Subgroups — #16831067

Order 3712.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745271	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₁₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₅₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271822	C₂	1	1	yes	maximal, characteristic
5	#14271557	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271421	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271392	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271386	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271374	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271327	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271321	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271321	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271304	C₂	1	1	yes	maximal, characteristic, complemented
14	#14271271	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₉×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#90805	C₂²	1	1	yes	characteristic
17	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
18	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
19	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
23	#11745301	—	2	1	no	—
24	D₄×Dic₂₉	—	2	1	no	—
25	#11745292	C₂²	1	1	yes	characteristic
26	#11745286	C₂²	1	1	yes	characteristic
27	#11745286	C₂²	1	1	yes	characteristic
28	#11745286	C₂²	1	1	yes	characteristic
29	#11745271	—	2	4	no	—
30	C₂×Dic₁₁₆	—	2	4	no	—
31	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
32	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
34	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
36	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
37	C₂×#90773	C₂²	1	1	yes	characteristic
38	C₂×#90773	C₂²	1	1	yes	characteristic
39	C₂×#90773	C₂²	1	1	yes	characteristic
40	#11745258	—	2	1	no	—
41	#11745250	—	2	1	no	—
42	#11745247	—	2	1	no	—
43	#11745243	—	2	1	no	—
44	#11745240	—	2	1	no	—
45	#11745240	—	2	1	no	—
46	#11745237	C₂²	1	1	yes	characteristic
47	#11745231	C₂²	1	1	yes	characteristic
48	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
49	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
50	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
52	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
53	C₂₉×#97	C₂²	1	1	yes	characteristic
54	#11745169	C₂²	1	1	yes	characteristic
55	#11745165	C₂²	1	1	yes	characteristic
56	#11745164	C₂²	1	1	yes	characteristic
57	#11745164	C₂²	1	1	yes	characteristic
58	#11745164	C₂²	1	1	yes	characteristic
59	#11745164	C₂²	1	1	yes	characteristic
60	#11745164	C₂²	1	1	yes	characteristic
61	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₂₉	—	2	1	no	—
65	C₂×C₂²⋊D₂₉	—	2	1	no	—
66	C₂²×Dic₂₉	—	2	1	no	—
67	C₂²×Dic₂₉	—	2	1	no	—
68	C₂²×Dic₂₉	—	2	1	no	—
69	#90805	—	2	2	no	—
70	#90805	—	4	1	no	—
71	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₂₉	—	2	1	no	—
73	C₂×Dic₅₈	—	2	2	no	—
74	C₂×Dic₅₈	—	2	2	no	—
75	C₂×Dic₅₈	—	2	2	no	—
76	C₂×Dic₅₈	—	2	1	no	—
77	C₂×Dic₅₈	—	2	1	no	—
78	C₂×Dic₅₈	—	2	1	no	—
79	C₂×Dic₅₈	—	4	1	no	—
