Subgroups — #16829295

Order 3712.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 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
2	C₄×Q₈×D₂₉	C₂	1	1	yes	maximal, characteristic
3	#14271471	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271470	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271467	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271457	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271456	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271447	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271446	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271204	C₂	1	1	yes	maximal, characteristic
11	#14271202	C₂	1	1	yes	maximal, characteristic
12	#14271185	C₂	1	1	yes	maximal, characteristic
13	#14271170	C₂	1	1	yes	maximal, characteristic
14	#14271170	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×#90772	C₂²	1	1	yes	characteristic
21	C₂×#90772	C₂²	1	1	yes	characteristic
22	#11745258	—	2	1	no	—
23	#11745258	—	2	1	no	—
24	#11745254	C₂²	1	1	yes	characteristic
25	#11745254	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×D₂₉	—	2	1	no	—
27	C₄⋊C₄×D₂₉	C₂²	1	1	yes	characteristic
28	#11745250	C₂²	1	1	yes	characteristic
29	#11745250	C₂²	1	1	yes	characteristic
30	#11745249	—	2	1	no	—
31	#11745249	—	2	1	no	—
32	#11745249	—	2	1	no	—
33	#11745249	C₂²	1	1	yes	characteristic
34	C₄²×D₂₉	—	2	1	no	—
35	C₄²×D₂₉	C₂²	1	1	yes	characteristic
36	#11745232	C₂²	1	1	yes	characteristic
37	C₄×Dic₅₈	—	2	1	no	—
38	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
39	C₂₉×#107	C₂²	1	1	yes	characteristic
40	C₂₉×#106	C₂²	1	1	yes	characteristic
41	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₂₉×#103	C₂²	1	1	yes	characteristic
43	C₂₉×#103	C₂²	1	1	yes	characteristic
44	#11745183	C₂²	1	1	yes	characteristic
45	#11745183	C₂²	1	1	yes	characteristic
46	#11745166	C₂²	1	1	yes	characteristic
47	#11745165	C₂²	1	1	yes	characteristic
48	#11745164	C₂²	1	1	yes	characteristic
49	#11745164	C₂²	1	1	yes	characteristic
50	#11745162	C₂²	1	1	yes	characteristic
51	#11745162	C₂²	1	1	yes	characteristic
52	#11745158	C₂²	1	1	yes	characteristic
53	#11745158	C₂²	1	1	yes	characteristic
54	#11745156	C₂²	1	1	yes	characteristic
55	#11745156	C₂²	1	1	yes	characteristic
56	#11745155	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	Q₈	1	4	yes	complemented
62	C₂×C₄×D₂₉	—	2	1	no	—
63	C₂×C₄×D₂₉	—	2	1	no	—
64	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
67	C₂×Dic₅₈	—	2	1	no	—
68	C₂×Dic₅₈	—	2	1	no	—
69	C₂×Dic₅₈	—	2	1	no	—
70	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₉	—	2	1	no	—
74	C₄⋊C₄×C₂₉	—	2	1	no	—
75	C₄⋊C₄×C₂₉	—	2	1	no	—
76	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
79	C₄²×C₂₉	—	2	1	no	—
80	C₄²×C₂₉	C₂³	1	1	yes	characteristic
81	#90781	—	2	1	no	—
82	#90781	—	2	1	no	—
83	C₄⋊Dic₂₉	—	2	1	no	—
84	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
87	#90779	—	2	1	no	—
88	#90779	—	2	1	no	—
89	#90779	—	2	1	no	—
90	#90779	—	2	1	no	—
91	#90779	—	2	1	no	—
92	#90779	—	2	1	no	—
