Subgroups — #16828148

Order 3712.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₂₉	C₂	1	1	yes	maximal, characteristic
2	#14271224	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271221	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271221	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271219	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271218	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271215	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271214	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271188	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271188	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271125	C₂	1	1	yes	maximal, characteristic
12	C₂²⋊C₈×D₂₉	C₂	1	1	yes	maximal, characteristic
13	#14271120	C₂	1	1	yes	maximal, characteristic, complemented
14	#14271117	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
17	C₂₉×#117	C₂²	1	1	yes	characteristic
18	#11745288	C₂²	1	1	yes	characteristic
19	C₂×#90772	C₂²	1	1	yes	characteristic
20	C₂×#90772	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
23	#11745254	C₄	1	1	yes	characteristic
24	#11745254	C₄	1	1	yes	characteristic
25	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
26	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₂₉	C₄	1	1	yes	characteristic
28	C₂²⋊C₄×D₂₉	C₄	1	1	yes	characteristic
29	#11745239	C₄	1	1	yes	characteristic
30	#11745239	C₄	1	1	yes	characteristic
31	#11745234	C₂²	1	1	yes	characteristic
32	#11745234	C₂²	1	1	yes	characteristic
33	C₄²×D₂₉	C₂²	1	1	yes	characteristic
34	C₄²×D₂₉	C₂²	1	1	yes	characteristic
35	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
36	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂₉×#97	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic, complemented
41	C₂²⋊#6597	C₂²	1	1	yes	characteristic
42	C₂²⋊#6597	C₂²	1	1	yes	characteristic
43	#11745168	C₂²	1	1	yes	characteristic
44	#11745168	C₂²	1	1	yes	characteristic
45	#11745168	C₂²	1	1	yes	characteristic
46	#11745168	C₂²	1	1	yes	characteristic
47	#11745163	C₂²	1	1	yes	characteristic, complemented
48	#11745163	C₂²	1	1	yes	characteristic, complemented
49	#11745162	C₂²	1	1	yes	characteristic, complemented
50	#11745162	C₂²	1	1	yes	characteristic, complemented
51	#11745162	C₂²	1	1	yes	characteristic, complemented
52	#11745162	C₂²	1	1	yes	characteristic, complemented
53	C₈×Dic₂₉	C₂²	1	1	yes	characteristic, complemented
54	C₈×Dic₂₉	C₂²	1	1	yes	characteristic, complemented
55	C₄⋊#6597	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊#6597	C₂²	1	1	yes	characteristic, complemented
57	#11745151	C₂²	1	1	yes	characteristic, complemented
58	C₄×#6597	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₂₉	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₂₉	—	2	2	no	—
63	C₂×C₄×D₂₉	—	2	2	no	—
64	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
80	C₄²×C₂₉	C₂³	1	1	yes	characteristic
81	C₄²×C₂₉	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
84	#90781	C₂×C₄	1	1	yes	characteristic
85	#90781	C₂×C₄	1	1	yes	characteristic
86	#90781	C₂×C₄	1	1	yes	characteristic
87	#90781	C₂×C₄	1	1	yes	characteristic
88	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
90	#90779	C₂×C₄	1	1	yes	characteristic
91	#90779	C₂×C₄	1	1	yes	characteristic
92	#90779	C₂×C₄	1	1	yes	characteristic
93	#90779	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
98	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
99	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
100	C₂×#6597	C₂³	1	1	yes	characteristic
101	C₂×#6597	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	C₈×D₂₉	—	2	2	no	—
107	C₈×D₂₉	—	2	2	no	—
