Subgroups — #16829287

Order 3712.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271719	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₂₉	C₂	1	1	yes	maximal, characteristic
3	#14271464	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271463	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271460	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271452	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271451	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271442	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271441	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271204	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271202	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271185	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271166	C₂	1	1	yes	maximal, characteristic
14	#14271166	C₂	1	1	yes	maximal, characteristic
15	C₂₉×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₂₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₉	—	2	1	no	—
18	C₂₉×#125	C₂²	1	1	yes	characteristic, complemented
19	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
20	D₄×Dic₂₉	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₂₉	—	2	1	no	—
22	C₂×#90772	C₂²	1	1	yes	characteristic
23	C₂×#90772	C₂²	1	1	yes	characteristic
24	#11745258	—	2	1	no	—
25	#11745258	—	2	1	no	—
26	#11745255	C₂²	1	1	yes	characteristic
27	#11745254	C₂²	1	1	yes	characteristic
28	#11745254	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×D₂₉	C₂²	1	1	yes	characteristic
30	#11745250	C₂²	1	1	yes	characteristic, complemented
31	#11745250	C₂²	1	1	yes	characteristic, complemented
32	#11745249	—	2	1	no	—
33	#11745249	—	2	1	no	—
34	#11745243	—	2	1	no	—
35	C₂²⋊C₄×D₂₉	—	2	1	no	—
36	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
37	C₄²×D₂₉	C₂²	1	1	yes	characteristic
38	#11745232	C₂²	1	1	yes	characteristic, complemented
39	C₂₉×#107	C₂²	1	1	yes	characteristic, complemented
40	C₂₉×#106	C₂²	1	1	yes	characteristic, complemented
41	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₂₉×#102	C₂²	1	1	yes	characteristic
43	C₂₉×#102	C₂²	1	1	yes	characteristic
44	#11745180	C₂²	1	1	yes	characteristic
45	#11745180	C₂²	1	1	yes	characteristic
46	#11745169	C₂²	1	1	yes	characteristic
47	#11745169	C₂²	1	1	yes	characteristic
48	#11745166	C₂²	1	1	yes	characteristic, complemented
49	#11745165	C₂²	1	1	yes	characteristic, complemented
50	#11745162	C₂²	1	1	yes	characteristic, complemented
51	#11745162	C₂²	1	1	yes	characteristic, complemented
52	#11745157	C₂²	1	1	yes	characteristic
53	#11745157	C₂²	1	1	yes	characteristic
54	#11745156	C₂²	1	1	yes	characteristic, complemented
55	#11745156	C₂²	1	1	yes	characteristic, complemented
56	#11745155	C₂²	1	1	yes	characteristic, complemented
57	#11745155	C₂²	1	1	yes	characteristic, complemented
58	C₄⋊#6597	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₂₉	—	2	1	no	—
