Subgroups — #16829967

Order 3712.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 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×#11745276	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11745281	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×D₂₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#14271738	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271731	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271556	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271552	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271549	C₂	1	1	yes	maximal, characteristic, complemented
10	#14271548	C₂	1	1	yes	maximal, characteristic, complemented
11	#14271545	C₂	1	1	yes	maximal, characteristic, complemented
12	#14271537	C₂	1	1	yes	maximal, characteristic, complemented
13	#14271534	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×#90809	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₂₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₂₉	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic, complemented
21	C₂×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic, complemented
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, complemented
25	C₂₉×#124	C₂²	1	1	yes	characteristic
26	#11745309	C₂²	1	1	yes	characteristic
27	#11745309	C₂²	1	1	yes	characteristic, complemented
28	#11745306	C₂²	1	1	yes	characteristic, complemented
29	C₂×#90785	C₂²	1	1	yes	characteristic, complemented
30	C₂×#90784	C₂²	1	1	yes	characteristic
31	C₂×#90784	C₂²	1	1	yes	characteristic
32	C₂×#90784	C₂²	1	1	yes	characteristic, complemented
33	C₂×#90784	C₂²	1	1	yes	characteristic, complemented
34	C₂×#90784	C₂²	1	1	yes	characteristic, complemented
35	#11745302	C₂²	1	1	yes	characteristic
36	C₂×#90783	C₂²	1	1	yes	characteristic
37	C₂×#90782	C₂²	1	1	yes	characteristic
38	#11745281	—	2	4	no	—
39	#11745280	—	2	4	no	—
40	#11745276	—	2	4	no	—
41	SD₁₆×D₂₉	—	2	4	no	—
42	C₂×D₂₃₂	C₂²	1	1	yes	characteristic
43	C₂×D₂₃₂	C₂²	1	1	yes	characteristic, complemented
44	C₂×#90773	C₂²	1	1	yes	characteristic
45	C₂×#90773	C₂²	1	1	yes	characteristic, complemented
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	#11745246	—	2	2	no	—
51	#11745244	—	2	2	no	—
52	#11745237	C₂²	1	1	yes	characteristic, complemented
53	#11745236	C₂²	1	1	yes	characteristic
54	#11745234	C₂²	1	1	yes	characteristic
55	C₂₉×#97	C₂²	1	1	yes	characteristic, complemented
56	#11745163	C₂²	1	1	yes	characteristic, complemented
57	C₈×Dic₂₉	C₂²	1	1	yes	characteristic, complemented
58	C₄×#6597	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₂₉	—	2	1	no	—
60	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₂₉	—	2	1	no	—
65	C₂×C₂²⋊D₂₉	—	2	2	no	—
66	#90809	—	2	2	no	—
67	#90809	—	2	2	no	—
68	#90809	—	2	2	no	—
69	#90809	—	2	2	no	—
70	Q₈×D₂₉	—	2	2	no	—
71	Q₈×D₂₉	—	2	2	no	—
72	D₄×D₂₉	—	2	2	no	—
73	D₄×D₂₉	—	2	2	no	—
74	C₂×D₁₁₆	—	2	1	no	—
75	C₂×D₁₁₆	—	2	1	no	—
76	C₂×D₁₁₆	—	2	2	no	—
77	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
