Subgroups — #16822224

Order 3712.

528 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745308	C₂	1	1	yes	maximal, characteristic
2	#14271070	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271060	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271058	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271048	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271047	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271046	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271045	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271037	C₂	1	1	yes	maximal, characteristic
10	#14271034	C₂	1	1	yes	maximal, characteristic
11	#14271022	C₂	1	1	yes	maximal, characteristic
12	C₄×#90781	C₂	1	1	yes	maximal, characteristic
13	#14271010	C₂	1	1	yes	maximal, characteristic
14	#95×D₂₉	C₂	1	1	yes	maximal, characteristic
15	C₂₉×#391	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₈×C₂₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
22	#11745308	C₄	1	8	yes	complemented
23	C₂×#90781	C₂²	1	1	yes	characteristic
24	C₂×#90781	C₂²	1	1	yes	characteristic
25	C₂×#90781	C₂²	1	1	yes	characteristic
26	C₂×#90781	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
29	C₂×#90779	C₂²	1	1	yes	characteristic
30	C₂×#90779	C₂²	1	1	yes	characteristic
31	C₂×#90779	C₂²	1	1	yes	characteristic
32	C₂×#90779	C₂²	1	1	yes	characteristic
33	C₂×#90779	C₂²	1	1	yes	characteristic
34	C₂×#90779	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	C₂₉×#95	C₂²	1	1	yes	characteristic
39	C₂₉×#95	C₂²	1	1	yes	characteristic
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	C₂³×D₂₉	C₂³	1	1	yes	characteristic
53	C₂×Q₈×C₂₉	—	4	1	no	—
54	C₂×Q₈×C₂₉	—	4	1	no	—
55	C₂×Q₈×C₂₉	C₂×C₄	1	4	yes	complemented
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×C₄×C₂₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	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	C₂×C₄×D₂₉	—	2	2	no	—
71	C₂×C₄×D₂₉	—	4	1	no	—
72	C₂×C₄×D₂₉	—	4	1	no	—
73	C₂×C₄×D₂₉	—	4	1	no	—
74	C₂×C₄×D₂₉	—	4	1	no	—
75	C₂×C₄×D₂₉	—	4	1	no	—
76	C₂×C₄×D₂₉	—	4	1	no	—
77	C₂×C₄×D₂₉	C₂×C₄	1	4	yes	—
78	C₂×C₄×D₂₉	Q₈	1	4	yes	complemented
79	C₄⋊C₄×C₂₉	—	4	1	no	—
80	C₄²×C₂₉	—	4	1	no	—
81	#90781	—	2	2	no	—
82	#90781	—	2	2	no	—
83	#90781	C₂×C₄	1	4	yes	—
84	#90781	C₂×C₄	1	4	yes	—
85	C₄⋊Dic₂₉	—	4	1	no	—
86	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	complemented
87	#90779	—	4	1	no	—
88	#90779	—	4	1	no	—
89	#90779	—	4	1	no	—
90	#90779	—	4	1	no	—
91	#90779	C₂×C₄	1	4	yes	complemented
92	#90779	C₂×C₄	1	4	yes	complemented
93	C₄×Dic₂₉	—	4	1	no	—
94	C₄×Dic₂₉	—	4	1	no	—
95	C₄×Dic₂₉	—	4	1	no	—
96	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
97	C₂²×D₂₉	—	2	1	no	—
98	C₂²×D₂₉	—	2	1	no	—
99	C₂²×D₂₉	—	2	1	no	—
100	C₂²×D₂₉	—	2	1	no	—
101	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
102	C₂²×D₂₉	C₂×Q₈	1	2	yes	—
103	C₂²×D₂₉	#41	1	2	yes	—
104	Q₈×C₂₉	—	4	4	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	—	2	1	no	—
115	C₂×C₄×C₂₉	—	2	1	no	—
116	C₂×C₄×C₂₉	—	2	1	no	—
117	C₂×C₄×C₂₉	—	2	1	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	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
124	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
125	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
126	C₂×Dic₂₉	—	2	1	no	—
127	C₂×Dic₂₉	—	2	1	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	—	2	1	no	—
133	C₂×Dic₂₉	—	2	1	no	—
