Subgroups — #16829794

Order 3712.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271776	C₂	1	1	yes	maximal, characteristic
2	#14271639	C₂	1	1	yes	maximal, characteristic
3	#14271507	C₂	1	1	yes	maximal, characteristic
4	#14271506	C₂	1	1	yes	maximal, characteristic
5	#14271467	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271374	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271373	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271320	C₂	1	1	yes	maximal, characteristic
9	#14271320	C₂	1	1	yes	maximal, characteristic
10	#14271269	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271267	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271257	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271255	C₂	1	1	yes	maximal, characteristic, complemented
14	#14271223	C₂	1	1	yes	maximal, characteristic
15	C₂₉×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₂₉	C₂²	1	1	yes	characteristic
17	C₂₉×#125	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
19	Q₈×Dic₂₉	C₂²	1	1	yes	characteristic
20	C₂×#90785	C₂²	1	1	yes	characteristic
21	C₂×#90785	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
24	#11745260	—	2	1	no	—
25	#11745260	—	2	1	no	—
26	#11745259	C₂²	1	1	yes	characteristic
27	#11745259	C₂²	1	1	yes	characteristic
28	#11745254	—	2	1	no	—
29	#11745254	C₂²	1	1	yes	characteristic
30	#11745252	—	2	1	no	—
31	#11745252	—	2	1	no	—
32	#11745249	—	2	1	no	—
33	#11745249	C₂²	1	1	yes	characteristic
34	#11745249	C₂²	1	1	yes	characteristic
35	#11745249	C₂²	1	1	yes	characteristic
36	#11745234	—	2	1	no	—
37	#11745234	C₂²	1	1	yes	characteristic
38	#11745231	C₂²	1	1	yes	characteristic
39	C₄×Dic₅₈	—	2	1	no	—
40	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
41	C₂₉×#107	C₂²	1	1	yes	characteristic
42	C₂₉×#106	C₂²	1	1	yes	characteristic
43	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₂₉×#103	C₂²	1	1	yes	characteristic
45	C₂₉×#103	C₂²	1	1	yes	characteristic
46	#11745168	C₂²	1	1	yes	characteristic
47	#11745168	C₂²	1	1	yes	characteristic
48	#11745164	C₂²	1	1	yes	characteristic
49	#11745164	C₂²	1	1	yes	characteristic
50	#11745163	C₂²	1	1	yes	characteristic
51	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
52	#11745158	C₂²	1	1	yes	characteristic
53	#11745158	C₂²	1	1	yes	characteristic
54	#11745158	C₂²	1	1	yes	characteristic
55	#11745158	C₂²	1	1	yes	characteristic
56	#11745156	C₂²	1	1	yes	characteristic
57	#11745155	C₂²	1	1	yes	characteristic
58	C₄⋊#6597	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
60	Q₈×D₂₉	—	2	2	no	—
61	Q₈×D₂₉	—	2	2	no	—
62	C₂×C₄×D₂₉	—	2	1	no	—
63	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
64	C₂×Dic₅₈	—	2	1	no	—
65	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
67	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₉	—	2	1	no	—
71	C₄⋊C₄×C₂₉	—	2	1	no	—
72	C₄⋊C₄×C₂₉	—	2	1	no	—
73	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
76	C₄²×C₂₉	—	2	1	no	—
77	C₄²×C₂₉	C₂³	1	1	yes	characteristic
78	#90785	—	2	2	no	—
79	#90785	—	2	2	no	—
80	#90781	—	2	1	no	—
81	#90781	—	2	1	no	—
82	#90781	—	2	1	no	—
83	#90781	D₄	1	1	yes	characteristic
84	#90781	D₄	1	1	yes	characteristic
85	C₄⋊Dic₂₉	—	2	1	no	—
86	C₄⋊Dic₂₉	—	2	1	no	—
