Subgroups — #16829404

Order 3712.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271673	C₂	1	1	yes	maximal, characteristic
2	#117×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271505	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271504	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271500	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271495	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271484	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271480	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271475	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271200	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271198	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271156	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271156	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₈×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₉×#482	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#90807	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
18	C₂₉×#121	C₂²	1	1	yes	characteristic
19	C₂₉×#117	C₂²	1	1	yes	characteristic
20	#11745300	C₂²	1	1	yes	characteristic
21	#11745299	—	2	1	no	—
22	D₄×Dic₂₉	—	2	1	no	—
23	#11745288	C₂²	1	1	yes	characteristic
24	#11745286	C₂²	1	1	yes	characteristic
25	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
27	#11745259	C₂²	1	1	yes	characteristic
28	#11745254	C₂²	1	1	yes	characteristic
29	#11745254	C₂²	1	1	yes	characteristic
30	#11745254	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₂₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₂₉	C₂²	1	1	yes	characteristic
33	#11745245	—	2	1	no	—
34	#11745243	—	2	1	no	—
35	C₂²⋊C₄×D₂₉	—	2	1	no	—
36	#11745239	—	2	1	no	—
37	#11745234	—	2	1	no	—
38	C₄²×D₂₉	—	2	1	no	—
39	C₂₉×#107	C₂²	1	1	yes	characteristic, complemented
40	C₂₉×#106	C₂²	1	1	yes	characteristic, complemented
41	C₂₉×#102	C₂²	1	1	yes	characteristic, complemented
42	C₂₉×#102	C₂²	1	1	yes	characteristic, complemented
43	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	#11745180	C₂²	1	1	yes	characteristic, complemented
45	#11745180	C₂²	1	1	yes	characteristic, complemented
46	C₂²⋊#6597	C₂²	1	1	yes	characteristic
47	#11745168	C₂²	1	1	yes	characteristic
48	#11745168	C₂²	1	1	yes	characteristic
49	#11745166	C₂²	1	1	yes	characteristic, complemented
50	#11745165	C₂²	1	1	yes	characteristic, complemented
51	#11745164	C₂²	1	1	yes	characteristic, complemented
52	#11745164	C₂²	1	1	yes	characteristic, complemented
53	#11745158	C₂²	1	1	yes	characteristic, complemented
54	#11745158	C₂²	1	1	yes	characteristic, complemented
55	#11745156	C₂²	1	1	yes	characteristic, complemented
56	#11745156	C₂²	1	1	yes	characteristic, complemented
57	#11745155	C₂²	1	1	yes	characteristic, complemented
58	#11745155	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₂₉	D₄	1	1	yes	characteristic
60	C₂×D₄×C₂₉	—	2	1	no	—
