—	58	2	no	—
260	C₂×C₈	—	58	2	no	—
261	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	29	1	no	—
265	C₄⋊C₄	—	29	1	no	—
266	C₄⋊C₄	—	58	1	no	—
267	C₄⋊C₄	—	58	1	no	—
268	C₄⋊C₄	—	58	1	no	—
269	C₄⋊C₄	—	58	1	no	—
270	C₄⋊C₄	—	58	1	no	—
271	C₄⋊C₄	—	58	1	no	—
272	C₄⋊C₄	—	58	1	no	—
273	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	58	1	no	—
277	C₂²⋊C₄	—	58	1	no	—
278	C₂²⋊C₄	—	58	1	no	—
279	C₂²⋊C₄	—	58	1	no	—
280	C₂²⋊C₄	—	58	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	29	1	no	—
283	C₄²	—	29	1	no	—
284	C₄²	—	29	1	no	—
285	C₄²	—	58	1	no	—
286	C₄²	C₂²×D₂₉	1	1	yes	characteristic
287	C₂³	—	2	1	no	—
288	C₂³	—	29	1	no	—
289	C₂³	—	58	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	58	1	no	—
293	Q₈	—	58	1	no	—
294	Q₈	—	116	2	no	—
295	Q₈	—	116	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	58	1	no	—
299	D₄	—	58	1	no	—
300	D₄	—	116	2	no	—
301	D₄	—	116	2	no	—
302	D₄	—	116	2	no	—
303	D₄	—	116	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	29	1	no	—
309	C₂×C₄	—	29	1	no	—
310	C₂×C₄	—	29	4	no	—
311	C₂×C₄	—	29	1	no	—
312	C₂×C₄	—	58	2	no	—
313	C₂×C₄	—	58	1	no	—
314	C₂×C₄	—	58	1	no	—
315	C₂×C₄	—	58	1	no	—
316	C₂×C₄	—	58	1	no	—
317	C₂×C₄	—	58	1	no	—
318	C₂×C₄	—	116	2	no	—
319	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
320	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	29	2	no	—
324	C₈	—	58	1	no	—
325	C₈	#90805	1	2	yes	complemented
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	29	2	no	—
330	C₂²	—	58	1	no	—
331	C₂²	—	58	1	no	—
332	C₂²	—	116	1	no	—
333	C₂²	—	116	1	no	—
334	C₂²	—	116	1	no	—
335	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	29	2	no	—
341	C₄	—	29	2	no	—
342	C₄	—	58	1	no	—
343	C₄	—	116	1	no	—
344	C₄	—	116	1	no	—
345	C₄	—	116	1	no	—
346	C₄	SD₁₆×D₂₉	1	2	yes	—
347	C₄	C₂×#90805	1	1	yes	characteristic
348	C₄	#41×D₂₉	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	58	2	no	—
351	C₂	—	232	1	no	—
352	C₂	#14271710	1	1	yes	characteristic
353	C₂	C₂×SD₁₆×D₂₉	1	1	yes	characteristic
354	C₂	#135×D₂₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₄×SD₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	#14271710	C₂
hypercenter	C₄×SD₁₆	D₂₉
hypocenter	C₂₉	#3167
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #14271717	C₂	semidirect
3. #14271560	C₂	semidirect
4. #14271534	C₂	semidirect
5. #14271512	C₂	semidirect
6. #14271508	C₂	semidirect
7. #14271395	C₂	semidirect
8. #14271393	C₂	semidirect
9. #14271391	C₂	semidirect
10. #14271389	C₂	semidirect
11. C₄×#90773	C₂	semidirect
12. C₄×#90784	C₂	semidirect
13. C₄×#90783	C₂	semidirect
14. C₄×C₈×D₂₉	C₂	semidirect
15. C₄×SD₁₆×C₂₉	C₂	semidirect
139. C₄×D₂₉	SD₁₆	semidirect
145. C₈×C₂₉	#41	semidirect
147. #6597	#41	semidirect
185. C₄×SD₁₆	D₂₉	semidirect
236. C₂₉	#3167	semidirect
325. C₈	#90805	semidirect

Maximal subgroups

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

subgroup	index
#14271717	2
#14271710	2
#14271560	2
#14271534	2
#14271512	2
#14271508	2
#14271395	2
#14271393	2
#14271391	2
#14271389	2
C₄×#90773	2
C₄×#90784	2
C₄×#90783	2
C₄×C₈×D₂₉	2
C₄×SD₁₆×C₂₉	2
#3167	29