93	#90779	—	2	1	no	—
94	C₄×Dic₂₉	—	2	1	no	—
95	C₄×Dic₂₉	—	2	1	no	—
96	C₄×Dic₂₉	—	2	1	no	—
97	C₄×Dic₂₉	—	2	1	no	—
98	C₄×Dic₂₉	—	2	1	no	—
99	C₄×Dic₂₉	D₄	1	1	yes	characteristic
100	C₄×Dic₂₉	D₄	1	1	yes	characteristic
101	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
102	C₂×#6597	C₂³	1	1	yes	characteristic
103	C₂×#6597	C₂³	1	1	yes	characteristic
104	#90772	—	2	2	no	—
105	#90772	—	2	2	no	—
106	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₂₉	—	2	1	no	—
108	Q₈×C₂₉	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₂₉	—	2	1	no	—
110	C₂×C₄×C₂₉	—	2	1	no	—
111	C₂×C₄×C₂₉	—	2	1	no	—
112	C₂×C₄×C₂₉	—	2	1	no	—
113	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₂₉	—	2	1	no	—
117	C₂×Dic₂₉	—	2	1	no	—
118	C₂×Dic₂₉	—	2	1	no	—
119	C₂×Dic₂₉	—	2	1	no	—
120	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
123	C₄×D₂₉	—	2	4	no	—
124	C₄×D₂₉	—	2	2	no	—
125	C₄×D₂₉	—	4	1	no	—
126	C₄×D₂₉	—	4	2	no	—
127	C₄×D₂₉	C₂×Q₈	1	2	yes	—
128	C₄×D₂₉	#41	1	2	yes	—
129	Dic₅₈	—	2	1	no	—
130	Dic₅₈	—	2	4	no	—
131	Dic₅₈	—	4	2	no	—
132	Dic₅₈	—	4	2	no	—
133	Dic₅₈	C₂×Q₈	1	2	yes	—
134	C₈×C₂₉	—	2	1	no	—
135	C₈×C₂₉	—	2	1	no	—
136	#6597	—	2	1	no	—
137	#6597	—	2	1	no	—
138	#3078	—	29	1	no	maximal, Sylow
139	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₂₉	—	2	1	no	—
141	C₂×D₂₉	—	2	1	no	—
142	C₂×D₂₉	C₂²⋊Q₈	1	2	yes	—
143	C₄×C₂₉	—	2	2	no	—
144	C₄×C₂₉	—	2	1	no	—
145	C₄×C₂₉	—	2	1	no	—
146	C₄×C₂₉	—	4	1	no	—
147	C₄×C₂₉	—	4	1	no	—
148	C₄×C₂₉	—	4	1	no	—
149	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
150	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
151	Dic₂₉	—	2	2	no	—
152	Dic₂₉	—	2	1	no	—
153	Dic₂₉	—	4	1	no	—
154	Dic₂₉	—	4	1	no	—
155	Dic₂₉	—	4	1	no	—
156	Dic₂₉	C₂²⋊Q₈	1	2	yes	—
157	Dic₂₉	#137	1	2	yes	—
158	#533	—	29	1	no	—
159	C₂×C₄×Q₈	—	29	1	no	—
160	#479	—	29	1	no	—
161	#479	—	29	1	no	—
162	#479	—	29	1	no	—
163	#479	D₂₉	1	1	yes	characteristic, complemented
164	#477	—	29	1	no	—
165	#477	—	29	1	no	—
166	#475	—	29	1	no	—
167	#475	—	29	1	no	—
168	#428	—	29	1	no	—
169	#428	—	29	1	no	—
170	#423	—	29	1	no	—
171	#419	—	29	1	no	—
172	#419	—	29	1	no	—
173	C₂×C₂₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₂₉	C₂×#137	1	1	yes	characteristic
175	C₂×C₂₉	#575	1	1	yes	characteristic
176	D₂₉	—	2	2	no	—
177	C₂²×Q₈	—	29	1	no	—
178	C₂×M₁₆	—	29	1	no	—
179	C₂×M₁₆	—	29	1	no	—
180	C₄⋊Q₈	—	29	1	no	—
181	C₄⋊Q₈	—	29	1	no	—
182	#125	—	29	1	no	—
183	#125	C₂×D₂₉	1	1	yes	characteristic
184	C₂²⋊Q₈	—	58	1	no	—
185	C₂²⋊Q₈	—	58	1	no	—
186	C₄×Q₈	—	29	1	no	—
187	C₄×Q₈	—	29	1	no	—
188	C₄×Q₈	—	29	1	no	—
189	C₄×Q₈	—	58	1	no	—
190	C₄×Q₈	—	58	1	no	—
191	C₄×Q₈	—	58	1	no	—
192	C₄×Q₈	—	58	1	no	—
193	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
194	#117	—	29	1	no	—
195	#117	—	29	1	no	—
196	C₂×C₄⋊C₄	—	29	1	no	—
197	C₂×C₄⋊C₄	—	58	1	no	—
198	C₂×C₄²	—	29	1	no	—
199	C₂×C₄²	—	58	1	no	—
200	#107	—	29	1	no	—
201	#107	—	29	1	no	—
202	#107	—	29	1	no	—
203	#107	C₂×D₂₉	1	1	yes	characteristic
204	#106	—	29	1	no	—
205	#106	—	29	1	no	—
206	#106	—	29	1	no	—