61	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₂₉	—	2	1	no	—
64	C₂×C₂²⋊D₂₉	—	2	1	no	—
65	C₂²×Dic₂₉	—	2	1	no	—
66	C₂²×Dic₂₉	D₄	1	1	yes	characteristic
67	#90807	—	2	2	no	—
68	#90807	—	2	2	no	—
69	C₂×C₄×D₂₉	—	2	2	no	—
70	C₂×C₄×D₂₉	—	2	2	no	—
71	C₂×C₄×D₂₉	—	2	1	no	—
72	C₂×C₄×D₂₉	—	2	1	no	—
73	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₂₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
77	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₂₉	—	2	1	no	—
85	C₄²×C₂₉	—	2	1	no	—
86	C₂²⋊Dic₂₉	—	2	1	no	—
87	C₂²⋊Dic₂₉	—	2	1	no	—
88	C₂²⋊Dic₂₉	—	2	1	no	—
89	#90781	—	2	1	no	—
90	#90781	—	2	1	no	—
91	#90781	—	2	1	no	—
92	#90781	—	2	1	no	—
93	C₄⋊Dic₂₉	—	2	1	no	—
94	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
97	#90779	—	2	1	no	—
98	#90779	—	2	1	no	—
99	#90779	—	2	1	no	—
100	C₄×Dic₂₉	—	2	1	no	—
101	C₄×Dic₂₉	—	2	1	no	—
102	C₄×Dic₂₉	—	2	1	no	—
103	C₄×Dic₂₉	—	2	1	no	—
104	C₂×#6597	C₂³	1	1	yes	characteristic
105	C₂×#6597	C₂³	1	1	yes	characteristic
106	C₈×D₂₉	—	2	2	no	—
107	C₈×D₂₉	—	2	2	no	—
108	C₂³×C₂₉	—	2	1	no	—
109	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₂₉	—	2	2	no	—
111	C₂²×D₂₉	—	2	2	no	—
112	C₂²×D₂₉	—	4	1	no	—
113	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
115	D₄×C₂₉	—	2	1	no	—
116	D₄×C₂₉	—	2	1	no	—
117	D₄×C₂₉	—	4	2	no	—
118	C₂×C₄×C₂₉	—	2	1	no	—
119	C₂×C₄×C₂₉	—	2	1	no	—
120	C₂×C₄×C₂₉	—	2	1	no	—
121	C₂×C₄×C₂₉	—	2	1	no	—
122	C₂×C₄×C₂₉	—	2	1	no	—
123	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₂₉	—	4	2	no	—
126	C₂²⋊D₂₉	—	4	2	no	—
127	C₂×Dic₂₉	—	2	2	no	—
128	C₂×Dic₂₉	—	2	1	no	—
129	C₂×Dic₂₉	—	2	1	no	—
130	C₂×Dic₂₉	—	2	1	no	—
131	C₂×Dic₂₉	—	2	1	no	—
132	C₂×Dic₂₉	—	4	2	no	—
133	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
135	C₄×D₂₉	—	2	2	no	—
136	C₄×D₂₉	—	2	1	no	—
137	C₄×D₂₉	—	2	1	no	—
138	C₄×D₂₉	—	2	2	no	—
139	C₄×D₂₉	—	4	2	no	—
140	C₄×D₂₉	—	4	2	no	—
141	C₄×D₂₉	#41	1	2	yes	—
142	C₄×D₂₉	#41	1	2	yes	—
143	Dic₅₈	—	2	1	no	—
144	Dic₅₈	—	2	1	no	—
145	C₈×C₂₉	—	2	1	no	—
146	C₈×C₂₉	—	2	1	no	—
147	#6597	—	2	1	no	—
148	#6597	—	2	1	no	—
149	#3091	—	29	1	no	maximal, Sylow
150	C₂²×C₂₉	—	2	1	no	—
151	C₂²×C₂₉	—	2	1	no	—
152	C₂²×C₂₉	—	2	1	no	—
153	C₂²×C₂₉	—	4	1	no	—
154	C₂²×C₂₉	—	4	1	no	—
155	C₂²×C₂₉	—	4	1	no	—
156	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₂₉	—	2	1	no	—
158	C₂×D₂₉	—	2	1	no	—
159	C₂×D₂₉	—	2	1	no	—
160	C₂×D₂₉	—	2	1	no	—
161	C₂×D₂₉	—	4	4	no	—
162	C₂×D₂₉	#123	1	2	yes	—
163	C₂×D₂₉	#135	1	2	yes	—
164	C₄×C₂₉	—	2	1	no	—
165	C₄×C₂₉	—	4	1	no	—
166	C₄×C₂₉	—	4	1	no	—
167	C₄×C₂₉	—	4	1	no	—
168	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
169	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
170	Dic₂₉	—	2	1	no	—
171	Dic₂₉	—	4	1	no	—
172	Dic₂₉	—	4	1	no	—
173	Dic₂₉	—	4	1	no	—
174	Dic₂₉	—	4	1	no	—
175	Dic₂₉	#123	1	2	yes	—
176	#525	—	29	1	no	—
177	C₂×#117	—	29	1	no	—
178	#485	—	29	1	no	—
179	#485	—	29	1	no	—
180	#484	—	29	1	no	—
181	#483	—	29	1	no	—
