Subgroups — #16822769

Order 3712.

528 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745249	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11745254	C₂	1	1	yes	maximal, characteristic
3	#14271105	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271098	C₂	1	1	yes	maximal, characteristic
5	#14271096	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271085	C₂	1	1	yes	maximal, characteristic
7	#14271085	C₂	1	1	yes	maximal, characteristic
8	#14271084	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271076	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×Dic₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊C₄×Dic₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271014	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271010	C₂	1	1	yes	maximal, characteristic
14	#14271010	C₂	1	1	yes	maximal, characteristic
15	C₂₉×#398	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₄×D₂₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
22	C₂×#90781	C₂²	1	1	yes	characteristic
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×#90781	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₂₉	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	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
41	#11745254	—	2	4	no	—
42	#11745249	—	2	4	no	—
43	C₂₉×#95	C₂²	1	1	yes	characteristic
44	C₂₉×#95	C₂²	1	1	yes	characteristic
45	C₂₉×#95	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	#11745179	C₂²	1	1	yes	characteristic
53	C₂³×D₂₉	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×C₄×C₂₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₉	C₂³	1	1	yes	characteristic
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	C₂×C₄×D₂₉	—	2	2	no	—
69	C₂×C₄×D₂₉	—	2	2	no	—
70	C₂×C₄×D₂₉	Q₈	1	4	yes	complemented
71	C₂×Dic₅₈	—	2	2	no	—
72	C₂×Dic₅₈	—	2	2	no	—
73	C₂×Dic₅₈	—	2	2	no	—
74	C₄⋊C₄×C₂₉	—	2	2	no	—
75	C₄⋊C₄×C₂₉	—	2	2	no	—
76	C₄⋊C₄×C₂₉	—	2	2	no	—
77	C₄⋊C₄×C₂₉	—	2	2	no	—
78	#90781	—	2	2	no	—
79	#90781	—	2	2	no	—
80	#90781	—	2	2	no	—
81	#90781	—	2	2	no	—
82	#90781	—	2	2	no	—
83	#90781	—	2	2	no	—
84	C₄⋊Dic₂₉	—	2	2	no	—
85	C₄⋊Dic₂₉	—	2	2	no	—
86	C₄⋊Dic₂₉	—	2	2	no	—
87	#90779	—	2	2	no	—
88	#90779	—	2	2	no	—
89	#90779	—	2	2	no	—
90	#90779	—	2	2	no	—
91	C₄×Dic₂₉	—	2	2	no	—
92	C₄×Dic₂₉	—	2	2	no	—
93	C₄×Dic₂₉	—	2	2	no	—
94	C₄×Dic₂₉	—	2	2	no	—
95	C₄×Dic₂₉	—	2	2	no	—
96	C₄×Dic₂₉	D₄	1	4	yes	—
97	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
98	C₂²×D₂₉	—	2	1	no	—
99	C₂²×D₂₉	—	2	1	no	—
100	C₂²×D₂₉	—	2	1	no	—
101	C₂²×D₂₉	—	2	1	no	—
102	C₂²×D₂₉	—	2	1	no	—
103	C₂²×D₂₉	—	2	1	no	—
104	C₂²×D₂₉	C₂×Q₈	1	2	yes	—
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	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