134	C₂×Dic₂₉	—	2	1	no	—
135	C₂×Dic₂₉	—	2	1	no	—
136	C₂×Dic₂₉	—	2	1	no	—
137	C₂×Dic₂₉	—	2	1	no	—
138	C₂×Dic₂₉	—	2	1	no	—
139	C₂×Dic₂₉	—	2	1	no	—
140	C₂×Dic₂₉	—	2	1	no	—
141	C₂×Dic₂₉	—	2	1	no	—
142	C₂×Dic₂₉	—	2	1	no	—
143	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
144	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
145	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
146	C₂×Dic₂₉	C₂×Q₈	1	2	yes	—
147	C₄×D₂₉	—	4	4	no	—
148	C₄×D₂₉	—	4	4	no	—
149	C₄×D₂₉	—	4	4	no	—
150	#2329	—	29	1	no	maximal, Sylow
151	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
152	C₂²×C₂₉	C₂²×Q₈	1	1	yes	characteristic
153	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
154	C₂²×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂²×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
158	C₂×D₂₉	—	2	2	no	—
159	C₂×D₂₉	—	2	2	no	—
160	C₂×D₂₉	—	2	2	no	—
161	C₂×D₂₉	—	2	2	no	—
162	C₂×D₂₉	—	2	2	no	—
163	C₂×D₂₉	—	2	2	no	—
164	C₂×D₂₉	C₄×Q₈	1	4	yes	—
165	C₄×C₂₉	—	2	2	no	—
166	C₄×C₂₉	—	2	2	no	—
167	C₄×C₂₉	—	2	2	no	—
168	C₄×C₂₉	—	2	2	no	—
169	C₄×C₂₉	—	4	1	no	—
170	C₄×C₂₉	—	4	1	no	—
171	C₄×C₂₉	—	4	1	no	—
172	Dic₂₉	—	2	2	no	—
173	Dic₂₉	—	2	2	no	—
174	Dic₂₉	—	2	2	no	—
175	Dic₂₉	—	2	2	no	—
176	Dic₂₉	—	4	1	no	—
177	Dic₂₉	—	4	1	no	—
178	Dic₂₉	—	4	1	no	—
179	C₂×C₂²⋊Q₈	—	29	1	no	—
180	#391	—	29	1	no	—
181	#391	D₂₉	1	1	yes	characteristic, complemented
182	#389	—	29	1	no	—
183	#389	—	29	1	no	—
184	#387	—	29	1	no	—
185	#387	—	29	1	no	—
186	#387	—	29	1	no	—
187	#387	—	29	1	no	—
188	#385	—	29	1	no	—
189	#385	—	29	1	no	—
190	#380	—	29	1	no	—
191	C₄×C₂²⋊C₄	—	29	1	no	—
192	C₄×C₂²⋊C₄	—	29	1	no	—
193	C₂×#95	—	29	1	no	—
194	C₂×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
195	C₂×C₂₉	#518	1	1	yes	characteristic
196	C₂×C₂₉	#520	1	1	yes	characteristic
197	C₂×C₂₉	#538	1	1	yes	characteristic
198	C₂×C₂₉	#539	1	1	yes	characteristic
199	C₂×C₂₉	#543	1	1	yes	characteristic
200	C₂×C₂₉	#544	1	1	yes	characteristic
201	D₂₉	—	2	4	no	—
202	C₂²×Q₈	C₂×D₂₉	1	1	yes	characteristic
203	C₂³×C₄	—	29	1	no	—
204	C₂³×C₄	—	29	1	no	—
205	C₂³×C₄	—	29	1	no	—
206	C₂²⋊Q₈	—	29	8	no	—
207	C₂×C₄⋊C₄	—	29	1	no	—
208	C₂×C₄⋊C₄	—	29	1	no	—
209	C₂×C₄⋊C₄	—	29	1	no	—
210	C₂×C₄⋊C₄	—	29	1	no	—
211	C₂×C₄⋊C₄	—	29	1	no	—
212	C₂×C₄⋊C₄	—	29	1	no	—
213	C₂×C₄⋊C₄	—	29	1	no	—
214	C₂×C₄⋊C₄	—	29	1	no	—
215	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	29	1	no	—
217	C₂×C₂²⋊C₄	—	29	1	no	—
218	C₂×C₂²⋊C₄	—	29	1	no	—
219	C₂×C₂²⋊C₄	—	29	1	no	—
220	C₂×C₄²	—	29	1	no	—
221	C₂×C₄²	—	29	1	no	—
222	C₂×C₄²	—	29	1	no	—
223	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
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	—	29	1	no	—
230	#95	—	29	1	no	—
231	#95	—	29	1	no	—
232	#95	—	29	1	no	—
233	#95	—	29	1	no	—
234	#95	C₂×D₂₉	1	1	yes	characteristic
235	#95	C₂×D₂₉	1	1	yes	characteristic
236	#95	C₂×D₂₉	1	1	yes	characteristic
237	#95	C₂×D₂₉	1	1	yes	characteristic
238	C₂₉	#2329	1	1	yes	characteristic, complemented, Sylow
239	C₂⁴	—	29	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	C₄×D₂₉	1	4	yes	complemented
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	—	29	4	no	—
248	C₂²×C₄	—	29	4	no	—
249	C₂²×C₄	—	29	1	no	—
250	C₂²×C₄	—	29	1	no	—
251	C₂²×C₄	—	29	1	no	—
252	C₂²×C₄	—	58	2	no	—
253	C₂²×C₄	—	116	1	no	—
254	C₂²×C₄	—	116	1	no	—
255	C₂²×C₄	—	116	1	no	—
256	C₂²×C₄	—	116	1	no	—
257	C₂²×C₄	—	116	1	no	—
258	C₂²×C₄	—	116	1	no	—