108	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₂₉	—	2	2	no	—
110	C₂²×D₂₉	—	2	2	no	—
111	C₂²×D₂₉	—	4	1	no	—
112	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
113	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₉	—	2	1	no	—
115	C₂×C₄×C₂₉	—	2	1	no	—
116	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₂₉	—	2	2	no	—
123	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
129	C₄×D₂₉	—	2	2	no	—
130	C₄×D₂₉	—	2	2	no	—
131	C₄×D₂₉	—	2	2	no	—
132	C₄×D₂₉	—	2	1	no	—
133	C₄×D₂₉	—	2	1	no	—
134	C₄×D₂₉	—	2	2	no	—
135	C₄×D₂₉	—	4	1	no	—
136	C₄×D₂₉	—	4	1	no	—
137	C₄×D₂₉	#41	1	2	yes	—
138	C₄×D₂₉	#41	1	2	yes	—
139	C₈×C₂₉	—	2	1	no	—
140	C₈×C₂₉	—	2	1	no	—
141	C₈×C₂₉	—	2	1	no	—
142	C₈×C₂₉	—	2	1	no	—
143	#6597	—	2	1	no	—
144	#6597	—	2	1	no	—
145	#6597	—	2	1	no	—
146	#6597	—	2	1	no	—
147	#2924	—	29	1	no	maximal, Sylow
148	C₂²×C₂₉	—	2	1	no	—
149	C₂²×C₂₉	—	2	1	no	—
150	C₂²×C₂₉	—	2	1	no	—
151	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
152	C₂×D₂₉	—	2	1	no	—
153	C₂×D₂₉	—	2	1	no	—
154	C₂×D₂₉	—	2	1	no	—
155	C₂×D₂₉	—	2	1	no	—
156	C₂×D₂₉	—	4	4	no	—
157	C₂×D₂₉	#117	1	2	yes	—
158	C₂×D₂₉	#131	1	2	yes	—
159	C₄×C₂₉	—	2	1	no	—
160	C₄×C₂₉	—	2	1	no	—
161	C₄×C₂₉	—	2	1	no	—
162	C₄×C₂₉	—	2	1	no	—
163	C₄×C₂₉	—	2	1	no	—
164	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
166	Dic₂₉	—	2	1	no	—
167	Dic₂₉	—	2	1	no	—
168	Dic₂₉	—	2	1	no	—
169	Dic₂₉	—	2	1	no	—
170	Dic₂₉	—	2	1	no	—
171	Dic₂₉	#117	1	2	yes	—
172	C₂×#117	—	29	1	no	—
173	#433	—	29	1	no	—
174	#433	—	29	1	no	—
175	#433	—	29	1	no	—
176	#433	D₂₉	1	1	yes	characteristic, complemented
177	#432	—	29	1	no	—
178	#432	—	29	1	no	—
179	#431	—	29	1	no	—
180	#431	—	29	1	no	—
181	#424	—	29	1	no	—
182	#424	—	29	1	no	—
183	#407	—	29	1	no	—
184	C₂×C₂²⋊C₈	—	29	1	no	—
185	#405	—	29	1	no	—
186	#405	—	29	1	no	—
187	C₂×C₂₉	—	4	1	no	—
188	C₂×C₂₉	C₂×#117	1	1	yes	characteristic
189	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
190	C₂×C₂₉	#568	1	1	yes	characteristic
191	D₂₉	—	2	2	no	—
192	D₂₉	—	2	2	no	—
193	C₂³×C₄	—	29	1	no	—
194	C₂×M₁₆	—	29	1	no	—
195	C₂×M₁₆	—	29	1	no	—
196	C₂²×C₈	—	29	1	no	—
197	C₂²×C₈	—	29	1	no	—
198	#117	—	29	1	no	—
199	#117	—	29	1	no	—
200	#117	—	29	1	no	—
201	#117	—	29	1	no	—
202	#117	—	29	1	no	—
203	#117	—	29	1	no	—
204	#117	—	29	1	no	—
205	#117	C₂×D₂₉	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	29	1	no	—
207	C₂×C₄⋊C₄	—	29	1	no	—
208	C₂×C₂²⋊C₄	—	29	1	no	—
209	C₂×C₂²⋊C₄	—	29	1	no	—
210	C₂×C₄²	—	29	1	no	—
211	C₂×C₄²	—	29	1	no	—
212	C₄⋊C₈	—	29	1	no	—
213	C₄⋊C₈	—	29	1	no	—
214	C₄⋊C₈	—	29	1	no	—
215	C₄⋊C₈	—	29	1	no	—
216	C₄⋊C₈	—	29	1	no	—
217	C₄⋊C₈	—	29	1	no	—
218	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic, complemented
219	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic, complemented
220	C₂²⋊C₈	—	29	1	no	—
221	C₂²⋊C₈	—	29	1	no	—
222	C₂²⋊C₈	—	29	1	no	—
223	C₂²⋊C₈	—	29	1	no	—
224	C₂²⋊C₈	—	29	1	no	—
225	C₂²⋊C₈	—	29	1	no	—
226	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
227	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
228	#97	—	29	1	no	—
229	#97	—	29	1	no	—
230	#97	—	29	1	no	—
231	#97	C₂×D₂₉	1	1	yes	characteristic, complemented
232	C₄×C₈	—	29	1	no	—
233	C₄×C₈	—	29	1	no	—
234	C₄×C₈	—	29	1	no	—
235	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic, complemented
236	C₂₉	#2924	1	1	yes	characteristic, complemented, Sylow
237	C₂⁴	—	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	—	29	1	no	—
243	C₂²×C₄	—	29	1	no	—
244	C₂²×C₄	—	29	1	no	—
245	C₂²×C₄	—	29	1	no	—
246	C₂²×C₄	—	29	1	no	—
247	C₂²×C₄	—	58	2	no	—
248	C₂²×C₄	—	58	2	no	—
249	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
250	M₁₆	—	58	2	no	—
251	M₁₆	—	58	2	no	—
252	C₂×C₈	—	29	1	no	—
253	C₂×C₈	—	29	1	no	—