78	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
79	C₂×D₁₁₆	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₂₉	—	2	1	no	—
81	C₂×C₄×D₂₉	—	2	1	no	—
82	C₂×C₄×D₂₉	—	2	1	no	—
83	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
84	C₂×Dic₅₈	—	2	1	no	—
85	C₂×Dic₅₈	C₂³	1	1	yes	characteristic
86	Q₁₆×C₂₉	—	2	2	no	—
87	Q₁₆×C₂₉	—	2	2	no	—
88	SD₁₆×C₂₉	—	2	2	no	—
89	SD₁₆×C₂₉	—	2	2	no	—
90	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₂₉	—	2	2	no	—
93	C₂²⋊C₄×C₂₉	—	2	2	no	—
94	C₄²×C₂₉	C₂³	1	1	yes	characteristic
95	#90785	—	2	2	no	—
96	#90784	—	2	2	no	—
97	#90784	—	2	2	no	—
98	#90784	—	2	2	no	—
99	#90784	—	2	2	no	—
100	#90784	—	2	2	no	—
101	#90783	—	2	2	no	—
102	#90782	—	2	2	no	—
103	#90781	—	2	2	no	—
104	#90781	—	2	2	no	—
105	#90781	—	2	2	no	—
106	#90781	D₄	1	2	yes	—
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	D₂₃₂	—	2	2	no	—
113	D₂₃₂	—	2	2	no	—
114	#90773	—	2	2	no	—
115	#90773	—	2	2	no	—
116	#90772	D₄	1	4	yes	complemented
117	C₈×D₂₉	D₄	1	4	yes	complemented
118	C₂³×C₂₉	—	2	1	no	—
119	C₂²×D₂₉	—	2	1	no	—
120	C₂²×D₂₉	—	2	1	no	—
121	C₂²×D₂₉	—	2	1	no	—
122	C₂²×D₂₉	—	4	2	no	—
123	C₂²×D₂₉	—	8	2	no	—
124	C₂²×D₂₉	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₂₉	—	2	1	no	—
126	Q₈×C₂₉	—	2	1	no	—
127	Q₈×C₂₉	—	2	1	no	—
128	Q₈×C₂₉	—	2	1	no	—
129	Q₈×C₂₉	—	2	1	no	—
130	Q₈×C₂₉	—	2	1	no	—
131	D₄×C₂₉	—	2	1	no	—
132	D₄×C₂₉	—	2	1	no	—
133	C₂×C₄×C₂₉	—	2	1	no	—
134	C₂×C₄×C₂₉	—	2	1	no	—
135	C₂×C₄×C₂₉	—	2	1	no	—
136	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
138	C₂²⋊D₂₉	—	4	2	no	—
139	C₂²⋊D₂₉	—	8	2	no	—
140	C₂×Dic₂₉	—	2	1	no	—
141	C₂×Dic₂₉	C₂×D₄	1	2	yes	—
142	C₂×Dic₂₉	C₂×D₄	1	1	yes	characteristic
143	D₁₁₆	—	2	1	no	—
144	D₁₁₆	—	2	1	no	—
145	D₁₁₆	—	2	1	no	—
146	D₁₁₆	—	2	1	no	—
147	D₁₁₆	—	2	1	no	—
148	D₁₁₆	—	2	1	no	—
149	D₁₁₆	—	4	2	no	—
150	D₁₁₆	—	4	2	no	—
151	D₁₁₆	—	8	2	no	—
152	C₄×D₂₉	—	4	2	no	—
153	C₄×D₂₉	—	4	2	no	—
154	C₄×D₂₉	—	4	2	no	—
155	C₄×D₂₉	C₂×D₄	1	2	yes	—
156	C₄×D₂₉	C₂×D₄	1	2	yes	—
157	Dic₅₈	—	2	1	no	—
158	Dic₅₈	—	2	1	no	—
159	Dic₅₈	—	4	2	no	—
160	C₈×C₂₉	C₂×D₄	1	2	yes	—
161	C₈×C₂₉	C₂×D₄	1	2	yes	—
162	#6597	C₂×D₄	1	2	yes	—
163	#6597	C₂×D₄	1	2	yes	—
164	#3154	—	29	1	no	maximal, Sylow
165	C₂²×C₂₉	—	4	1	no	—
166	C₂²×C₂₉	—	4	2	no	—
167	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
168	C₂×D₂₉	—	2	2	no	—
169	C₂×D₂₉	—	4	1	no	—
170	C₂×D₂₉	—	4	1	no	—
171	C₂×D₂₉	—	4	1	no	—
172	C₂×D₂₉	—	4	2	no	—
173	C₂×D₂₉	—	4	2	no	—
174	C₂×D₂₉	—	4	2	no	—
175	C₂×D₂₉	—	8	4	no	—
176	C₂×D₂₉	#127	1	2	yes	—
177	C₄×C₂₉	—	2	2	no	—
178	C₄×C₂₉	—	4	1	no	—
179	C₄×C₂₉	—	4	1	no	—
180	C₄×C₂₉	—	4	1	no	—
181	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
182	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
183	Dic₂₉	—	2	2	no	—
184	Dic₂₉	—	4	1	no	—
185	Dic₂₉	#127	1	2	yes	—
186	C₂×#137	—	29	1	no	—
187	C₂×AGL(1,ℤ/8)	—	29	1	no	—
188	C₂×#135	—	29	1	no	—
189	C₂²×SD₁₆	—	29	1	no	—
190	#536	—	29	1	no	—
191	#535	—	29	1	no	—
192	#497	—	29	1	no	—
193	#497	D₂₉	1	1	yes	characteristic, complemented
194	#496	—	29	1	no	—