182	#482	—	29	1	no	—
183	#482	D₂₉	1	1	yes	characteristic, complemented
184	#481	—	29	1	no	—
185	#480	—	29	1	no	—
186	#427	—	29	1	no	—
187	#427	—	29	1	no	—
188	#416	—	29	1	no	—
189	#416	—	29	1	no	—
190	C₂×C₂²⋊C₈	—	29	1	no	—
191	C₂×C₂₉	—	4	1	no	—
192	C₂×C₂₉	—	8	1	no	—
193	C₂×C₂₉	C₂×#123	1	1	yes	characteristic
194	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
195	C₂×C₂₉	#575	1	1	yes	characteristic
196	D₂₉	—	2	2	no	—
197	D₂₉	—	2	2	no	—
198	C₂×#41	—	29	1	no	—
199	C₂³×C₄	—	29	1	no	—
200	C₂²×C₈	—	29	1	no	—
201	C₂²×C₈	—	29	1	no	—
202	#123	—	58	1	no	—
203	C₂²⋊Q₈	—	29	1	no	—
204	C₂²⋊Q₈	—	29	1	no	—
205	#121	—	29	1	no	—
206	#121	C₂×D₂₉	1	1	yes	characteristic
207	#120	—	58	1	no	—
208	C₄×D₄	—	58	1	no	—
209	C₄×D₄	—	58	1	no	—
210	#117	—	29	1	no	—
211	#117	—	29	1	no	—
212	#117	—	29	1	no	—
213	#117	—	29	1	no	—
214	#117	—	58	1	no	—
215	#117	—	58	1	no	—
216	#117	C₂×D₂₉	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	29	1	no	—
218	C₂×C₄⋊C₄	—	29	1	no	—
219	C₂×C₂²⋊C₄	—	58	1	no	—
220	C₂×C₄²	—	58	1	no	—
221	#107	—	29	1	no	—
222	#107	—	29	1	no	—
223	#107	—	29	1	no	—
224	#107	C₂×D₂₉	1	1	yes	characteristic, complemented
225	#106	—	29	1	no	—
226	#106	—	29	1	no	—
227	#106	—	29	1	no	—
228	#106	C₂×D₂₉	1	1	yes	characteristic, complemented
229	#103	—	29	1	no	—
230	#103	—	29	1	no	—
231	#103	—	29	1	no	—
232	#103	—	29	1	no	—
233	#102	—	29	1	no	—
234	#102	—	29	1	no	—
235	#102	C₂×D₂₉	1	1	yes	characteristic, complemented
236	#102	C₂×D₂₉	1	1	yes	characteristic, complemented
237	C₂²⋊C₈	—	29	1	no	—
238	C₂²⋊C₈	—	29	1	no	—
239	C₂²⋊C₈	—	29	1	no	—
240	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
241	C₂₉	#3091	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	29	1	no	—
243	#41	—	58	2	no	—
244	#41	—	58	2	no	—
245	C₂×Q₈	—	29	1	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	58	1	no	—
248	C₂×D₄	—	58	1	no	—
249	C₂×D₄	C₂²×D₂₉	1	1	yes	characteristic
250	C₂²×C₄	—	29	1	no	—
251	C₂²×C₄	—	29	1	no	—
252	C₂²×C₄	—	29	1	no	—
253	C₂²×C₄	—	29	1	no	—
254	C₂²×C₄	—	29	1	no	—
255	C₂²×C₄	—	58	2	no	—
256	C₂²×C₄	—	58	2	no	—
257	C₂²×C₄	—	58	1	no	—
258	C₂²×C₄	—	58	1	no	—
259	C₂²×C₄	—	58	1	no	—
260	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
261	C₂×C₈	—	29	1	no	—
262	C₂×C₈	—	29	1	no	—
263	C₂×C₈	—	58	2	no	—
264	C₂×C₈	—	58	2	no	—
265	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
267	C₄⋊C₄	—	29	1	no	—
268	C₄⋊C₄	—	29	1	no	—
269	C₄⋊C₄	—	29	1	no	—
270	C₄⋊C₄	—	58	1	no	—
271	C₄⋊C₄	—	58	1	no	—
272	C₄⋊C₄	—	58	1	no	—
273	C₄⋊C₄	—	58	1	no	—
274	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₂₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	58	1	no	—
280	C₂²⋊C₄	—	58	1	no	—
281	C₂²⋊C₄	—	58	1	no	—
282	C₂²⋊C₄	—	58	1	no	—
283	C₂²⋊C₄	—	58	1	no	—
284	C₂²⋊C₄	—	58	1	no	—
285	C₂²⋊C₄	—	58	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	58	1	no	—
288	C₄²	—	58	1	no	—
289	C₄²	—	58	1	no	—
290	C₄²	—	58	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	29	1	no	—
