Subgroups — #16821986

Order 3712.

524 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745239	C₂	1	1	yes	maximal, characteristic
2	#14271057	C₂	1	1	yes	maximal, characteristic
3	#14271056	C₂	1	1	yes	maximal, characteristic
4	#14271051	C₂	1	1	yes	maximal, characteristic
5	#14271050	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271047	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271045	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271044	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₄×Dic₂₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271014	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊C₄×Dic₂₉	C₂	1	1	yes	maximal, characteristic
12	C₂²⋊C₄×Dic₂₉	C₂	1	1	yes	maximal, characteristic
13	#14271008	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄²×Dic₂₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₉×#388	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₂₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₂₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₂₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₂₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
26	C₂×#90779	C₂²	1	1	yes	characteristic
27	C₂×#90779	C₂²	1	1	yes	characteristic
28	C₂×#90779	C₂²	1	1	yes	characteristic
29	C₂×#90779	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
39	#11745239	C₄	1	8	yes	complemented
40	C₂₉×#95	C₂²	1	1	yes	characteristic
41	C₂₉×#95	C₂²	1	1	yes	characteristic
42	#11745179	C₂²	1	1	yes	characteristic
43	#11745179	C₂²	1	1	yes	characteristic
44	#11745179	C₂²	1	1	yes	characteristic
45	#11745179	C₂²	1	1	yes	characteristic
46	#11745179	C₂²	1	1	yes	characteristic
47	#11745179	C₂²	1	1	yes	characteristic
48	#11745179	C₂²	1	1	yes	characteristic
49	#11745179	C₂²	1	1	yes	characteristic
50	#11745179	C₂²	1	1	yes	characteristic
51	#11745179	C₂²	1	1	yes	characteristic
52	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₂₉	—	2	2	no	—
60	C₂²×Dic₂₉	—	2	2	no	—
61	C₂²×Dic₂₉	C₂×C₄	1	4	yes	—
62	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
65	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
66	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
67	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
68	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
69	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₉	—	2	2	no	—
71	C₂²⋊C₄×C₂₉	—	2	2	no	—
72	C₂²⋊C₄×C₂₉	—	2	2	no	—
73	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
74	C₄²×C₂₉	—	2	2	no	—
75	C₂²⋊Dic₂₉	—	2	2	no	—
76	C₂²⋊Dic₂₉	—	2	2	no	—
77	C₂²⋊Dic₂₉	C₂×C₄	1	4	yes	—
78	C₄⋊Dic₂₉	—	2	2	no	—
79	#90779	—	2	2	no	—
80	#90779	—	2	2	no	—
81	#90779	C₂×C₄	1	4	yes	—
82	#90779	C₂×C₄	1	4	yes	—
83	C₄×Dic₂₉	—	2	2	no	—
84	C₄×Dic₂₉	—	2	2	no	—
85	C₄×Dic₂₉	—	2	2	no	—
86	C₄×Dic₂₉	—	2	2	no	—
87	C₄×Dic₂₉	—	2	2	no	—
88	C₄×Dic₂₉	—	2	2	no	—
89	C₄×Dic₂₉	—	2	2	no	—
90	C₄×Dic₂₉	C₂×C₄	1	4	yes	—
91	C₄×Dic₂₉	C₂×C₄	1	4	yes	—
92	C₂³×C₂₉	—	2	1	no	—
93	C₂³×C₂₉	—	2	1	no	—
94	C₂³×C₂₉	—	2	1	no	—
95	C₂³×C₂₉	—	2	1	no	—
96	C₂³×C₂₉	—	2	1	no	—
97	C₂³×C₂₉	—	2	1	no	—
98	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
99	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₂₉	—	2	1	no	—
101	C₂×C₄×C₂₉	—	2	1	no	—
102	C₂×C₄×C₂₉	—	2	1	no	—
103	C₂×C₄×C₂₉	—	2	1	no	—
104	C₂×C₄×C₂₉	—	2	1	no	—
105	C₂×C₄×C₂₉	—	2	1	no	—
106	C₂×C₄×C₂₉	—	2	1	no	—
107	C₂×C₄×C₂₉	—	2	1	no	—
108	C₂×C₄×C₂₉	—	2	1	no	—