254	C₂×C₈	—	29	1	no	—
255	C₂×C₈	—	29	1	no	—
256	C₂×C₈	—	58	2	no	—
257	C₂×C₈	—	58	2	no	—
258	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
259	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
262	C₄⋊C₄	—	29	1	no	—
263	C₄⋊C₄	—	29	1	no	—
264	C₄⋊C₄	—	29	1	no	—
265	C₄⋊C₄	—	29	1	no	—
266	C₄⋊C₄	—	29	1	no	—
267	C₄⋊C₄	—	29	1	no	—
268	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
269	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	29	1	no	—
271	C₂²⋊C₄	—	29	1	no	—
272	C₂²⋊C₄	—	29	1	no	—
273	C₂²⋊C₄	—	29	1	no	—
274	C₂²⋊C₄	—	29	1	no	—
275	C₂²⋊C₄	—	29	1	no	—
276	C₂²⋊C₄	C₄×D₂₉	1	1	yes	characteristic
277	C₂²⋊C₄	C₄×D₂₉	1	1	yes	characteristic
278	C₄²	—	29	1	no	—
279	C₄²	—	29	1	no	—
280	C₄²	—	29	1	no	—
281	C₄²	—	29	1	no	—
282	C₄²	—	29	1	no	—
283	C₄²	—	29	1	no	—
284	C₄²	C₂²×D₂₉	1	1	yes	characteristic
285	C₄²	C₂²×D₂₉	1	1	yes	characteristic
286	C₂³	—	29	1	no	—
287	C₂³	—	29	1	no	—
288	C₂³	—	58	2	no	—
289	C₂³	—	58	2	no	—
290	C₂³	—	116	1	no	—
291	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	29	1	no	—
295	C₂×C₄	—	29	2	no	—
296	C₂×C₄	—	29	2	no	—
297	C₂×C₄	—	29	1	no	—
298	C₂×C₄	—	29	1	no	—
299	C₂×C₄	—	29	1	no	—
300	C₂×C₄	—	29	1	no	—
301	C₂×C₄	—	29	1	no	—
302	C₂×C₄	—	58	2	no	—
303	C₂×C₄	—	58	2	no	—
304	C₂×C₄	—	58	2	no	—
305	C₂×C₄	—	58	2	no	—
306	C₂×C₄	—	58	2	no	—
307	C₂×C₄	—	58	1	no	—
308	C₂×C₄	—	58	1	no	—
309	C₂×C₄	—	116	1	no	—
310	C₂×C₄	—	116	1	no	—
311	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×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	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	58	1	no	—
322	C₈	—	58	1	no	—
323	C₈	—	58	1	no	—
324	C₈	—	58	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	29	2	no	—
329	C₂²	—	29	2	no	—
330	C₂²	—	58	1	no	—
331	C₂²	—	58	1	no	—
332	C₂²	—	58	1	no	—
333	C₂²	—	58	1	no	—
334	C₂²	—	116	4	no	—
335	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	29	2	no	—
342	C₄	—	58	1	no	—
343	C₄	—	58	1	no	—
344	C₄	—	58	1	no	—
345	C₄	—	58	1	no	—
346	C₄	—	58	1	no	—
347	C₄	#41×D₂₉	1	1	yes	characteristic
348	C₄	#41×D₂₉	1	1	yes	characteristic
349	C₂	—	4	1	no	—
350	C₂	—	58	2	no	—
351	C₂	—	58	2	no	—
352	C₂	#117×D₂₉	1	1	yes	characteristic
353	C₂	#131×D₂₉	1	1	yes	characteristic
354	C₂	#14271928	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#433	C₂
prime radical	C₂³×D₂₉	C₂×C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#433	D₂₉
hypocenter	C₂₉	#2924
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #14271224	C₂	semidirect
3. #14271221	C₂	semidirect
4. #14271221	C₂	semidirect
5. #14271219	C₂	semidirect
6. #14271218	C₂	semidirect
7. #14271215	C₂	semidirect
8. #14271214	C₂	semidirect
9. #14271188	C₂	semidirect
10. #14271188	C₂	semidirect
13. #14271120	C₂	semidirect
14. #14271117	C₂	semidirect
15. C₂₉×#433	C₂	semidirect
35. C₂₉×C₄⋊C₈	C₂²	semidirect
36. C₂₉×C₄⋊C₈	C₂²	semidirect
39. C₂₉×#97	C₂²	semidirect
40. C₄×C₈×C₂₉	C₂²	semidirect
47. #11745163	C₂²	semidirect
48. #11745163	C₂²	semidirect
49. #11745162	C₂²	semidirect
50. #11745162	C₂²	semidirect
51. #11745162	C₂²	semidirect
52. #11745162	C₂²	semidirect
53. C₈×Dic₂₉	C₂²	semidirect
54. C₈×Dic₂₉	C₂²	semidirect
55. C₄⋊#6597	C₂²	semidirect
56. C₄⋊#6597	C₂²	semidirect
57. #11745151	C₂²	semidirect
58. C₄×#6597	C₂²	semidirect
176. #433	D₂₉	semidirect
218. C₄⋊C₈	C₂×D₂₉	semidirect
219. C₄⋊C₈	C₂×D₂₉	semidirect
231. #97	C₂×D₂₉	semidirect
235. C₄×C₈	C₂×D₂₉	semidirect
236. C₂₉	#2924	semidirect

Maximal subgroups

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

subgroup	index
#117×D₂₉	2
#14271224	2
#14271221	2
#14271219	2
#14271218	2
#14271215	2
#14271214	2
#14271188	2
#14271125	2
C₂²⋊C₈×D₂₉	2
#14271120	2
#14271117	2
C₂₉×#433	2
#2924	29