293	C₂³	—	29	1	no	—
294	C₂³	—	58	2	no	—
295	C₂³	—	58	2	no	—
296	C₂³	—	116	1	no	—
297	C₂³	D₄×D₂₉	1	1	yes	characteristic
298	Q₈	—	58	1	no	—
299	Q₈	—	58	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	4	2	no	—
303	D₄	—	116	2	no	—
304	D₄	—	116	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	29	1	no	—
311	C₂×C₄	—	29	2	no	—
312	C₂×C₄	—	29	2	no	—
313	C₂×C₄	—	29	1	no	—
314	C₂×C₄	—	58	2	no	—
315	C₂×C₄	—	58	2	no	—
316	C₂×C₄	—	58	2	no	—
317	C₂×C₄	—	58	1	no	—
318	C₂×C₄	—	58	1	no	—
319	C₂×C₄	—	58	1	no	—
320	C₂×C₄	—	58	1	no	—
321	C₂×C₄	—	58	1	no	—
322	C₂×C₄	—	58	1	no	—
323	C₂×C₄	—	116	2	no	—
324	C₂×C₄	—	116	2	no	—
325	C₂×C₄	—	116	2	no	—
326	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	58	1	no	—
331	C₈	—	58	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	29	2	no	—
339	C₂²	—	29	2	no	—
340	C₂²	—	58	1	no	—
341	C₂²	—	58	1	no	—
342	C₂²	—	58	1	no	—
343	C₂²	—	58	1	no	—
344	C₂²	—	116	4	no	—
345	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	29	2	no	—
351	C₄	—	58	1	no	—
352	C₄	—	116	1	no	—
353	C₄	—	116	1	no	—
354	C₄	—	116	1	no	—
355	C₄	—	116	1	no	—
356	C₄	#41×D₂₉	1	1	yes	characteristic
357	C₄	#41×D₂₉	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	8	1	no	—
360	C₂	—	58	2	no	—
361	C₂	—	58	2	no	—
362	C₂	#123×D₂₉	1	1	yes	characteristic
363	C₂	#135×D₂₉	1	1	yes	characteristic
364	C₂	#14271959	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×#482	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	#14271673	C₂
hypercenter	#482	D₂₉
hypocenter	C₂₉	#3091
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
2. #117×D₂₉	C₂	semidirect
3. #14271505	C₂	semidirect
4. #14271504	C₂	semidirect
5. #14271500	C₂	semidirect
6. #14271495	C₂	semidirect
7. #14271484	C₂	semidirect
8. #14271480	C₂	semidirect
9. #14271475	C₂	semidirect
10. #14271200	C₂	semidirect
11. #14271198	C₂	semidirect
12. #14271156	C₂	semidirect
13. #14271156	C₂	semidirect
14. C₂²⋊C₈×D₂₉	C₂	semidirect
15. C₂₉×#482	C₂	semidirect
39. C₂₉×#107	C₂²	semidirect
40. C₂₉×#106	C₂²	semidirect
41. C₂₉×#102	C₂²	semidirect
42. C₂₉×#102	C₂²	semidirect
44. #11745180	C₂²	semidirect
45. #11745180	C₂²	semidirect
49. #11745166	C₂²	semidirect
50. #11745165	C₂²	semidirect
51. #11745164	C₂²	semidirect
52. #11745164	C₂²	semidirect
53. #11745158	C₂²	semidirect
54. #11745158	C₂²	semidirect
55. #11745156	C₂²	semidirect
56. #11745156	C₂²	semidirect
57. #11745155	C₂²	semidirect
58. #11745155	C₂²	semidirect
183. #482	D₂₉	semidirect
224. #107	C₂×D₂₉	semidirect
228. #106	C₂×D₂₉	semidirect
235. #102	C₂×D₂₉	semidirect
236. #102	C₂×D₂₉	semidirect
241. C₂₉	#3091	semidirect

Maximal subgroups

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

subgroup	index
#14271673	2
#117×D₂₉	2
#14271505	2
#14271504	2
#14271500	2
#14271495	2
#14271484	2
#14271480	2
#14271475	2
#14271200	2
#14271198	2
#14271156	2
C₂²⋊C₈×D₂₉	2
C₂₉×#482	2
#3091	29