Subgroups — #16829984

Order 3712.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11745280	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11745277	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11745278	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271738	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271733	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271557	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271556	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271554	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271545	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271544	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271542	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271541	C₂	1	1	yes	maximal, characteristic, complemented
14	#14271120	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₉×#497	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#90809	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₂₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₂₉	C₂²	1	1	yes	characteristic
19	C₂×#90807	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
22	C₂×SD₁₆×C₂₉	C₂²	1	1	yes	characteristic
23	C₂×SD₁₆×C₂₉	C₂²	1	1	yes	characteristic
24	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂₉×#124	C₂²	1	1	yes	characteristic
26	#11745309	C₂²	1	1	yes	characteristic
27	#11745306	C₂²	1	1	yes	characteristic
28	#11745306	C₂²	1	1	yes	characteristic
29	C₂×#90785	C₂²	1	1	yes	characteristic
30	C₂×#90785	C₂²	1	1	yes	characteristic
31	C₂×#90785	C₂²	1	1	yes	characteristic
32	C₂×#90785	C₂²	1	1	yes	characteristic
33	C₂×#90785	C₂²	1	1	yes	characteristic
34	C₂×#90784	C₂²	1	1	yes	characteristic
35	#11745298	C₂²	1	1	yes	characteristic
36	C₂×#90783	C₂²	1	1	yes	characteristic
37	C₂×#90782	C₂²	1	1	yes	characteristic
38	#11745280	—	2	4	no	—
39	Q₁₆×D₂₉	—	2	4	no	—
40	#11745278	—	2	4	no	—
41	#11745277	—	2	4	no	—
42	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
43	C₂×Dic₁₁₆	C₂²	1	1	yes	characteristic
44	C₂×#90773	C₂²	1	1	yes	characteristic
45	C₂×#90773	C₂²	1	1	yes	characteristic
46	C₂×#90772	C₂²	1	1	yes	characteristic
47	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
48	#11745258	—	2	2	no	—
49	#11745256	—	2	2	no	—
50	#11745245	—	2	2	no	—
51	#11745240	—	2	2	no	—
52	#11745237	C₂²	1	1	yes	characteristic
53	#11745234	C₂²	1	1	yes	characteristic
54	#11745231	C₂²	1	1	yes	characteristic
55	C₂₉×#97	C₂²	1	1	yes	characteristic
56	#11745163	C₂²	1	1	yes	characteristic
57	C₈×Dic₂₉	C₂²	1	1	yes	characteristic
58	C₄×#6597	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₂₉	—	2	1	no	—
64	C₂²×Dic₂₉	—	2	1	no	—
65	#90809	—	2	2	no	—
66	#90809	—	2	2	no	—
67	Q₈×D₂₉	—	2	2	no	—
68	Q₈×D₂₉	—	2	2	no	—
69	Q₈×D₂₉	—	2	2	no	—
70	Q₈×D₂₉	—	2	2	no	—
71	#90807	—	2	2	no	—