80	C₂×Dic₅₈	—	4	1	no	—
81	C₂×Dic₅₈	—	4	1	no	—
82	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
83	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
84	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
85	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
86	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
87	M₁₆×C₂₉	—	2	2	no	—
88	C₂×C₈×C₂₉	—	2	2	no	—
89	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
90	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
94	C₂²⋊C₄×C₂₉	D₄	1	1	yes	characteristic
95	C₂²⋊C₄×C₂₉	D₄	1	1	yes	characteristic
96	C₄²×C₂₉	C₂³	1	1	yes	characteristic
97	C₂²⋊Dic₂₉	—	2	1	no	—
98	C₂²⋊Dic₂₉	—	2	1	no	—
99	C₂²⋊Dic₂₉	—	2	1	no	—
100	#90781	—	2	1	no	—
101	#90781	—	2	1	no	—
102	#90781	—	2	1	no	—
103	C₄⋊Dic₂₉	—	2	1	no	—
104	C₄⋊Dic₂₉	—	2	1	no	—
105	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
106	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
107	#90779	—	2	1	no	—
108	#90779	—	2	1	no	—
109	#90779	—	2	1	no	—
110	#90779	—	2	1	no	—
111	#90779	—	2	1	no	—
112	C₄×Dic₂₉	—	2	1	no	—
113	C₄×Dic₂₉	—	2	1	no	—
114	Dic₁₁₆	—	2	2	no	—
115	Dic₁₁₆	—	2	2	no	—
116	Dic₁₁₆	—	2	2	no	—
117	Dic₁₁₆	—	2	2	no	—
118	Dic₁₁₆	—	4	1	no	—
119	Dic₁₁₆	D₄	1	4	yes	complemented
120	#90773	—	2	2	no	—
121	#90773	—	2	2	no	—
122	#90773	—	4	1	no	—
123	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
124	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
125	C₂²×D₂₉	—	2	1	no	—
126	D₄×C₂₉	—	2	2	no	—
127	C₂×C₄×C₂₉	—	2	1	no	—
128	C₂×C₄×C₂₉	—	2	1	no	—
129	C₂×C₄×C₂₉	—	2	1	no	—
130	C₂×C₄×C₂₉	—	2	1	no	—
131	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
136	C₂²⋊D₂₉	—	4	2	no	—
137	C₂²⋊D₂₉	—	4	2	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	—	2	1	no	—
144	C₂×Dic₂₉	—	2	1	no	—
145	C₂×Dic₂₉	—	4	2	no	—
146	C₂×Dic₂₉	—	4	2	no	—
147	C₂×Dic₂₉	—	4	2	no	—
148	D₁₁₆	—	2	1	no	—
149	D₁₁₆	—	2	1	no	—
150	C₄×D₂₉	—	4	2	no	—
151	Dic₅₈	—	2	1	no	—
152	Dic₅₈	—	2	1	no	—
153	Dic₅₈	—	2	1	no	—
154	Dic₅₈	—	2	1	no	—
155	Dic₅₈	—	2	1	no	—
156	Dic₅₈	—	2	1	no	—
157	Dic₅₈	—	2	1	no	—
158	Dic₅₈	—	2	1	no	—
159	Dic₅₈	—	4	2	no	—
160	Dic₅₈	—	4	2	no	—
161	Dic₅₈	—	4	2	no	—
162	Dic₅₈	C₂×D₄	1	2	yes	—
163	Dic₅₈	C₂×D₄	1	2	yes	—
164	C₈×C₂₉	—	2	1	no	—
165	C₈×C₂₉	—	2	1	no	—
166	C₈×C₂₉	—	2	1	no	—
167	C₈×C₂₉	C₂×D₄	1	2	yes	—
168	#3270	—	29	1	no	maximal, Sylow
169	C₂²×C₂₉	—	2	1	no	—
170	C₂²×C₂₉	—	2	1	no	—
171	C₂²×C₂₉	—	2	1	no	—
172	C₂²×C₂₉	—	2	1	no	—
173	C₂²×C₂₉	—	2	1	no	—
174	C₂²×C₂₉	#137	1	2	yes	complemented
175	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
176	C₂×D₂₉	—	4	1	no	—
177	C₂×D₂₉	—	4	1	no	—
178	C₂×D₂₉	—	4	1	no	—
179	C₄×C₂₉	—	2	1	no	—
180	C₄×C₂₉	—	2	1	no	—
181	C₄×C₂₉	—	2	1	no	—
182	C₄×C₂₉	—	2	1	no	—
183	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
184	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
185	Dic₂₉	—	2	2	no	—
186	Dic₂₉	—	2	2	no	—
187	Dic₂₉	—	4	1	no	—
188	Dic₂₉	—	4	1	no	—
189	Dic₂₉	—	4	1	no	—
190	Dic₂₉	—	4	1	no	—
191	Dic₂₉	—	4	1	no	—
192	C₂×#137	—	29	1	no	—
193	C₂²×Q₁₆	—	29	1	no	—