259	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
260	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
264	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
265	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	29	4	no	—
268	C₄⋊C₄	—	29	4	no	—
269	C₄⋊C₄	—	29	4	no	—
270	C₄⋊C₄	—	116	1	no	—
271	C₄⋊C₄	—	116	1	no	—
272	C₄⋊C₄	—	116	1	no	—
273	C₄⋊C₄	—	116	1	no	—
274	C₄⋊C₄	—	116	1	no	—
275	C₂²⋊C₄	—	29	4	no	—
276	C₂²⋊C₄	—	29	4	no	—
277	C₂²⋊C₄	—	58	2	no	—
278	C₂²⋊C₄	—	58	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	116	1	no	—
281	C₄²	—	116	1	no	—
282	C₄²	—	116	1	no	—
283	C₂³	—	29	2	no	—
284	C₂³	—	29	2	no	—
285	C₂³	—	29	2	no	—
286	C₂³	—	58	1	no	—
287	C₂³	—	58	1	no	—
288	C₂³	—	58	1	no	—
289	C₂³	—	58	1	no	—
290	C₂³	C₂³×D₂₉	1	1	yes	characteristic
291	Q₈	—	4	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	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	—	29	2	no	—
310	C₂×C₄	—	29	2	no	—
311	C₂×C₄	—	29	2	no	—
312	C₂×C₄	—	29	2	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	—	58	1	no	—
320	C₂×C₄	—	58	1	no	—
321	C₂×C₄	—	58	1	no	—
322	C₂×C₄	—	58	1	no	—
323	C₂×C₄	—	58	1	no	—
324	C₂×C₄	—	58	1	no	—
325	C₂×C₄	—	58	1	no	—
326	C₂×C₄	—	58	1	no	—
327	C₂×C₄	—	58	1	no	—
328	C₂×C₄	—	58	1	no	—
329	C₂×C₄	—	58	1	no	—
330	C₂×C₄	—	116	4	no	—
331	C₂×C₄	—	116	4	no	—
332	C₂×C₄	—	116	4	no	—
333	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
334	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
335	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
336	C₂×C₄	Q₈×D₂₉	1	2	yes	—
337	C₂²	—	29	4	no	—
338	C₂²	—	58	2	no	—
339	C₂²	—	58	2	no	—
340	C₂²	—	58	2	no	—
341	C₂²	—	58	2	no	—
342	C₂²	—	58	2	no	—
343	C₂²	—	58	2	no	—
344	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
345	C₂²	#11745359	1	1	yes	characteristic
346	C₂²	#11745359	1	1	yes	characteristic
347	C₂²	C₂×Q₈×D₂₉	1	1	yes	characteristic
348	C₂²	#11745362	1	1	yes	characteristic
349	C₂²	#41×D₂₉	1	1	yes	characteristic
350	C₂²	#11745364	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	58	2	no	—
359	C₄	—	58	2	no	—
360	C₄	—	58	2	no	—
361	C₄	—	58	2	no	—
362	C₄	—	116	1	no	—
363	C₄	—	116	1	no	—
364	C₄	—	116	1	no	—
365	C₂	—	58	4	no	—
366	C₂	C₄×Q₈×D₂₉	1	1	yes	characteristic
367	C₂	#14271636	1	1	yes	characteristic
368	C₂	#14271645	1	1	yes	characteristic
369	C₂	#14271747	1	1	yes	characteristic
370	C₂	#14271759	1	1	yes	characteristic
371	C₂	#14271788	1	1	yes	characteristic
372	C₂	#14271790	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₉
commutator	C₂²×C₂₉	C₂³×C₄
Fitting	C₂₉×#391	C₂
socle, cosocle	C₂³×C₂₉	C₂⁴
prime radical	C₂³×D₂₉	C₂³
hypercenter	#391	D₂₉
hypocenter	C₂₉	#2329
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
2. #14271070	C₂	semidirect
3. #14271060	C₂	semidirect
4. #14271058	C₂	semidirect
5. #14271048	C₂	semidirect
6. #14271047	C₂	semidirect
7. #14271046	C₂	semidirect
8. #14271045	C₂	semidirect
15. C₂₉×#391	C₂	semidirect
22. #11745308	C₄	semidirect
55. C₂×Q₈×C₂₉	C₂×C₄	semidirect
78. C₂×C₄×D₂₉	Q₈	semidirect
86. C₄⋊Dic₂₉	C₂×C₄	semidirect
91. #90779	C₂×C₄	semidirect
92. #90779	C₂×C₄	semidirect
181. #391	D₂₉	semidirect
238. C₂₉	#2329	semidirect
242. C₂×Q₈	C₄×D₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745308	2
#14271070	2
#14271060	2
#14271058	2
#14271048	2
#14271047	2
#14271046	2
#14271045	2
#14271037	2
#14271034	2
#14271022	2
C₄×#90781	2
#14271010	2
#95×D₂₉	2
C₂₉×#391	2
#2329	29