60	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	—	2	1	no	—
62	C₂×C₂²⋊D₂₉	—	2	1	no	—
63	C₂²×Dic₂₉	—	2	1	no	—
64	D₄×D₂₉	—	2	2	no	—
65	D₄×D₂₉	Q₈	1	4	yes	complemented
66	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₂₉	—	2	1	no	—
68	C₂×C₄×D₂₉	—	2	1	no	—
69	C₂×C₄×D₂₉	—	2	1	no	—
70	C₂×C₄×D₂₉	—	4	4	no	—
71	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
72	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
74	C₂×Dic₅₈	—	2	1	no	—
75	C₂×Dic₅₈	—	2	1	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	—	2	1	no	—
80	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₂₉	—	2	1	no	—
84	C₄²×C₂₉	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₂₉	—	2	1	no	—
86	#90781	—	2	1	no	—
87	#90781	—	2	1	no	—
88	#90781	—	2	1	no	—
89	#90781	—	2	1	no	—
90	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
93	#90779	—	2	1	no	—
94	#90779	—	2	1	no	—
95	#90779	—	2	1	no	—
96	#90779	—	2	1	no	—
97	#90779	—	2	1	no	—
98	C₄×Dic₂₉	—	2	1	no	—
99	C₄×Dic₂₉	—	2	1	no	—
100	C₄×Dic₂₉	D₄	1	1	yes	characteristic
101	C₄×Dic₂₉	D₄	1	1	yes	characteristic
102	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
103	C₂×#6597	C₂³	1	1	yes	characteristic
104	C₂×#6597	C₂³	1	1	yes	characteristic
105	#90772	—	2	2	no	—
106	#90772	—	2	2	no	—
107	C₂³×C₂₉	—	2	1	no	—
108	C₂²×D₂₉	—	2	1	no	—
109	C₂²×D₂₉	—	2	4	no	—
110	C₂²×D₂₉	—	4	2	no	—
111	C₂²×D₂₉	—	4	2	no	—
112	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
113	D₄×C₂₉	—	2	1	no	—
114	D₄×C₂₉	C₂×Q₈	1	2	yes	—
115	C₂×C₄×C₂₉	—	2	1	no	—
116	C₂×C₄×C₂₉	—	2	1	no	—
117	C₂×C₄×C₂₉	—	2	1	no	—
118	C₂×C₄×C₂₉	—	4	2	no	—
119	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₂₉	—	2	4	no	—
123	C₂×Dic₂₉	—	2	1	no	—
124	C₂×Dic₂₉	—	2	1	no	—
125	C₂×Dic₂₉	—	2	1	no	—
126	C₂×Dic₂₉	—	4	2	no	—
127	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
130	D₁₁₆	—	2	1	no	—
131	D₁₁₆	C₂×Q₈	1	2	yes	—
132	C₄×D₂₉	—	2	2	no	—
133	C₄×D₂₉	—	4	1	no	—
134	C₄×D₂₉	—	4	2	no	—
135	C₄×D₂₉	—	4	2	no	—
136	C₄×D₂₉	—	4	2	no	—
137	C₄×D₂₉	C₂×Q₈	1	2	yes	—
138	C₄×D₂₉	#41	1	2	yes	—
139	Dic₅₈	—	4	2	no	—
140	Dic₅₈	—	4	2	no	—
141	C₈×C₂₉	—	2	1	no	—
142	C₈×C₂₉	—	2	1	no	—
143	#6597	—	2	1	no	—
144	#6597	—	2	1	no	—
145	#3077	—	29	1	no	maximal, Sylow
146	C₂²×C₂₉	—	2	2	no	—
147	C₂²×C₂₉	—	4	1	no	—
148	C₂²×C₂₉	—	4	1	no	—
149	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₂₉	—	2	1	no	—
151	C₂×D₂₉	—	2	1	no	—
152	C₂×D₂₉	—	2	2	no	—
153	C₂×D₂₉	—	4	8	no	—
154	C₂×D₂₉	—	4	1	no	—
155	C₂×D₂₉	—	4	1	no	—
156	C₂×D₂₉	C₂²⋊Q₈	1	2	yes	—
157	C₄×C₂₉	—	2	1	no	—
158	C₄×C₂₉	—	2	1	no	—
159	C₄×C₂₉	—	4	1	no	—
160	C₄×C₂₉	—	4	1	no	—
161	C₄×C₂₉	—	4	1	no	—
162	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
163	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
164	Dic₂₉	—	2	1	no	—
165	Dic₂₉	—	4	1	no	—
166	Dic₂₉	—	4	1	no	—
167	Dic₂₉	—	4	1	no	—
168	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