194	D₄×Q₈	—	29	1	no	—
195	#548	—	29	1	no	—
196	#497	—	29	1	no	—
197	#470	—	29	1	no	—
198	#465	—	29	1	no	—
199	#464	—	29	1	no	—
200	#462	—	29	1	no	—
201	#452	—	29	1	no	—
202	#451	—	29	1	no	—
203	#451	—	29	1	no	—
204	#448	—	29	1	no	—
205	#441	—	29	1	no	—
206	#435	D₂₉	1	1	yes	characteristic, complemented
207	C₂×C₂₉	—	2	2	no	—
208	C₂×C₂₉	—	4	1	no	—
209	C₂×C₂₉	D₄²	1	1	yes	characteristic
210	C₂×C₂₉	C₂×#137	1	1	yes	characteristic
211	C₂×C₂₉	#578	1	1	yes	characteristic
212	D₂₉	—	8	1	no	—
213	C₂×#41	—	29	1	no	—
214	C₂²×Q₈	—	29	1	no	—
215	C₂²×Q₈	—	29	1	no	—
216	C₂²×Q₈	—	29	1	no	—
217	#137	—	58	4	no	—
218	C₂×Q₁₆	—	29	1	no	—
219	C₂×Q₁₆	—	29	1	no	—
220	C₂×Q₁₆	—	29	1	no	—
221	C₂×Q₁₆	—	29	1	no	—
222	C₂×Q₁₆	—	29	1	no	—
223	C₂×Q₁₆	—	29	1	no	—
224	C₂×Q₁₆	—	58	4	no	—
225	C₂×SD₁₆	—	29	1	no	—
226	C₂×SD₁₆	—	29	1	no	—
227	C₂×SD₁₆	—	29	1	no	—
228	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
229	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
230	C₄⋊Q₈	—	29	1	no	—
231	C₄⋊Q₈	—	58	1	no	—
232	#124	—	29	1	no	—
233	#124	—	58	1	no	—
234	C₂²⋊Q₈	—	29	1	no	—
235	C₂²⋊Q₈	—	29	1	no	—
236	C₂²⋊Q₈	—	29	1	no	—
237	C₂²⋊Q₈	—	58	1	no	—
238	C₂²⋊Q₈	—	58	1	no	—
239	C₂²⋊Q₈	—	58	1	no	—
240	#121	—	29	1	no	—
241	#121	—	58	1	no	—
242	C₄×Q₈	—	29	1	no	—
243	C₄×Q₈	—	29	1	no	—
244	C₄×D₄	—	58	1	no	—
245	C₄×D₄	—	58	1	no	—
246	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
247	#106	—	29	1	no	—
248	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
249	#103	—	29	1	no	—
250	#103	—	29	1	no	—
251	#103	—	29	1	no	—
252	#103	—	29	1	no	—
253	#103	—	29	1	no	—
254	#102	—	29	1	no	—
255	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
256	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
257	#97	C₂×D₂₉	1	1	yes	characteristic
258	C₂₉	#3270	1	1	yes	characteristic, complemented, Sylow
259	#41	—	58	2	no	—
260	#41	—	116	1	no	—
261	C₂×Q₈	—	29	1	no	—
262	C₂×Q₈	—	29	1	no	—
263	C₂×Q₈	—	29	1	no	—
264	C₂×Q₈	—	29	1	no	—
265	C₂×Q₈	—	29	1	no	—
266	C₂×Q₈	—	58	2	no	—
267	C₂×Q₈	—	58	2	no	—
268	C₂×Q₈	—	58	2	no	—
269	C₂×Q₈	—	58	1	no	—
270	C₂×Q₈	—	58	1	no	—
271	C₂×Q₈	—	58	1	no	—
272	C₂×Q₈	—	116	1	no	—
273	C₂×Q₈	—	116	1	no	—
274	C₂×Q₈	—	116	1	no	—
275	C₂×D₄	—	29	1	no	—
276	C₂×D₄	—	58	1	no	—
277	C₂×D₄	—	58	1	no	—
278	C₂×D₄	D₁₁₆	1	1	yes	characteristic
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	C₂²×D₂₉	1	1	yes	characteristic
284	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
285	Q₁₆	—	29	4	no	—
286	Q₁₆	—	58	2	no	—
287	Q₁₆	—	58	2	no	—
288	Q₁₆	—	58	2	no	—
289	Q₁₆	—	58	2	no	—
290	Q₁₆	—	116	1	no	—
291	SD₁₆	—	58	2	no	—
292	SD₁₆	—	58	2	no	—
293	SD₁₆	—	116	1	no	—
294	M₁₆	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
297	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
298	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
300	C₄⋊C₄	—	29	1	no	—
301	C₄⋊C₄	—	29	1	no	—
302	C₄⋊C₄	—	58	1	no	—
303	C₄⋊C₄	—	58	1	no	—
304	C₄⋊C₄	—	58	1	no	—
305	C₄⋊C₄	—	58	1	no	—
306	C₄⋊C₄	—	58	1	no	—
307	C₄⋊C₄	—	58	1	no	—
308	C₄⋊C₄	—	58	1	no	—
309	C₄⋊C₄	D₁₁₆	1	1	yes	characteristic