72	#90807	—	2	2	no	—
73	C₂×D₁₁₆	—	2	1	no	—
74	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₂₉	—	2	1	no	—
76	C₂×C₄×D₂₉	—	2	1	no	—
77	C₂×C₄×D₂₉	—	2	1	no	—
78	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₅₈	—	2	1	no	—
80	C₂×Dic₅₈	—	2	1	no	—
81	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
82	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
83	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
84	Q₁₆×C₂₉	—	2	2	no	—
85	Q₁₆×C₂₉	—	2	2	no	—
86	SD₁₆×C₂₉	—	2	2	no	—
87	SD₁₆×C₂₉	—	2	2	no	—
88	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
89	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₂₉	—	2	2	no	—
91	C₂²⋊C₄×C₂₉	—	2	2	no	—
92	C₄²×C₂₉	C₂³	1	1	yes	characteristic
93	C₂²⋊Dic₂₉	—	2	2	no	—
94	#90785	—	2	2	no	—
95	#90785	—	2	2	no	—
96	#90785	—	2	2	no	—
97	#90785	—	2	2	no	—
98	#90785	—	2	2	no	—
99	#90784	—	2	2	no	—
100	#90783	—	2	2	no	—
101	#90782	—	2	2	no	—
102	#90781	—	2	2	no	—
103	#90781	—	2	2	no	—
104	#90781	D₄	1	2	yes	—
105	C₄⋊Dic₂₉	—	2	2	no	—
106	#90779	—	2	2	no	—
107	#90779	—	2	2	no	—
108	#90779	D₄	1	2	yes	—
109	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
110	C₂×#6597	C₂³	1	1	yes	characteristic
111	C₂×#6597	C₂³	1	1	yes	characteristic
112	Dic₁₁₆	—	2	2	no	—
113	Dic₁₁₆	—	2	2	no	—
114	#90773	—	2	2	no	—
115	#90773	—	2	2	no	—
116	#90772	D₄	1	4	yes	—
117	C₈×D₂₉	D₄	1	4	yes	—
118	C₂³×C₂₉	—	2	1	no	—
119	C₂²×D₂₉	—	2	1	no	—
120	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
121	Q₈×C₂₉	—	2	1	no	—
122	Q₈×C₂₉	—	2	1	no	—
123	Q₈×C₂₉	—	2	1	no	—
124	Q₈×C₂₉	—	2	1	no	—
125	Q₈×C₂₉	—	2	1	no	—
126	Q₈×C₂₉	—	2	1	no	—
127	D₄×C₂₉	—	2	1	no	—
128	D₄×C₂₉	—	2	1	no	—
129	C₂×C₄×C₂₉	—	2	1	no	—
130	C₂×C₄×C₂₉	—	2	1	no	—
131	C₂×C₄×C₂₉	—	2	1	no	—
132	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
133	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₂₉	—	4	2	no	—
135	C₂×Dic₂₉	—	2	1	no	—
136	C₂×Dic₂₉	—	2	1	no	—
137	C₂×Dic₂₉	—	2	1	no	—
138	C₂×Dic₂₉	—	4	2	no	—
139	C₂×Dic₂₉	C₂×D₄	1	2	yes	—
140	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
141	D₁₁₆	—	2	1	no	—
142	D₁₁₆	—	2	1	no	—
143	D₁₁₆	—	4	2	no	—
144	C₄×D₂₉	—	4	2	no	—
145	C₄×D₂₉	—	4	2	no	—
146	C₄×D₂₉	—	4	2	no	—
147	C₄×D₂₉	C₂×D₄	1	2	yes	—
148	C₄×D₂₉	C₂×D₄	1	2	yes	—
149	Dic₅₈	—	2	1	no	—
150	Dic₅₈	—	2	1	no	—
151	Dic₅₈	—	2	1	no	—
152	Dic₅₈	—	2	1	no	—
153	Dic₅₈	—	2	1	no	—
154	Dic₅₈	—	2	1	no	—
155	Dic₅₈	—	4	2	no	—
156	Dic₅₈	—	4	2	no	—
157	C₈×C₂₉	C₂×D₄	1	2	yes	—
158	C₈×C₂₉	C₂×D₄	1	2	yes	—
159	#6597	C₂×D₄	1	2	yes	—
160	#6597	C₂×D₄	1	2	yes	—
161	#3156	—	29	1	no	maximal, Sylow
162	C₂²×C₂₉	—	4	1	no	—
163	C₂²×C₂₉	—	4	2	no	—
164	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×D₂₉	—	2	2	no	—
166	C₂×D₂₉	—	4	1	no	—
167	C₂×D₂₉	—	4	2	no	—
168	C₂×D₂₉	#127	1	2	yes	—
169	C₄×C₂₉	—	2	2	no	—
170	C₄×C₂₉	—	4	1	no	—
171	C₄×C₂₉	—	4	1	no	—
172	C₄×C₂₉	—	4	1	no	—