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	—	2	1	no	—
114	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₂₉	#41	1	2	yes	—
117	C₂×C₄×C₂₉	#41	1	2	yes	—
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	—	2	1	no	—
123	C₂×Dic₂₉	—	2	1	no	—
124	C₂×Dic₂₉	—	2	1	no	—
125	C₂×Dic₂₉	—	2	1	no	—
126	C₂×Dic₂₉	—	2	1	no	—
127	C₂×Dic₂₉	—	2	1	no	—
128	C₂×Dic₂₉	—	2	8	no	—
129	C₂×Dic₂₉	—	2	1	no	—
130	C₂×Dic₂₉	—	2	1	no	—
131	C₂×Dic₂₉	—	2	1	no	—
132	C₂×Dic₂₉	—	2	1	no	—
133	C₂×Dic₂₉	—	2	1	no	—
134	C₂×Dic₂₉	—	2	1	no	—
135	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
136	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
137	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
138	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
139	C₂×Dic₂₉	#41	1	2	yes	—
140	C₂×Dic₂₉	#41	1	2	yes	—
141	C₂×Dic₂₉	#41	1	2	yes	—
142	C₂×Dic₂₉	#41	1	2	yes	—
143	#2298	—	29	1	no	maximal, Sylow
144	C₂²×C₂₉	—	2	2	no	—
145	C₂²×C₂₉	—	2	2	no	—
146	C₂²×C₂₉	—	2	2	no	—
147	C₂²×C₂₉	—	4	1	no	—
148	C₂²×C₂₉	—	4	1	no	—
149	C₂²×C₂₉	—	4	1	no	—
150	C₂²×C₂₉	—	4	1	no	—
151	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
152	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
153	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
154	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
155	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
156	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
157	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₂₉	—	2	2	no	—
159	C₄×C₂₉	—	2	2	no	—
160	C₄×C₂₉	—	2	2	no	—
161	C₄×C₂₉	—	2	2	no	—
162	C₄×C₂₉	—	4	1	no	—
163	C₄×C₂₉	—	4	1	no	—
164	Dic₂₉	—	2	2	no	—
165	Dic₂₉	—	2	2	no	—
166	Dic₂₉	—	2	2	no	—
167	Dic₂₉	—	2	2	no	—
168	Dic₂₉	—	2	2	no	—
169	Dic₂₉	—	4	1	no	—
170	Dic₂₉	—	4	1	no	—
171	Dic₂₉	#117	1	4	yes	—
172	C₂×#117	—	29	1	no	—
173	#388	—	29	1	no	—
174	#388	—	29	1	no	—
175	#388	—	29	1	no	—
176	#388	D₂₉	1	1	yes	characteristic
177	#387	—	29	1	no	—
178	#387	—	29	1	no	—
179	#387	—	29	1	no	—
180	#387	—	29	1	no	—
181	C₄×C₄⋊C₄	—	29	1	no	—
182	C₄×C₄⋊C₄	—	29	1	no	—
183	C₄×C₂²⋊C₄	—	29	1	no	—
184	C₄×C₂²⋊C₄	—	29	1	no	—
185	#376	—	29	1	no	—
186	C₄³	—	29	1	no	—
187	C₂×C₂₉	—	4	2	no	—
188	C₂×C₂₉	C₂×#117	1	1	yes	characteristic
189	C₂×C₂₉	C₄×#41	1	1	yes	characteristic
190	C₂×C₂₉	C₄×#41	1	1	yes	characteristic
191	C₂×C₂₉	#529	1	1	yes	characteristic
192	C₂×C₂₉	#529	1	1	yes	characteristic
193	C₂×C₂₉	#529	1	1	yes	characteristic
194	C₂×C₂₉	#529	1	1	yes	characteristic
195	C₂³×C₄	—	29	1	no	—
196	#117	—	29	8	no	—
197	C₂×C₄⋊C₄	—	29	1	no	—
198	C₂×C₄⋊C₄	—	29	1	no	—
199	C₂×C₄⋊C₄	—	29	1	no	—
200	C₂×C₄⋊C₄	—	29	1	no	—
201	C₂×C₄⋊C₄	—	29	1	no	—
202	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	29	1	no	—
204	C₂×C₂²⋊C₄	—	29	1	no	—
205	C₂×C₂²⋊C₄	—	29	1	no	—
206	C₂×C₂²⋊C₄	C₂×D₂₉	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	C₂×D₂₉	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	C₂×D₂₉	1	1	yes	characteristic
209	C₂×C₄²	—	29	1	no	—
210	C₂×C₄²	—	29	1	no	—
211	C₂×C₄²	—	29	1	no	—
212	C₂×C₄²	—	29	1	no	—
213	C₂×C₄²	—	29	1	no	—
214	C₂×C₄²	—	29	1	no	—
215	C₂×C₄²	—	29	1	no	—
216	C₂×C₄²	—	29	1	no	—
217	C₂×C₄²	—	29	1	no	—
218	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
219	#95	—	29	1	no	—
220	#95	—	29	1	no	—
221	#95	—	29	1	no	—
222	#95	—	29	1	no	—
223	#95	—	29	1	no	—
224	#95	—	29	1	no	—
225	#95	—	29	1	no	—
226	#95	—	29	1	no	—
227	#95	—	29	1	no	—
228	#95	—	29	1	no	—
