Subgroups — #16831485

Order 3712.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271822	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271817	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271523	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271499	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271477	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271459	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271409	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271399	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271379	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271310	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271304	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271265	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271165	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#11745164	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₉×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#90807	—	2	1	no	—
17	C₂×#90805	C₂²	1	1	yes	characteristic
18	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
21	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
22	#11745301	—	2	1	no	—
23	#11745292	C₂²	1	1	yes	characteristic
24	#11745292	C₂²	1	1	yes	characteristic
25	#11745292	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
28	#11745286	C₂²	1	1	yes	characteristic
29	C₂×D₂₃₂	C₂²	1	1	yes	characteristic
30	C₂×#90773	C₂²	1	1	yes	characteristic
31	C₂×#90773	C₂²	1	1	yes	characteristic
32	#11745259	—	2	1	no	—
33	#11745258	—	2	1	no	—
34	#11745248	—	2	1	no	—
35	#11745248	—	2	1	no	—
36	#11745247	—	2	1	no	—
37	#11745243	—	2	1	no	—
38	#11745237	C₂²	1	1	yes	characteristic
39	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
40	#11745231	C₂²	1	1	yes	characteristic
41	C₄×Dic₅₈	C₂²	1	1	yes	characteristic
42	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₂₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₂₉×#97	C₂²	1	1	yes	characteristic
46	#11745169	C₂²	1	1	yes	characteristic
47	#11745169	C₂²	1	1	yes	characteristic
48	#11745169	C₂²	1	1	yes	characteristic
49	#11745169	C₂²	1	1	yes	characteristic
50	#11745166	C₂²	1	1	yes	characteristic
51	#11745166	C₂²	1	1	yes	characteristic
52	#11745166	C₂²	1	1	yes	characteristic
53	#11745164	C₂²	1	1	yes	characteristic
54	#11745164	C₂²	1	1	yes	characteristic
55	#11745164	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	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₂₉	—	2	1	no	—
63	C₂×C₂²⋊D₂₉	—	2	1	no	—
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	#90807	—	4	4	no	—
70	#90805	—	2	2	no	—
71	#90805	—	4	1	no	—
72	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
73	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₂₉	—	2	1	no	—
75	C₂×C₄×D₂₉	—	2	1	no	—
76	C₂×Dic₅₈	—	2	2	no	—
77	C₂×Dic₅₈	—	2	1	no	—
78	C₂×Dic₅₈	—	2	1	no	—
79	C₂×Dic₅₈	—	4	1	no	—
80	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
81	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
82	M₁₆×C₂₉	—	4	1	no	—
83	C₂×C₈×C₂₉	—	4	1	no	—
84	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
85	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
87	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
89	C₂²⋊C₄×C₂₉	D₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₂₉	D₄	1	1	yes	characteristic
91	C₄²×C₂₉	C₂³	1	1	yes	characteristic
92	C₂²⋊Dic₂₉	—	2	1	no	—
93	#90781	—	2	1	no	—
94	#90781	—	2	1	no	—
95	#90781	—	2	1	no	—
96	#90781	—	2	1	no	—
97	#90781	—	2	1	no	—
98	#90781	—	2	1	no	—
99	C₄⋊Dic₂₉	—	2	1	no	—
100	C₄⋊Dic₂₉	—	2	1	no	—
101	C₄⋊Dic₂₉	—	2	1	no	—
102	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
103	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