195	#496	—	29	1	no	—
196	#495	—	29	1	no	—
197	#495	—	29	1	no	—
198	#492	—	29	1	no	—
199	#492	—	29	1	no	—
200	#405	—	29	1	no	—
201	C₂×C₂₉	—	8	1	no	—
202	C₂×C₂₉	C₂×#127	1	1	yes	characteristic
203	C₂×C₂₉	#577	1	2	yes	—
204	D₂₉	—	2	2	no	—
205	D₂₉	—	8	1	no	—
206	D₂₉	—	8	1	no	—
207	D₂₉	—	8	1	no	—
208	C₂×#41	—	29	1	no	—
209	C₂×#41	—	29	1	no	—
210	C₂²×Q₈	—	29	1	no	—
211	C₂²×D₄	—	29	1	no	—
212	#137	—	58	4	no	—
213	AGL(1,ℤ/8)	—	58	4	no	—
214	#135	—	58	4	no	—
215	C₂×Q₁₆	—	29	1	no	—
216	C₂×Q₁₆	C₂×D₂₉	1	1	yes	characteristic, complemented
217	C₂×Q₁₆	C₂×D₂₉	1	1	yes	characteristic, complemented
218	C₂×SD₁₆	—	29	1	no	—
219	C₂×SD₁₆	—	29	1	no	—
220	C₂×SD₁₆	—	29	1	no	—
221	C₂×SD₁₆	—	29	1	no	—
222	C₂×SD₁₆	—	29	1	no	—
223	C₂×SD₁₆	—	29	1	no	—
224	C₂×SD₁₆	—	29	1	no	—
225	C₂×SD₁₆	—	29	1	no	—
226	C₂×SD₁₆	—	58	4	no	—
227	C₂×SD₁₆	C₂×D₂₉	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×D₂₉	1	1	yes	characteristic
229	C₂×D₈	—	29	1	no	—
230	C₂×D₈	—	29	1	no	—
231	C₂×D₈	—	29	1	no	—
232	C₂×M₁₆	—	29	1	no	—
233	C₂²×C₈	—	29	1	no	—
234	C₄⋊Q₈	—	29	1	no	—
235	C₄⋊Q₈	C₂×D₂₉	1	1	yes	characteristic, complemented
236	#127	—	29	1	no	—
237	#127	—	29	1	no	—
238	#124	—	29	1	no	—
239	#124	—	29	1	no	—
240	#124	—	29	1	no	—
241	#124	C₂×D₂₉	1	1	yes	characteristic
242	#123	—	58	2	no	—
243	C₂²⋊Q₈	—	58	2	no	—
244	#121	—	58	2	no	—
245	#120	—	58	2	no	—
246	#117	—	29	1	no	—
247	#97	—	29	1	no	—
248	#97	C₂×D₂₉	1	1	yes	characteristic, complemented
249	C₄×C₈	—	29	1	no	—
250	C₄×C₈	—	29	1	no	—
251	C₂₉	#3154	1	1	yes	characteristic, complemented, Sylow
252	C₂⁴	—	58	1	no	—
253	#41	—	58	2	no	—
254	#41	—	58	2	no	—
255	#41	—	58	2	no	—
256	#41	—	58	2	no	—
257	C₂×Q₈	—	29	1	no	—
258	C₂×Q₈	—	58	1	no	—
259	C₂×Q₈	—	58	2	no	—
260	C₂×Q₈	—	58	2	no	—
261	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
262	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
263	C₂×Q₈	C₂²×D₂₉	1	1	yes	characteristic
264	C₂×D₄	—	29	1	no	—
265	C₂×D₄	—	29	1	no	—
266	C₂×D₄	—	29	1	no	—
267	C₂×D₄	—	58	1	no	—
268	C₂×D₄	—	58	1	no	—
269	C₂×D₄	—	58	2	no	—
270	C₂×D₄	—	58	2	no	—
271	C₂×D₄	—	58	1	no	—
272	C₂×D₄	—	58	2	no	—
273	C₂×D₄	—	58	2	no	—
274	C₂×D₄	C₂²×D₂₉	1	1	yes	characteristic
275	C₂²×C₄	—	29	1	no	—
276	C₂²×C₄	—	58	1	no	—
277	C₂²×C₄	—	58	1	no	—
278	C₂²×C₄	—	58	1	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	2	2	no	—
281	Q₁₆	—	58	2	no	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	58	2	no	—
285	SD₁₆	—	58	2	no	—
286	SD₁₆	—	58	2	no	—
287	SD₁₆	—	58	2	no	—
288	SD₁₆	—	58	2	no	—
289	SD₁₆	—	58	2	no	—
290	SD₁₆	—	58	2	no	—
291	SD₁₆	—	58	2	no	—
292	D₈	—	58	2	no	—
293	D₈	—	58	2	no	—
294	D₈	—	58	2	no	—
295	M₁₆	—	29	4	no	—
296	C₂×C₈	—	29	1	no	—
297	C₂×C₈	—	29	1	no	—
298	C₂×C₈	—	29	4	no	—
299	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
300	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	29	2	no	—
303	C₄⋊C₄	—	58	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	29	2	no	—
306	C₂²⋊C₄	—	58	2	no	—
307	C₂²⋊C₄	—	58	2	no	—
308	C₂²⋊C₄	—	58	2	no	—
309	C₄²	—	29	1	no	—