207	#106	C₂×D₂₉	1	1	yes	characteristic
208	C₄⋊C₈	—	29	1	no	—
209	C₄⋊C₈	—	29	1	no	—
210	C₄⋊C₈	—	29	1	no	—
211	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
212	#103	—	29	1	no	—
213	#103	—	29	1	no	—
214	#103	—	29	1	no	—
215	#103	—	29	1	no	—
216	#103	—	29	1	no	—
217	#103	—	29	1	no	—
218	#103	C₂×D₂₉	1	1	yes	characteristic
219	#103	C₂×D₂₉	1	1	yes	characteristic
220	C₂₉	#3078	1	1	yes	characteristic, complemented, Sylow
221	C₂×Q₈	—	29	4	no	—
222	C₂×Q₈	—	29	1	no	—
223	C₂×Q₈	—	58	1	no	—
224	C₂×Q₈	—	58	1	no	—
225	C₂×Q₈	—	58	1	no	—
226	C₂×Q₈	—	58	2	no	—
227	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
228	C₂²×C₄	—	29	1	no	—
229	C₂²×C₄	—	29	1	no	—
230	C₂²×C₄	—	29	1	no	—
231	C₂²×C₄	—	58	1	no	—
232	C₂²×C₄	—	58	1	no	—
233	M₁₆	—	58	2	no	—
234	M₁₆	—	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	C₂²×D₂₉	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
256	C₂²⋊C₄	—	58	1	no	—
257	C₂²⋊C₄	—	58	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	29	1	no	—
260	C₄²	—	29	1	no	—
261	C₄²	—	29	1	no	—
262	C₄²	—	58	1	no	—
263	C₄²	—	58	1	no	—
264	C₄²	—	58	1	no	—
265	C₄²	—	58	1	no	—
266	C₄²	—	58	1	no	—
267	C₄²	C₂²×D₂₉	1	1	yes	characteristic
268	C₂³	—	29	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	29	2	no	—
271	Q₈	—	58	4	no	—
272	Q₈	—	58	1	no	—
273	Q₈	—	116	2	no	—
274	Q₈	—	116	2	no	—
275	Q₈	Q₈×D₂₉	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	29	1	no	—
281	C₂×C₄	—	29	1	no	—
282	C₂×C₄	—	29	2	no	—
283	C₂×C₄	—	29	2	no	—
284	C₂×C₄	—	29	1	no	—
285	C₂×C₄	—	58	2	no	—
286	C₂×C₄	—	58	4	no	—
287	C₂×C₄	—	58	1	no	—
288	C₂×C₄	—	58	1	no	—
289	C₂×C₄	—	58	1	no	—
290	C₂×C₄	—	58	1	no	—
291	C₂×C₄	—	116	1	no	—
292	C₂×C₄	—	116	2	no	—
293	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
294	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	58	1	no	—
299	C₈	—	58	1	no	—
300	C₂²	—	29	2	no	—
301	C₂²	—	58	1	no	—
302	C₂²	—	58	1	no	—
303	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	29	2	no	—
311	C₄	—	29	2	no	—
312	C₄	—	58	2	no	—
313	C₄	—	58	1	no	—
314	C₄	—	116	1	no	—
315	C₄	—	116	1	no	—
316	C₄	—	116	1	no	—
317	C₄	C₂×Q₈×D₂₉	1	1	yes	characteristic
318	C₄	#41×D₂₉	1	1	yes	characteristic
319	C₂	—	58	2	no	—
320	C₂	C₂²⋊Q₈×D₂₉	1	1	yes	characteristic
321	C₂	#137×D₂₉	1	1	yes	characteristic
322	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×#479	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂²×D₂₉	C₂×D₄
hypercenter	#479	D₂₉
hypocenter	C₂₉	#3078
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
3. #14271471	C₂	semidirect
4. #14271470	C₂	semidirect
5. #14271467	C₂	semidirect
6. #14271457	C₂	semidirect
7. #14271456	C₂	semidirect
8. #14271447	C₂	semidirect
9. #14271446	C₂	semidirect
15. C₂₉×#479	C₂	semidirect
61. Q₈×D₂₉	Q₈	semidirect
163. #479	D₂₉	semidirect
220. C₂₉	#3078	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×Q₈×D₂₉	2
#14271471	2
#14271470	2
#14271467	2
#14271457	2
#14271456	2
#14271447	2
#14271446	2
#14271204	2
#14271202	2
#14271185	2
#14271170	2
C₂₉×#479	2
#3078	29