173	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
174	C₄×C₂₉	C₂²×D₄	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	#127	1	2	yes	—
180	C₂×#137	—	29	1	no	—
181	C₂×#137	—	29	1	no	—
182	C₂×#135	—	29	1	no	—
183	C₂²×Q₁₆	—	29	1	no	—
184	#536	—	29	1	no	—
185	#536	—	29	1	no	—
186	#497	—	29	1	no	—
187	#497	—	29	1	no	—
188	#497	—	29	1	no	—
189	#497	D₂₉	1	1	yes	characteristic, complemented
190	#495	—	29	1	no	—
191	#495	—	29	1	no	—
192	#494	—	29	1	no	—
193	#494	—	29	1	no	—
194	#405	—	29	1	no	—
195	C₂×C₂₉	—	8	1	no	—
196	C₂×C₂₉	C₂×#127	1	1	yes	characteristic
197	C₂×C₂₉	#578	1	2	yes	—
198	D₂₉	—	2	2	no	—
199	D₂₉	—	8	1	no	—
200	C₂×#41	—	29	1	no	—
201	C₂×#41	—	29	1	no	—
202	C₂²×Q₈	—	29	1	no	—
203	C₂²×Q₈	—	29	1	no	—
204	#137	—	58	4	no	—
205	#137	—	58	4	no	—
206	#135	—	58	4	no	—
207	C₂×Q₁₆	—	29	1	no	—
208	C₂×Q₁₆	—	29	1	no	—
209	C₂×Q₁₆	—	29	1	no	—
210	C₂×Q₁₆	—	29	1	no	—
211	C₂×Q₁₆	—	29	1	no	—
212	C₂×Q₁₆	—	29	1	no	—
213	C₂×Q₁₆	—	29	1	no	—
214	C₂×Q₁₆	—	58	4	no	—
215	C₂×Q₁₆	C₂×D₂₉	1	1	yes	characteristic
216	C₂×Q₁₆	C₂×D₂₉	1	1	yes	characteristic
217	C₂×SD₁₆	—	29	1	no	—
218	C₂×SD₁₆	—	29	1	no	—
219	C₂×SD₁₆	—	29	1	no	—
220	C₂×SD₁₆	—	29	1	no	—
221	C₂×SD₁₆	C₂×D₂₉	1	1	yes	characteristic
222	C₂×SD₁₆	C₂×D₂₉	1	1	yes	characteristic
223	C₂×D₈	—	29	1	no	—
224	C₂×M₁₆	—	29	1	no	—
225	C₂²×C₈	—	29	1	no	—
226	C₄⋊Q₈	—	29	1	no	—
227	C₄⋊Q₈	—	29	1	no	—
228	C₄⋊Q₈	—	29	1	no	—
229	C₄⋊Q₈	C₂×D₂₉	1	1	yes	characteristic
230	#124	—	29	1	no	—
231	#124	—	29	1	no	—
232	#124	—	29	1	no	—
233	#124	C₂×D₂₉	1	1	yes	characteristic
234	#123	—	58	2	no	—
235	#123	—	58	2	no	—
236	C₂²⋊Q₈	—	58	2	no	—
237	C₂²⋊Q₈	—	58	2	no	—
238	#117	—	29	1	no	—
239	#97	—	29	1	no	—
240	#97	C₂×D₂₉	1	1	yes	characteristic
241	C₄×C₈	—	29	1	no	—
242	C₄×C₈	—	29	1	no	—
243	C₂₉	#3156	1	1	yes	characteristic, complemented, Sylow
244	#41	—	58	2	no	—
245	#41	—	58	2	no	—
246	#41	—	58	2	no	—
247	#41	—	58	2	no	—
248	C₂×Q₈	—	29	1	no	—
249	C₂×Q₈	—	29	1	no	—
250	C₂×Q₈	—	29	1	no	—
251	C₂×Q₈	—	58	1	no	—
252	C₂×Q₈	—	58	1	no	—
253	C₂×Q₈	—	58	2	no	—
254	C₂×Q₈	—	58	2	no	—
255	C₂×Q₈	—	58	2	no	—
256	C₂×Q₈	—	58	2	no	—
257	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
258	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
260	C₂×D₄	—	29	1	no	—
261	C₂×D₄	—	58	1	no	—
262	C₂×D₄	—	58	1	no	—
263	C₂×D₄	C₂²×D₂₉	1	1	yes	characteristic
264	C₂²×C₄	—	29	1	no	—
265	C₂²×C₄	—	58	1	no	—
266	C₂²×C₄	—	58	1	no	—
267	C₂²×C₄	—	58	1	no	—
268	C₂²×C₄	—	58	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	58	2	no	—
272	Q₁₆	—	58	2	no	—
273	Q₁₆	—	58	2	no	—
274	Q₁₆	—	58	2	no	—
275	Q₁₆	—	58	2	no	—
276	Q₁₆	—	58	2	no	—
277	Q₁₆	—	58	2	no	—
278	SD₁₆	—	2	2	no	—
279	SD₁₆	—	2	2	no	—
280	SD₁₆	—	58	2	no	—
281	SD₁₆	—	58	2	no	—
282	SD₁₆	—	58	2	no	—
283	SD₁₆	—	58	2	no	—
284	D₈	—	58	2	no	—
285	M₁₆	—	29	4	no	—
286	C₂×C₈	—	29	1	no	—