124	C₂×C₄×C₂₉	#41	1	2	yes	—
125	C₂×C₄×C₂₉	#41	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	C₂×D₄	1	2	yes	—
142	C₂×Dic₂₉	C₂×D₄	1	2	yes	—
143	C₂×Dic₂₉	C₂×Q₈	1	2	yes	—
144	C₂×Dic₂₉	#41	1	2	yes	—
145	C₂×Dic₂₉	#41	1	2	yes	—
146	C₂×Dic₂₉	#41	1	2	yes	—
147	C₄×D₂₉	—	2	8	no	—
148	Dic₅₈	—	2	8	no	—
149	#2396	—	29	1	no	maximal, Sylow
150	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
151	C₂²×C₂₉	C₂²×Q₈	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	C₂×D₂₉	—	2	2	no	—
158	C₂×D₂₉	—	2	2	no	—
159	C₂×D₂₉	—	2	2	no	—
160	C₂×D₂₉	—	4	1	no	—
161	C₂×D₂₉	—	4	1	no	—
162	C₂×D₂₉	—	4	1	no	—
163	C₂×D₂₉	—	4	1	no	—
164	C₄×C₂₉	—	2	2	no	—
165	C₄×C₂₉	—	2	2	no	—
166	C₄×C₂₉	—	2	2	no	—
167	C₄×C₂₉	—	2	2	no	—
168	C₄×C₂₉	—	4	1	no	—
169	C₄×C₂₉	—	4	1	no	—
170	C₄×C₂₉	—	4	1	no	—
171	Dic₂₉	—	2	2	no	—
172	Dic₂₉	—	2	2	no	—
173	Dic₂₉	—	2	2	no	—
174	Dic₂₉	—	4	1	no	—
175	Dic₂₉	—	4	1	no	—
176	Dic₂₉	—	4	1	no	—
177	Dic₂₉	C₂²⋊Q₈	1	4	yes	—
178	C₂×C₄×Q₈	—	29	1	no	—
179	C₂×#117	—	29	1	no	—
180	#400	—	29	1	no	—
181	#399	—	29	1	no	—
182	#398	—	29	1	no	—
183	#398	D₂₉	1	1	yes	characteristic, complemented
184	#396	—	29	1	no	—
185	#396	—	29	1	no	—
186	#395	—	29	1	no	—
187	#393	—	29	1	no	—
188	C₄×C₄⋊C₄	—	29	1	no	—
189	C₄×C₄⋊C₄	—	29	1	no	—
190	C₄×C₄⋊C₄	—	29	1	no	—
191	C₄×C₂²⋊C₄	—	29	1	no	—
192	C₄×C₂²⋊C₄	—	29	1	no	—
193	C₂×C₂₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
194	C₂×C₂₉	#525	1	1	yes	characteristic
195	C₂×C₂₉	#529	1	1	yes	characteristic
196	C₂×C₂₉	#529	1	1	yes	characteristic
197	C₂×C₂₉	#532	1	1	yes	characteristic
198	C₂×C₂₉	#533	1	1	yes	characteristic
199	C₂×C₂₉	#533	1	1	yes	characteristic
200	D₂₉	—	4	2	no	—
201	C₂²×Q₈	—	29	1	no	—
202	C₂³×C₄	—	29	1	no	—
203	C₄×Q₈	—	58	4	no	—
204	#117	—	58	4	no	—
205	C₂×C₄⋊C₄	—	29	1	no	—
206	C₂×C₄⋊C₄	—	29	1	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	C₂×D₂₉	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂×D₂₉	1	1	yes	characteristic
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×C₂²⋊C₄	—	29	1	no	—
221	C₂×C₂²⋊C₄	—	29	1	no	—
222	C₂×C₄²	—	29	1	no	—
223	C₂×C₄²	—	29	1	no	—
224	C₂×C₄²	—	29	1	no	—
225	C₂×C₄²	—	29	1	no	—
226	C₂×C₄²	—	29	1	no	—
227	C₂×C₄²	—	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	—	29	1	no	—
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	#2396	1	1	yes	characteristic, complemented, Sylow
239	C₂⁴	—	29	1	no	—
240	C₂×Q₈	—	58	2	no	—
241	C₂×Q₈	—	58	2	no	—
242	C₂×Q₈	—	58	2	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	—	29	4	no	—
248	C₂²×C₄	—	29	1	no	—
249	C₂²×C₄	—	29	1	no	—
250	C₂²×C₄	—	29	1	no	—
251	C₂²×C₄	—	58	2	no	—
252	C₂²×C₄	—	58	2	no	—
253	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
257	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