229	#95	C₂×D₂₉	1	1	yes	characteristic
230	#95	C₂×D₂₉	1	1	yes	characteristic
231	C₂₉	#2298	1	1	yes	characteristic, complemented, Sylow
232	C₂⁴	C₂²×D₂₉	1	1	yes	characteristic
233	C₂²×C₄	—	29	1	no	—
234	C₂²×C₄	—	29	4	no	—
235	C₂²×C₄	—	29	1	no	—
236	C₂²×C₄	—	29	1	no	—
237	C₂²×C₄	—	29	1	no	—
238	C₂²×C₄	—	29	1	no	—
239	C₂²×C₄	—	29	1	no	—
240	C₂²×C₄	—	29	1	no	—
241	C₂²×C₄	—	29	1	no	—
242	C₂²×C₄	—	58	2	no	—
243	C₂²×C₄	—	58	2	no	—
244	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	29	4	no	—
252	C₄⋊C₄	—	29	4	no	—
253	C₄⋊C₄	—	58	2	no	—
254	C₄⋊C₄	—	58	2	no	—
255	C₄⋊C₄	—	58	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	29	4	no	—
259	C₂²⋊C₄	—	58	2	no	—
260	C₂²⋊C₄	—	58	2	no	—
261	C₂²⋊C₄	C₄×D₂₉	1	4	yes	—
262	C₄²	—	2	2	no	—
263	C₄²	—	29	4	no	—
264	C₄²	—	29	4	no	—
265	C₄²	—	58	2	no	—
266	C₄²	—	58	2	no	—
267	C₄²	—	58	2	no	—
268	C₄²	—	58	2	no	—
269	C₄²	—	58	2	no	—
270	C₄²	—	58	2	no	—
271	C₄²	—	58	2	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	C₂×C₄×D₂₉	1	2	yes	—
279	C₂³	C₂³×D₂₉	1	1	yes	characteristic
280	C₂×C₄	—	2	1	no	—
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	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	29	2	no	—
295	C₂×C₄	—	29	2	no	—
296	C₂×C₄	—	29	2	no	—
297	C₂×C₄	—	29	2	no	—
298	C₂×C₄	—	29	2	no	—
299	C₂×C₄	—	29	2	no	—
300	C₂×C₄	—	29	2	no	—
301	C₂×C₄	—	29	2	no	—
302	C₂×C₄	—	58	8	no	—
303	C₂×C₄	—	58	1	no	—
304	C₂×C₄	—	58	1	no	—
305	C₂×C₄	—	58	1	no	—
306	C₂×C₄	—	58	1	no	—
307	C₂×C₄	—	58	1	no	—
308	C₂×C₄	—	58	1	no	—
309	C₂×C₄	—	58	1	no	—
310	C₂×C₄	—	58	1	no	—
311	C₂×C₄	—	58	1	no	—
312	C₂×C₄	—	58	1	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	—	58	1	no	—
319	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
320	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
321	C₂×C₄	#90807	1	2	yes	—
322	C₂×C₄	#90807	1	2	yes	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
331	C₂²	C₂×#90807	1	1	yes	characteristic
332	C₂²	C₂×#90807	1	1	yes	characteristic
333	C₂²	#41×D₂₉	1	1	yes	characteristic
334	C₂²	#41×D₂₉	1	1	yes	characteristic
335	C₂²	#41×D₂₉	1	1	yes	characteristic
336	C₂²	#41×D₂₉	1	1	yes	characteristic
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	29	4	no	—
344	C₄	—	58	2	no	—
345	C₄	—	58	2	no	—
346	C₄	—	58	2	no	—
347	C₄	—	58	2	no	—
348	C₄	—	58	2	no	—
349	C₄	—	116	1	no	—
350	C₄	—	116	1	no	—
351	C₂	—	4	2	no	—
352	C₂	#117×D₂₉	1	1	yes	characteristic
353	C₂	C₄×#90807	1	1	yes	characteristic
354	C₂	C₄×#90807	1	1	yes	characteristic
355	C₂	#14271690	1	1	yes	characteristic
356	C₂	#14271692	1	1	yes	characteristic
357	C₂	#14271694	1	1	yes	characteristic
358	C₂	#14271696	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₉
commutator	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#388	C₂
socle, cosocle	C₂³×C₂₉	C₂⁴
prime radical	C₂⁴×C₂₉	C₂³
hypercenter	#388	D₂₉
hypocenter	C₂₉	#2298
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
5. #14271050	C₂	semidirect
6. #14271047	C₂	semidirect
7. #14271045	C₂	semidirect
8. #14271044	C₂	semidirect
9. C₄⋊C₄×Dic₂₉	C₂	semidirect
10. #14271014	C₂	semidirect
13. #14271008	C₂	semidirect
14. C₄²×Dic₂₉	C₂	semidirect
39. #11745239	C₄	semidirect
231. C₂₉	#2298	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745239	2
#14271057	2
#14271056	2
#14271051	2
#14271050	2
#14271047	2
#14271045	2
#14271044	2
C₄⋊C₄×Dic₂₉	2
#14271014	2
C₂²⋊C₄×Dic₂₉	2
#14271008	2
C₄²×Dic₂₉	2
C₂₉×#388	2
#2298	29