104	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₂₉	C₂³	1	1	yes	characteristic
106	#90779	—	2	1	no	—
107	#90779	—	2	1	no	—
108	C₄×Dic₂₉	—	2	1	no	—
109	D₂₃₂	—	4	1	no	—
110	#90773	—	2	2	no	—
111	#90773	—	4	1	no	—
112	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₂₉	—	2	1	no	—
115	C₂²×D₂₉	—	2	1	no	—
116	D₄×C₂₉	—	2	2	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	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₂₉	—	4	2	no	—
127	C₂²⋊D₂₉	—	4	2	no	—
128	C₂²⋊D₂₉	—	4	2	no	—
129	C₂²⋊D₂₉	—	4	2	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	—	4	2	no	—
137	C₂×Dic₂₉	—	4	2	no	—
138	C₂×Dic₂₉	—	4	2	no	—
139	D₁₁₆	—	2	1	no	—
140	D₁₁₆	—	2	1	no	—
141	D₁₁₆	—	2	1	no	—
142	D₁₁₆	—	2	1	no	—
143	C₄×D₂₉	—	4	2	no	—
144	C₄×D₂₉	—	4	2	no	—
145	Dic₅₈	—	2	1	no	—
146	Dic₅₈	—	2	1	no	—
147	Dic₅₈	—	2	1	no	—
148	Dic₅₈	—	4	2	no	—
149	Dic₅₈	—	4	2	no	—
150	Dic₅₈	#41	1	2	yes	complemented
151	C₈×C₂₉	—	2	1	no	—
152	C₈×C₂₉	—	2	1	no	—
153	C₈×C₂₉	—	2	1	no	—
154	C₈×C₂₉	—	2	1	no	—
155	#3308	—	29	1	no	maximal, Sylow
156	C₂²×C₂₉	—	2	1	no	—
157	C₂²×C₂₉	—	2	1	no	—
158	C₂²×C₂₉	—	2	1	no	—
159	C₂²×C₂₉	—	2	1	no	—
160	C₂²×C₂₉	—	2	1	no	—
161	C₂²×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
162	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₂₉	—	4	1	no	—
164	C₂×D₂₉	—	4	1	no	—
165	C₂×D₂₉	—	4	1	no	—
166	C₂×D₂₉	—	4	1	no	—
167	C₂×D₂₉	—	4	1	no	—
168	C₂×D₂₉	—	4	1	no	—
169	C₄×C₂₉	—	2	1	no	—
170	C₄×C₂₉	—	2	1	no	—
171	C₄×C₂₉	—	2	1	no	—
172	C₄×C₂₉	—	2	1	no	—
173	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
174	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
175	Dic₂₉	—	2	2	no	—
176	Dic₂₉	—	4	1	no	—
177	Dic₂₉	—	4	1	no	—
178	Dic₂₉	—	4	1	no	—
179	Dic₂₉	—	4	1	no	—
180	Dic₂₉	—	4	1	no	—
181	#548	—	29	1	no	—
182	#547	—	29	1	no	—
183	#489	—	29	1	no	—
184	#484	—	29	1	no	—
185	#480	—	29	1	no	—
186	#477	—	29	1	no	—
187	#468	—	29	1	no	—
188	#466	—	29	1	no	—
189	#463	—	29	1	no	—
190	#449	—	29	1	no	—
191	#448	—	29	1	no	—
192	#440	—	29	1	no	—
193	#435	D₂₉	1	1	yes	characteristic, complemented
194	#417	—	29	1	no	—
195	C₂×#103	—	29	1	no	—
196	C₂×C₂₉	—	2	2	no	—
197	C₂×C₂₉	—	4	1	no	—
198	C₂×C₂₉	#546	1	1	yes	characteristic
199	C₂×C₂₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂×C₂₉	#578	1	1	yes	characteristic
201	D₂₉	—	8	1	no	—
202	D₂₉	—	8	1	no	—
203	C₂×#41	—	29	1	no	—
204	C₂×#41	—	58	1	no	—
205	C₂²×Q₈	—	29	1	no	—
206	C₂×SD₁₆	—	29	1	no	—
207	C₂×SD₁₆	—	29	1	no	—
208	C₂×D₈	—	29	1	no	—
209	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
210	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
211	C₄⋊Q₈	—	29	1	no	—
212	#124	—	29	1	no	—
213	#124	—	58	1	no	—
214	#123	—	58	1	no	—
215	#123	—	58	1	no	—
216	C₂²⋊Q₈	—	29	1	no	—
217	C₂²⋊Q₈	—	58	1	no	—
218	C₂²⋊Q₈	—	58	1	no	—
219	#121	—	29	1	no	—
220	#121	—	29	1	no	—
221	#121	—	29	1	no	—
222	#121	—	58	1	no	—
223	C₄×Q₈	—	29	1	no	—
224	C₄×D₄	—	29	1	no	—
225	C₄×D₄	—	58	1	no	—
226	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	29	1	no	—
228	C₂×C₄⋊C₄	—	29	1	no	—
229	#107	—	29	1	no	—
230	#107	—	29	1	no	—
231	#107	—	29	1	no	—
232	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
233	#103	—	29	1	no	—
234	#103	—	29	1	no	—
235	#103	—	29	1	no	—
236	#103	—	29	1	no	—
237	#103	—	29	1	no	—
238	#103	—	29	1	no	—
239	#102	—	29	1	no	—
240	#102	—	29	1	no	—
241	#102	—	29	1	no	—
242	#102	—	29	1	no	—
243	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
244	C₂²⋊C₈	C₂×D₂₉	1	1	yes	characteristic
245	#97	C₂×D₂₉	1	1	yes	characteristic
246	C₂₉	#3308	1	1	yes	characteristic, complemented, Sylow
247	#41	—	58	2	no	—
248	#41	—	116	1	no	—
249	#41	—	116	4	no	—
250	C₂×Q₈	—	29	1	no	—
251	C₂×Q₈	—	29	1	no	—
252	C₂×Q₈	—	58	2	no	—
253	C₂×Q₈	—	58	1	no	—