310	C₂²⋊C₄	—	58	1	no	—
311	C₂²⋊C₄	—	58	1	no	—
312	C₂²⋊C₄	—	58	1	no	—
313	C₂²⋊C₄	—	58	1	no	—
314	C₂²⋊C₄	—	58	1	no	—
315	C₂²⋊C₄	—	58	1	no	—
316	C₂²⋊C₄	D₁₁₆	1	1	yes	characteristic
317	C₂²⋊C₄	D₁₁₆	1	1	yes	characteristic
318	C₄²	—	58	1	no	—
319	C₄²	—	58	1	no	—
320	C₄²	C₂²×D₂₉	1	1	yes	characteristic
321	C₂³	—	58	1	no	—
322	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
323	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
324	Q₈	—	29	2	no	—
325	Q₈	—	29	2	no	—
326	Q₈	—	58	1	no	—
327	Q₈	—	58	1	no	—
328	Q₈	—	58	1	no	—
329	Q₈	—	58	1	no	—
330	Q₈	—	58	1	no	—
331	Q₈	—	58	1	no	—
332	Q₈	—	58	1	no	—
333	Q₈	—	58	1	no	—
334	Q₈	—	116	2	no	—
335	Q₈	—	116	2	no	—
336	Q₈	—	116	2	no	—
337	D₄	—	2	2	no	—
338	D₄	—	58	1	no	—
339	D₄	—	58	1	no	—
340	D₄	—	116	2	no	—
341	D₄	—	116	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	58	1	no	—
347	C₂×C₄	—	58	1	no	—
348	C₂×C₄	—	58	1	no	—
349	C₂×C₄	—	58	1	no	—
350	C₂×C₄	—	58	1	no	—
351	C₂×C₄	—	58	1	no	—
352	C₂×C₄	—	58	1	no	—
353	C₂×C₄	—	116	2	no	—
354	C₂×C₄	—	116	2	no	—
355	C₂×C₄	—	116	2	no	—
356	C₂×C₄	—	116	2	no	—
357	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
358	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
359	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
360	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
361	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	2	1	no	—
365	C₈	D₄×D₂₉	1	2	yes	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	116	1	no	—
372	C₂²	—	116	1	no	—
373	C₂²	—	116	1	no	—
374	C₂²	#11745271	1	2	yes	complemented
375	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	58	2	no	—
381	C₄	—	58	2	no	—
382	C₄	—	116	1	no	—
383	C₄	—	116	1	no	—
384	C₄	—	116	1	no	—
385	C₄	—	116	1	no	—
386	C₄	—	116	1	no	—
387	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
388	C₄	#11745364	1	1	yes	characteristic
389	C₂	—	2	2	no	—
390	C₂	—	4	1	no	—
391	C₂	—	232	1	no	—
392	C₂	D₄×D₁₁₆	1	1	yes	characteristic
393	C₂	C₂×#11745271	1	1	yes	characteristic
394	C₂	#14271980	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₁₆
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#435	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	#14271822	C₂
hypercenter	#435	D₂₉
hypocenter	C₂₉	#3270
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. C₂×#11745271	C₂	semidirect
2. C₂²×Dic₁₁₆	C₂	semidirect
3. D₄×Dic₅₈	C₂	semidirect
5. #14271557	C₂	semidirect
6. #14271421	C₂	semidirect
7. #14271392	C₂	semidirect
8. #14271386	C₂	semidirect
9. #14271374	C₂	semidirect
10. #14271327	C₂	semidirect
11. #14271321	C₂	semidirect
12. #14271321	C₂	semidirect
13. #14271304	C₂	semidirect
14. #14271271	C₂	semidirect
15. C₂₉×#435	C₂	semidirect
119. Dic₁₁₆	D₄	semidirect
174. C₂²×C₂₉	#137	semidirect
206. #435	D₂₉	semidirect
258. C₂₉	#3270	semidirect
374. C₂²	#11745271	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745271	2
C₂²×Dic₁₁₆	2
D₄×Dic₅₈	2
#14271822	2
#14271557	2
#14271421	2
#14271392	2
#14271386	2
#14271374	2
#14271327	2
#14271321	2
#14271304	2
#14271271	2
C₂₉×#435	2
#3270	29