169	Dic₂₉	AGL(1,ℤ/8)	1	2	yes	—
170	#533	—	29	1	no	—
171	C₂×C₄×D₄	—	29	1	no	—
172	#478	—	29	1	no	—
173	#478	—	29	1	no	—
174	#478	—	29	1	no	—
175	#478	D₂₉	1	1	yes	characteristic, complemented
176	#476	—	29	1	no	—
177	#476	—	29	1	no	—
178	#474	—	29	1	no	—
179	#474	—	29	1	no	—
180	#428	—	29	1	no	—
181	#428	—	29	1	no	—
182	#423	—	29	1	no	—
183	#418	—	29	1	no	—
184	#418	—	29	1	no	—
185	C₂×C₂₉	—	4	2	no	—
186	C₂×C₂₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
187	C₂×C₂₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂×C₂₉	#575	1	1	yes	characteristic
189	D₂₉	—	2	2	no	—
190	D₂₉	—	4	2	no	—
191	C₂²×D₄	—	29	1	no	—
192	C₂³×C₄	—	58	1	no	—
193	C₂×M₁₆	—	29	1	no	—
194	C₂×M₁₆	—	29	1	no	—
195	C₄⋊Q₈	—	29	1	no	—
196	C₄⋊Q₈	—	29	1	no	—
197	#125	—	29	1	no	—
198	#125	C₂×D₂₉	1	1	yes	characteristic, complemented
199	C₂²⋊Q₈	—	58	1	no	—
200	C₂²⋊Q₈	—	58	1	no	—
201	C₄×Q₈	—	58	1	no	—
202	C₄×Q₈	—	58	1	no	—
203	C₄×D₄	—	29	1	no	—
204	C₄×D₄	—	29	1	no	—
205	C₄×D₄	—	29	1	no	—
206	C₄×D₄	—	58	1	no	—
207	C₄×D₄	—	58	1	no	—
208	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
209	#117	—	29	1	no	—
210	#117	—	29	1	no	—
211	C₂×C₄⋊C₄	—	29	1	no	—
212	C₂×C₂²⋊C₄	—	58	1	no	—
213	C₂×C₄²	—	29	1	no	—
214	#107	—	29	1	no	—
215	#107	—	29	1	no	—
216	#107	—	29	1	no	—
217	#107	C₂×D₂₉	1	1	yes	characteristic, complemented
218	#106	—	29	1	no	—
219	#106	—	29	1	no	—
220	#106	—	29	1	no	—
221	#106	C₂×D₂₉	1	1	yes	characteristic, complemented
222	C₄⋊C₈	—	29	1	no	—
223	C₄⋊C₈	—	29	1	no	—
224	C₄⋊C₈	—	29	1	no	—
225	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic, complemented
226	#102	—	29	1	no	—
227	#102	—	29	1	no	—
228	#102	—	29	1	no	—
229	#102	—	29	1	no	—
230	#102	—	29	1	no	—
231	#102	—	29	1	no	—
232	#102	C₂×D₂₉	1	1	yes	characteristic
233	#102	C₂×D₂₉	1	1	yes	characteristic
234	C₂₉	#3077	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	—	58	1	no	—
236	C₂×Q₈	—	58	1	no	—
237	C₂×Q₈	—	58	1	no	—
238	C₂×D₄	—	29	4	no	—
239	C₂×D₄	—	29	1	no	—
240	C₂×D₄	—	58	2	no	—
241	C₂×D₄	—	58	1	no	—
242	C₂×D₄	C₂²×D₂₉	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	29	1	no	—
245	C₂²×C₄	—	29	1	no	—
246	C₂²×C₄	—	29	1	no	—
247	C₂²×C₄	—	58	1	no	—
248	C₂²×C₄	—	58	1	no	—
249	C₂²×C₄	—	58	1	no	—
250	C₂²×C₄	—	58	1	no	—
251	C₂²×C₄	—	116	4	no	—
252	M₁₆	—	58	2	no	—
253	M₁₆	—	58	2	no	—
254	C₂×C₈	—	29	1	no	—
255	C₂×C₈	—	29	1	no	—
256	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	29	1	no	—
261	C₄⋊C₄	—	29	1	no	—
262	C₄⋊C₄	—	29	1	no	—
263	C₄⋊C₄	—	58	1	no	—
264	C₄⋊C₄	—	58	1	no	—
265	C₄⋊C₄	—	58	1	no	—
266	C₄⋊C₄	—	58	1	no	—
267	C₄⋊C₄	—	58	1	no	—
268	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	58	1	no	—
273	C₂²⋊C₄	—	58	1	no	—
274	C₂²⋊C₄	—	58	1	no	—
275	C₂²⋊C₄	—	58	1	no	—
276	C₂²⋊C₄	—	58	1	no	—
277	C₄²	—	29	1	no	—
278	C₄²	—	29	1	no	—
279	C₄²	—	29	1	no	—
280	C₄²	—	58	1	no	—
281	C₄²	—	58	1	no	—