87	C₄⋊Dic₂₉	—	2	1	no	—
88	C₄⋊Dic₂₉	—	2	1	no	—
89	C₄⋊Dic₂₉	—	2	1	no	—
90	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
91	#90779	—	2	1	no	—
92	#90779	—	2	1	no	—
93	#90779	—	2	1	no	—
94	#90779	—	2	1	no	—
95	#90779	D₄	1	1	yes	characteristic
96	#90779	D₄	1	1	yes	characteristic
97	C₄×Dic₂₉	—	2	1	no	—
98	C₄×Dic₂₉	—	2	1	no	—
99	C₄×Dic₂₉	—	2	1	no	—
100	C₄×Dic₂₉	—	2	1	no	—
101	C₄×Dic₂₉	—	2	1	no	—
102	C₄×Dic₂₉	—	2	1	no	—
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	Dic₁₁₆	—	2	2	no	—
107	Dic₁₁₆	—	2	2	no	—
108	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₂₉	—	2	1	no	—
110	Q₈×C₂₉	#41	1	2	yes	—
111	C₂×C₄×C₂₉	—	2	1	no	—
112	C₂×C₄×C₂₉	—	2	1	no	—
113	C₂×C₄×C₂₉	—	2	1	no	—
114	C₂×C₄×C₂₉	—	2	1	no	—
115	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₂₉	—	2	1	no	—
119	C₂×Dic₂₉	—	2	1	no	—
120	C₂×Dic₂₉	—	2	1	no	—
121	C₂×Dic₂₉	—	2	1	no	—
122	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
125	C₄×D₂₉	—	2	4	no	—
126	C₄×D₂₉	—	2	1	no	—
127	C₄×D₂₉	—	2	1	no	—
128	Dic₅₈	—	2	4	no	—
129	Dic₅₈	—	2	1	no	—
130	Dic₅₈	—	2	1	no	—
131	Dic₅₈	—	2	1	no	—
132	Dic₅₈	—	2	1	no	—
133	Dic₅₈	—	2	1	no	—
134	Dic₅₈	#41	1	2	yes	—
135	C₈×C₂₉	—	2	1	no	—
136	C₈×C₂₉	—	2	1	no	—
137	#6597	—	2	1	no	—
138	#6597	—	2	1	no	—
139	#3127	—	29	1	no	maximal, Sylow
140	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₂₉	—	2	1	no	—
142	C₂×D₂₉	—	2	1	no	—
143	C₂×D₂₉	—	2	1	no	—
144	C₄×C₂₉	—	2	2	no	—
145	C₄×C₂₉	—	2	1	no	—
146	C₄×C₂₉	—	2	1	no	—
147	C₄×C₂₉	—	4	1	no	—
148	C₄×C₂₉	—	4	1	no	—
149	C₄×C₂₉	—	4	1	no	—
150	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
151	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
152	Dic₂₉	—	2	1	no	—
153	Dic₂₉	—	2	2	no	—
154	Dic₂₉	—	2	1	no	—
155	Dic₂₉	—	2	1	no	—
156	Dic₂₉	—	4	1	no	—
157	Dic₂₉	—	4	1	no	—
158	Dic₂₉	—	4	1	no	—
159	#541	—	29	1	no	—
160	#519	—	29	1	no	—
161	#485	—	29	1	no	—
162	#485	—	29	1	no	—
163	#479	—	29	1	no	—
164	#479	D₂₉	1	1	yes	characteristic, complemented
165	#462	—	29	1	no	—
166	#462	—	29	1	no	—
167	#451	—	29	1	no	—
168	#451	—	29	1	no	—
169	#441	—	29	1	no	—
170	#441	—	29	1	no	—
171	C₄×Q₁₆	—	29	1	no	—
172	C₄×Q₁₆	—	29	1	no	—
173	#433	—	29	1	no	—
174	C₂×C₂₉	#525	1	1	yes	characteristic
175	C₂×C₂₉	#578	1	1	yes	characteristic
176	C₂×C₂₉	#578	1	1	yes	characteristic
177	D₂₉	—	4	1	no	—
178	C₂²×Q₈	—	29	1	no	—
179	C₂×Q₁₆	—	29	1	no	—
180	C₂×Q₁₆	—	29	1	no	—
181	C₂×Q₁₆	—	29	1	no	—
182	C₂×Q₁₆	—	29	1	no	—
183	C₄⋊Q₈	—	29	1	no	—
184	#126	—	58	1	no	—
185	#126	—	58	1	no	—
186	#125	—	58	1	no	—
187	#125	—	58	1	no	—
188	#125	C₂×D₂₉	1	1	yes	characteristic
189	C₂²⋊Q₈	—	29	1	no	—
190	C₂²⋊Q₈	—	29	1	no	—
191	C₄×Q₈	—	29	1	no	—
192	C₄×Q₈	—	29	1	no	—
193	C₄×Q₈	—	29	1	no	—
194	C₄×Q₈	—	29	1	no	—
195	C₄×Q₈	—	29	1	no	—
196	C₄×Q₈	—	58	1	no	—
197	C₄×Q₈	—	58	1	no	—
198	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
199	#117	—	29	1	no	—
200	#117	—	29	1	no	—
201	#117	—	58	1	no	—
202	#117	—	58	1	no	—
203	#107	—	29	1	no	—
204	#107	C₂×D₂₉	1	1	yes	characteristic
205	#106	—	29	1	no	—
206	#106	C₂×D₂₉	1	1	yes	characteristic
207	C₄⋊C₈	—	29	1	no	—
208	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
209	#103	—	29	1	no	—
210	#103	—	29	1	no	—
211	#103	—	29	1	no	—
212	#103	—	29	1	no	—