258	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
259	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	58	2	no	—
265	C₄⋊C₄	—	58	2	no	—
266	C₄⋊C₄	—	58	2	no	—
267	C₄⋊C₄	—	58	2	no	—
268	C₄⋊C₄	—	58	2	no	—
269	C₄⋊C₄	—	58	2	no	—
270	C₄⋊C₄	—	58	2	no	—
271	C₂²⋊C₄	—	58	2	no	—
272	C₂²⋊C₄	—	58	2	no	—
273	C₂²⋊C₄	—	58	2	no	—
274	C₂²⋊C₄	—	58	2	no	—
275	C₂²⋊C₄	—	58	2	no	—
276	C₂²⋊C₄	—	58	2	no	—
277	C₄²	—	29	4	no	—
278	C₄²	—	58	2	no	—
279	C₄²	—	58	2	no	—
280	C₄²	—	58	2	no	—
281	C₄²	—	58	2	no	—
282	C₄²	—	58	2	no	—
283	C₂³	—	29	2	no	—
284	C₂³	—	58	1	no	—
285	C₂³	—	58	1	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	—	58	8	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	—	29	2	no	—
314	C₂×C₄	—	29	2	no	—
315	C₂×C₄	—	58	8	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	—	58	1	no	—
331	C₂×C₄	D₄×D₂₉	1	2	yes	—
332	C₂×C₄	#90807	1	2	yes	—
333	C₂×C₄	Q₈×D₂₉	1	2	yes	—
334	C₂×C₄	#90809	1	2	yes	—
335	C₂²	—	58	2	no	—
336	C₂²	—	58	2	no	—
337	C₂²	—	58	2	no	—
338	C₂²	—	116	1	no	—
339	C₂²	—	116	1	no	—
340	C₂²	—	116	1	no	—
341	C₂²	—	116	1	no	—
342	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
343	C₂²	C₂×#90807	1	1	yes	characteristic
344	C₂²	C₂×Q₈×D₂₉	1	1	yes	characteristic
345	C₂²	C₂×#90809	1	1	yes	characteristic
346	C₂²	#41×D₂₉	1	1	yes	characteristic
347	C₂²	#41×D₂₉	1	1	yes	characteristic
348	C₂²	#41×D₂₉	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	29	4	no	—
357	C₄	—	58	2	no	—
358	C₄	—	58	2	no	—
359	C₄	—	58	2	no	—
360	C₄	—	116	1	no	—
361	C₄	—	116	1	no	—
362	C₄	—	116	1	no	—
363	C₂	—	116	2	no	—
364	C₂	C₂²⋊Q₈×D₂₉	1	1	yes	characteristic
365	C₂	#14271676	1	1	yes	characteristic
366	C₂	#14271690	1	1	yes	characteristic
367	C₂	#14271698	1	1	yes	characteristic
368	C₂	#14271708	1	1	yes	characteristic
369	C₂	#14271718	1	1	yes	characteristic
370	C₂	#14271719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₉
commutator, socle, cosocle	C₂³×C₂₉	C₂⁴
Fitting	C₂₉×#398	C₂
prime radical	C₂³×D₂₉	C₂³
hypercenter	#398	D₂₉
hypocenter	C₂₉	#2396
LCS	G ▷ C₂³×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ #398

Complements

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

N	H ≅ G/N	type
1. C₂×#11745249	C₂	semidirect
3. #14271105	C₂	semidirect
5. #14271096	C₂	semidirect
8. #14271084	C₂	semidirect
10. C₄⋊C₄×Dic₂₉	C₂	semidirect
11. C₄⋊C₄×Dic₂₉	C₂	semidirect
12. #14271014	C₂	semidirect
15. C₂₉×#398	C₂	semidirect
70. C₂×C₄×D₂₉	Q₈	semidirect
183. #398	D₂₉	semidirect
238. C₂₉	#2396	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745249	2
C₂×#11745254	2
#14271105	2
#14271098	2
#14271096	2
#14271085	2
#14271084	2
#14271076	2
C₄⋊C₄×Dic₂₉	2
#14271014	2
#14271010	2
C₂₉×#398	2
#2396	29