282	C₄²	C₂²×D₂₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	29	1	no	—
285	C₂³	—	58	4	no	—
286	C₂³	—	58	1	no	—
287	C₂³	—	116	2	no	—
288	C₂³	—	116	2	no	—
289	Q₈	—	116	2	no	—
290	Q₈	—	116	2	no	—
291	D₄	—	2	1	no	—
292	D₄	—	29	2	no	—
293	D₄	—	58	4	no	—
294	D₄	—	58	1	no	—
295	D₄	Q₈×D₂₉	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	29	1	no	—
301	C₂×C₄	—	29	1	no	—
302	C₂×C₄	—	29	2	no	—
303	C₂×C₄	—	29	2	no	—
304	C₂×C₄	—	29	1	no	—
305	C₂×C₄	—	58	2	no	—
306	C₂×C₄	—	58	1	no	—
307	C₂×C₄	—	58	1	no	—
308	C₂×C₄	—	58	1	no	—
309	C₂×C₄	—	116	1	no	—
310	C₂×C₄	—	116	2	no	—
311	C₂×C₄	—	116	2	no	—
312	C₂×C₄	—	116	2	no	—
313	C₂×C₄	—	116	2	no	—
314	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
315	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	58	1	no	—
320	C₈	—	58	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	29	2	no	—
325	C₂²	—	58	1	no	—
326	C₂²	—	58	1	no	—
327	C₂²	—	58	2	no	—
328	C₂²	—	116	8	no	—
329	C₂²	—	116	1	no	—
330	C₂²	—	116	1	no	—
331	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	29	2	no	—
338	C₄	—	29	2	no	—
339	C₄	—	58	1	no	—
340	C₄	—	116	1	no	—
341	C₄	—	116	1	no	—
342	C₄	—	116	1	no	—
343	C₄	C₂×Q₈×D₂₉	1	1	yes	characteristic
344	C₄	#41×D₂₉	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	58	2	no	—
347	C₂	—	116	2	no	—
348	C₂	C₂²⋊Q₈×D₂₉	1	1	yes	characteristic
349	C₂	AGL(1,ℤ/8)×D₂₉	1	1	yes	characteristic
350	C₂	#14271960	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#478	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×D₄×D₂₉	C₂²
hypercenter	#478	D₂₉
hypocenter	C₂₉	#3077
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #14271719	C₂	semidirect
3. #14271464	C₂	semidirect
4. #14271463	C₂	semidirect
5. #14271460	C₂	semidirect
6. #14271452	C₂	semidirect
7. #14271451	C₂	semidirect
8. #14271442	C₂	semidirect
9. #14271441	C₂	semidirect
10. #14271204	C₂	semidirect
11. #14271202	C₂	semidirect
12. #14271185	C₂	semidirect
15. C₂₉×#478	C₂	semidirect
18. C₂₉×#125	C₂²	semidirect
30. #11745250	C₂²	semidirect
31. #11745250	C₂²	semidirect
38. #11745232	C₂²	semidirect
39. C₂₉×#107	C₂²	semidirect
40. C₂₉×#106	C₂²	semidirect
41. C₂₉×C₄⋊C₈	C₂²	semidirect
48. #11745166	C₂²	semidirect
49. #11745165	C₂²	semidirect
50. #11745162	C₂²	semidirect
51. #11745162	C₂²	semidirect
54. #11745156	C₂²	semidirect
55. #11745156	C₂²	semidirect
56. #11745155	C₂²	semidirect
57. #11745155	C₂²	semidirect
58. C₄⋊#6597	C₂²	semidirect
65. D₄×D₂₉	Q₈	semidirect
175. #478	D₂₉	semidirect
198. #125	C₂×D₂₉	semidirect
217. #107	C₂×D₂₉	semidirect
221. #106	C₂×D₂₉	semidirect
225. C₄⋊C₈	C₂×D₂₉	semidirect
234. C₂₉	#3077	semidirect

Maximal subgroups

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

subgroup	index
#14271719	2
C₄×D₄×D₂₉	2
#14271464	2
#14271463	2
#14271460	2
#14271452	2
#14271451	2
#14271442	2
#14271441	2
#14271204	2
#14271202	2
#14271185	2
#14271166	2
C₂₉×#478	2
#3077	29