213	#103	—	29	1	no	—
214	#103	—	29	1	no	—
215	#103	C₂×D₂₉	1	1	yes	characteristic
216	#103	C₂×D₂₉	1	1	yes	characteristic
217	C₂²⋊C₈	—	29	1	no	—
218	C₂²⋊C₈	—	29	1	no	—
219	#97	—	29	1	no	—
220	C₄×C₈	—	29	1	no	—
221	C₂₉	#3127	1	1	yes	characteristic, complemented, Sylow
222	C₂×Q₈	—	29	1	no	—
223	C₂×Q₈	—	29	1	no	—
224	C₂×Q₈	—	29	1	no	—
225	C₂×Q₈	—	58	1	no	—
226	C₂×Q₈	—	58	2	no	—
227	C₂×Q₈	—	58	2	no	—
228	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
229	C₂²×C₄	—	29	1	no	—
230	C₂²×C₄	—	58	1	no	—
231	Q₁₆	—	58	2	no	—
232	Q₁₆	—	58	2	no	—
233	Q₁₆	—	58	2	no	—
234	Q₁₆	—	58	2	no	—
235	C₂×C₈	—	29	1	no	—
236	C₂×C₈	—	29	1	no	—
237	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	29	1	no	—
243	C₄⋊C₄	—	29	1	no	—
244	C₄⋊C₄	—	29	1	no	—
245	C₄⋊C₄	—	58	1	no	—
246	C₄⋊C₄	—	58	1	no	—
247	C₄⋊C₄	—	58	1	no	—
248	C₄⋊C₄	—	58	1	no	—
249	C₄⋊C₄	—	58	1	no	—
250	C₄⋊C₄	—	58	1	no	—
251	C₄⋊C₄	—	58	1	no	—
252	C₄⋊C₄	—	58	1	no	—
253	C₄⋊C₄	—	58	1	no	—
254	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	29	1	no	—
258	C₂²⋊C₄	—	29	1	no	—
259	C₂²⋊C₄	—	58	1	no	—
260	C₂²⋊C₄	—	58	1	no	—
261	C₂²⋊C₄	—	58	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	29	1	no	—
264	C₄²	—	58	1	no	—
265	C₄²	—	58	1	no	—
266	C₄²	—	58	1	no	—
267	C₄²	—	58	1	no	—
268	C₄²	—	58	1	no	—
269	C₄²	—	58	1	no	—
270	C₄²	C₂²×D₂₉	1	1	yes	characteristic
271	C₂³	—	29	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	29	2	no	—
274	Q₈	—	58	4	no	—
275	Q₈	—	58	1	no	—
276	Q₈	—	58	1	no	—
277	Q₈	—	58	1	no	—
278	Q₈	—	58	1	no	—
279	Q₈	—	58	1	no	—
280	Q₈	#90809	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	29	1	no	—
286	C₂×C₄	—	29	1	no	—
287	C₂×C₄	—	29	1	no	—
288	C₂×C₄	—	58	4	no	—
289	C₂×C₄	—	58	1	no	—
290	C₂×C₄	—	58	1	no	—
291	C₂×C₄	—	58	1	no	—
292	C₂×C₄	—	58	1	no	—
293	C₂×C₄	—	58	1	no	—
294	C₂×C₄	—	58	1	no	—
295	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
296	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	58	1	no	—
301	C₈	—	58	1	no	—
302	C₂²	—	58	1	no	—
303	C₂²	—	58	1	no	—
304	C₂²	—	58	1	no	—
305	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	58	1	no	—
313	C₄	—	58	2	no	—
314	C₄	—	58	1	no	—
315	C₄	—	58	1	no	—
316	C₄	—	116	1	no	—
317	C₄	—	116	1	no	—
318	C₄	—	116	1	no	—
319	C₄	C₂×#90809	1	1	yes	characteristic
320	C₄	#41×D₂₉	1	1	yes	characteristic
321	C₂	—	116	1	no	—
322	C₂	#14271676	1	1	yes	characteristic
323	C₂	#14271977	1	1	yes	characteristic
324	C₂	#14271979	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×#479	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂²×D₂₉	C₂×D₄
hypercenter	#479	D₂₉
hypocenter	C₂₉	#3127
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
5. #14271467	C₂	semidirect
6. #14271374	C₂	semidirect
7. #14271373	C₂	semidirect
10. #14271269	C₂	semidirect
11. #14271267	C₂	semidirect
12. #14271257	C₂	semidirect
13. #14271255	C₂	semidirect
15. C₂₉×#479	C₂	semidirect
164. #479	D₂₉	semidirect
221. C₂₉	#3127	semidirect

Maximal subgroups

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

subgroup	index
#14271776	2
#14271639	2
#14271507	2
#14271506	2
#14271467	2
#14271374	2
#14271373	2
#14271320	2
#14271269	2
#14271267	2
#14271257	2
#14271255	2
#14271223	2
C₂₉×#479	2
#3127	29