310	C₄²	C₂²×D₂₉	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	29	1	no	—
313	C₂³	—	58	1	no	—
314	C₂³	—	58	1	no	—
315	C₂³	—	58	1	no	—
316	C₂³	—	116	2	no	—
317	C₂³	—	232	2	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	58	1	no	—
325	Q₈	—	58	1	no	—
326	Q₈	—	116	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	58	1	no	—
330	D₄	—	58	1	no	—
331	D₄	—	58	1	no	—
332	D₄	—	58	1	no	—
333	D₄	—	58	1	no	—
334	D₄	—	58	1	no	—
335	D₄	—	116	2	no	—
336	D₄	—	116	2	no	—
337	D₄	—	116	2	no	—
338	D₄	—	232	2	no	—
339	D₄	—	232	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	29	1	no	—
344	C₂×C₄	—	29	2	no	—
345	C₂×C₄	—	29	2	no	—
346	C₂×C₄	—	29	2	no	—
347	C₂×C₄	—	58	1	no	—
348	C₂×C₄	—	116	2	no	—
349	C₂×C₄	—	116	2	no	—
350	C₂×C₄	—	116	2	no	—
351	C₂×C₄	D₄×D₂₉	1	2	yes	—
352	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
353	C₈	—	29	2	no	—
354	C₈	—	29	2	no	—
355	C₈	D₄×D₂₉	1	2	yes	—
356	C₈	D₄×D₂₉	1	2	yes	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	29	2	no	—
360	C₂²	—	58	2	no	—
361	C₂²	—	116	1	no	—
362	C₂²	—	116	1	no	—
363	C₂²	—	116	1	no	—
364	C₂²	—	116	2	no	—
365	C₂²	—	116	2	no	—
366	C₂²	—	116	2	no	—
367	C₂²	—	232	4	no	—
368	C₂²	C₂×D₄×D₂₉	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	29	2	no	—
374	C₄	—	58	2	no	—
375	C₄	—	116	1	no	—
376	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
377	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	58	2	no	—
380	C₂	—	232	1	no	—
381	C₂	—	232	1	no	—
382	C₂	—	232	1	no	—
383	C₂	#127×D₂₉	1	1	yes	characteristic
384	C₂	#14271972	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
hypercenter	#497	D₂₉
hypocenter	C₂₉	#3154
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×#11745276	C₂	semidirect
3. C₂×#11745281	C₂	semidirect
4. C₂×SD₁₆×D₂₉	C₂	semidirect
5. #14271738	C₂	semidirect
6. #14271731	C₂	semidirect
7. #14271556	C₂	semidirect
8. #14271552	C₂	semidirect
9. #14271549	C₂	semidirect
10. #14271548	C₂	semidirect
11. #14271545	C₂	semidirect
12. #14271537	C₂	semidirect
13. #14271534	C₂	semidirect
14. #14271120	C₂	semidirect
15. C₂₉×#497	C₂	semidirect
20. C₂×Q₁₆×C₂₉	C₂²	semidirect
21. C₂×Q₁₆×C₂₉	C₂²	semidirect
24. C₂₉×C₄⋊Q₈	C₂²	semidirect
27. #11745309	C₂²	semidirect
28. #11745306	C₂²	semidirect
29. C₂×#90785	C₂²	semidirect
32. C₂×#90784	C₂²	semidirect
33. C₂×#90784	C₂²	semidirect
34. C₂×#90784	C₂²	semidirect
43. C₂×D₂₃₂	C₂²	semidirect
45. C₂×#90773	C₂²	semidirect
52. #11745237	C₂²	semidirect
55. C₂₉×#97	C₂²	semidirect
56. #11745163	C₂²	semidirect
57. C₈×Dic₂₉	C₂²	semidirect
58. C₄×#6597	C₂²	semidirect
116. #90772	D₄	semidirect
117. C₈×D₂₉	D₄	semidirect
193. #497	D₂₉	semidirect
216. C₂×Q₁₆	C₂×D₂₉	semidirect
217. C₂×Q₁₆	C₂×D₂₉	semidirect
235. C₄⋊Q₈	C₂×D₂₉	semidirect
248. #97	C₂×D₂₉	semidirect
251. C₂₉	#3154	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×#11745276	2
C₂×#11745281	2
C₂×SD₁₆×D₂₉	2
#14271738	2
#14271731	2
#14271556	2
#14271552	2
#14271549	2
#14271548	2
#14271545	2
#14271537	2
#14271534	2
#14271120	2
C₂₉×#497	2
#3154	29