254	C₂×Q₈	—	58	1	no	—
255	C₂×Q₈	—	116	1	no	—
256	C₂×D₄	—	29	1	no	—
257	C₂×D₄	—	29	1	no	—
258	C₂×D₄	—	58	1	no	—
259	C₂×D₄	—	58	1	no	—
260	C₂×D₄	—	58	1	no	—
261	C₂×D₄	—	58	1	no	—
262	C₂×D₄	D₁₁₆	1	1	yes	characteristic
263	C₂²×C₄	—	58	1	no	—
264	C₂²×C₄	—	58	1	no	—
265	C₂²×C₄	—	58	1	no	—
266	C₂²×C₄	—	58	1	no	—
267	C₂²×C₄	—	58	1	no	—
268	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
269	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
270	SD₁₆	—	58	2	no	—
271	SD₁₆	—	116	1	no	—
272	D₈	—	116	1	no	—
273	M₁₆	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
276	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
277	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
278	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
279	C₄⋊C₄	—	29	1	no	—
280	C₄⋊C₄	—	29	1	no	—
281	C₄⋊C₄	—	29	1	no	—
282	C₄⋊C₄	—	29	1	no	—
283	C₄⋊C₄	—	58	1	no	—
284	C₄⋊C₄	—	58	1	no	—
285	C₄⋊C₄	—	58	1	no	—
286	C₄⋊C₄	—	58	1	no	—
287	C₄⋊C₄	—	58	1	no	—
288	C₄⋊C₄	D₁₁₆	1	1	yes	characteristic
289	C₂²⋊C₄	—	58	1	no	—
290	C₂²⋊C₄	—	58	1	no	—
291	C₂²⋊C₄	—	58	1	no	—
292	C₂²⋊C₄	—	58	1	no	—
293	C₂²⋊C₄	—	58	1	no	—
294	C₂²⋊C₄	—	58	1	no	—
295	C₂²⋊C₄	—	58	1	no	—
296	C₂²⋊C₄	D₁₁₆	1	1	yes	characteristic
297	C₂²⋊C₄	D₁₁₆	1	1	yes	characteristic
298	C₄²	—	58	1	no	—
299	C₄²	C₂²×D₂₉	1	1	yes	characteristic
300	C₂³	—	58	1	no	—
301	C₂³	—	58	1	no	—
302	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
303	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
304	Q₈	—	29	2	no	—
305	Q₈	—	58	1	no	—
306	Q₈	—	58	1	no	—
307	Q₈	—	58	1	no	—
308	Q₈	—	116	2	no	—
309	Q₈	—	116	2	no	—
310	D₄	—	2	2	no	—
311	D₄	—	58	1	no	—
312	D₄	—	58	1	no	—
313	D₄	—	58	1	no	—
314	D₄	—	58	1	no	—
315	D₄	—	116	2	no	—
316	D₄	—	116	2	no	—
317	D₄	—	116	2	no	—
318	D₄	—	116	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	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	—	116	2	no	—
330	C₂×C₄	—	116	2	no	—
331	C₂×C₄	—	116	2	no	—
332	C₂×C₄	—	116	2	no	—
333	C₂×C₄	—	116	2	no	—
334	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
335	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
336	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
337	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	116	1	no	—
349	C₂²	—	116	1	no	—
350	C₂²	—	116	1	no	—
351	C₂²	—	116	1	no	—
352	C₂²	—	116	1	no	—
353	C₂²	—	116	1	no	—
354	C₂²	#11745270	1	2	yes	—
355	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	58	2	no	—
361	C₄	—	116	1	no	—
362	C₄	—	116	1	no	—
363	C₄	—	116	1	no	—
364	C₄	—	116	1	no	—
365	C₄	—	116	1	no	—
366	C₄	#11745359	1	1	yes	characteristic
367	C₄	#41×D₂₉	1	1	yes	characteristic
368	C₂	—	2	2	no	—
369	C₂	—	4	1	no	—
370	C₂	—	232	1	no	—
371	C₂	—	232	1	no	—
372	C₂	#14271808	1	1	yes	characteristic
373	C₂	C₂×#11745270	1	1	yes	characteristic
374	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
hypercenter	#435	D₂₉
hypocenter	C₂₉	#3308
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. #14271822	C₂	semidirect
2. #14271817	C₂	semidirect
3. #14271523	C₂	semidirect
4. #14271499	C₂	semidirect
5. #14271477	C₂	semidirect
6. #14271459	C₂	semidirect
7. #14271409	C₂	semidirect
8. #14271399	C₂	semidirect
9. #14271379	C₂	semidirect
10. #14271310	C₂	semidirect
11. #14271304	C₂	semidirect
12. #14271265	C₂	semidirect
13. #14271165	C₂	semidirect
14. C₂×#11745164	C₂	semidirect
15. C₂₉×#435	C₂	semidirect
150. Dic₅₈	#41	semidirect
193. #435	D₂₉	semidirect
246. C₂₉	#3308	semidirect

Maximal subgroups

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

subgroup	index
#14271822	2
#14271817	2
#14271523	2
#14271499	2
#14271477	2
#14271459	2
#14271409	2
#14271399	2
#14271379	2
#14271310	2
#14271304	2
#14271265	2
#14271165	2
C₂×#11745164	2
C₂₉×#435	2
#3308	29