287	C₂×C₈	—	29	1	no	—
288	C₂×C₈	—	29	4	no	—
289	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
290	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	29	2	no	—
293	C₄⋊C₄	—	58	2	no	—
294	C₄⋊C₄	—	58	2	no	—
295	C₄⋊C₄	—	58	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	29	2	no	—
298	C₂²⋊C₄	—	58	2	no	—
299	C₂²⋊C₄	—	58	2	no	—
300	C₂²⋊C₄	—	58	2	no	—
301	C₄²	—	29	1	no	—
302	C₄²	C₂²×D₂₉	1	1	yes	characteristic
303	C₂³	—	2	1	no	—
304	C₂³	—	29	1	no	—
305	C₂³	—	58	1	no	—
306	Q₈	—	2	1	no	—
307	Q₈	—	2	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	2	1	no	—
312	Q₈	—	58	1	no	—
313	Q₈	—	58	1	no	—
314	Q₈	—	58	1	no	—
315	Q₈	—	58	1	no	—
316	Q₈	—	58	1	no	—
317	Q₈	—	58	1	no	—
318	Q₈	—	116	2	no	—
319	Q₈	—	116	2	no	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	58	1	no	—
323	D₄	—	58	1	no	—
324	D₄	—	116	2	no	—
325	D₄	—	116	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	29	1	no	—
330	C₂×C₄	—	29	2	no	—
331	C₂×C₄	—	29	2	no	—
332	C₂×C₄	—	29	2	no	—
333	C₂×C₄	—	58	1	no	—
334	C₂×C₄	—	58	1	no	—
335	C₂×C₄	—	58	1	no	—
336	C₂×C₄	—	116	2	no	—
337	C₂×C₄	—	116	2	no	—
338	C₂×C₄	—	116	2	no	—
339	C₂×C₄	—	116	2	no	—
340	C₂×C₄	D₄×D₂₉	1	2	yes	—
341	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
342	C₈	—	29	2	no	—
343	C₈	—	29	2	no	—
344	C₈	D₄×D₂₉	1	2	yes	—
345	C₈	D₄×D₂₉	1	2	yes	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	2	no	—
348	C₂²	—	29	2	no	—
349	C₂²	—	58	2	no	—
350	C₂²	—	116	1	no	—
351	C₂²	—	116	2	no	—
352	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	29	2	no	—
358	C₄	—	58	2	no	—
359	C₄	—	116	1	no	—
360	C₄	—	116	1	no	—
361	C₄	—	116	1	no	—
362	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
363	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
364	C₂	—	8	1	no	—
365	C₂	—	58	2	no	—
366	C₂	—	232	1	no	—
367	C₂	#127×D₂₉	1	1	yes	characteristic
368	C₂	#14271979	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₉
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#497	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#11745278	C₂
hypercenter	#497	D₂₉
hypocenter	C₂₉	#3156
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
1. C₂×#11745280	C₂	semidirect
2. C₂×#11745277	C₂	semidirect
4. C₂×Q₁₆×D₂₉	C₂	semidirect
5. #14271738	C₂	semidirect
6. #14271733	C₂	semidirect
7. #14271557	C₂	semidirect
8. #14271556	C₂	semidirect
9. #14271554	C₂	semidirect
10. #14271545	C₂	semidirect
11. #14271544	C₂	semidirect
12. #14271542	C₂	semidirect
13. #14271541	C₂	semidirect
14. #14271120	C₂	semidirect
15. C₂₉×#497	C₂	semidirect
189. #497	D₂₉	semidirect
243. C₂₉	#3156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11745280	2
C₂×#11745277	2
C₂×#11745278	2
C₂×Q₁₆×D₂₉	2
#14271738	2
#14271733	2
#14271557	2
#14271556	2
#14271554	2
#14271545	2
#14271544	2
#14271542	2
#14271541	2
#14271120	2
C₂₉×#497	2
#3156	29