Subgroups — C₂₉×#33871 #20328771

Order 7424.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₉×#3287	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#2234	C₂	1	2	yes	maximal, complemented
5	C₂₉×#2225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#2217	C₂	1	2	yes	maximal, complemented
7	C₂₉×#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₉×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#2138	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₉×#135	C₂²	1	2	yes	—
13	C₂₉×#556	—	2	1	no	—
14	C₂₉×#550	—	2	1	no	—
15	C₂×C₂₉×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×SD₃₂	C₂²	1	2	yes	—
17	C₂²×C₁₆×C₂₉	C₂²	1	2	yes	—
18	C₂₉×#500	C₂²	1	1	yes	characteristic
19	C₂₉×#493	C₂²	1	1	yes	characteristic
20	C₂₉×#485	—	2	2	no	—
21	C₂₉×#475	—	2	1	no	—
22	C₂₉×#467	C₂²	1	2	yes	—
23	C₂₉×#466	C₂²	1	2	yes	—
24	C₂₉×#460	—	2	1	no	—
25	C₂₉×#449	—	2	2	no	—
26	C₄×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₂₉	C₂²	1	1	yes	characteristic
29	C₂₉×#427	C₂²	1	2	yes	—
30	C₂₉×#367	C₂²	1	2	yes	—
31	C₂₉×#366	C₂²	1	1	yes	characteristic
32	C₂₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₂₉×#358	C₂²	1	1	yes	characteristic
34	C₂₉×#358	C₂²	1	1	yes	characteristic
35	C₂₉×#358	C₂²	1	2	yes	—
36	C₂₉×#357	C₂²	1	2	yes	—
37	C₂₉×#357	C₂²	1	2	yes	—
38	C₂₉×#357	C₂²	1	2	yes	—
39	C₂₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
41	C₂×#41×C₂₉	—	2	2	no	—
42	C₂₉×#135	—	2	4	no	—
43	C₂₉×#135	—	4	2	no	—
44	C₂×Q₁₆×C₂₉	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₂₉	C₂³	1	1	yes	characteristic
46	C₂×SD₁₆×C₂₉	—	2	2	no	—
47	C₂×D₈×C₂₉	—	2	1	no	—
48	C₂×D₈×C₂₉	C₂³	1	2	yes	—
49	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
50	C₂₉×C₄⋊Q₈	—	2	1	no	—
51	C₂₉×#127	—	2	1	no	—
52	C₂₉×#127	—	4	1	no	—
53	C₂₉×#126	—	4	2	no	—
54	C₂₉×#124	—	4	1	no	—
55	C₂₉×C₂²⋊Q₈	—	2	2	no	—
56	C₂₉×#121	—	2	2	no	—
57	C₂₉×#121	—	4	2	no	—
58	C₄×Q₈×C₂₉	—	2	1	no	—
59	C₄×Q₈×C₂₉	—	2	1	no	—
60	C₄×Q₈×C₂₉	—	4	1	no	—
61	C₄×D₄×C₂₉	—	4	1	no	—
62	C₄×D₄×C₂₉	D₄	1	1	yes	characteristic
63	C₂₉×#117	—	2	2	no	—
64	C₂₉×Q₃₂	—	2	2	no	—
65	C₂₉×SD₃₂	—	4	2	no	—
66	C₂×C₁₆×C₂₉	—	4	2	no	—
67	C₂×C₁₆×C₂₉	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
69	C₂×C₁₆×C₂₉	C₂³	1	1	yes	characteristic
70	C₂₉×#107	—	2	1	no	—
71	C₂₉×#107	C₂³	1	1	yes	characteristic
72	C₂₉×#107	C₂³	1	2	yes	—
73	C₂₉×#107	C₂³	1	1	yes	characteristic
74	C₂₉×#106	—	2	2	no	—
75	C₂₉×C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₂₉×#103	—	2	1	no	—
77	C₂₉×#103	—	2	1	no	—
78	C₂₉×#103	—	2	2	no	—
79	C₂₉×#102	—	2	2	no	—
80	C₂₉×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
82	#41×C₂₉	—	4	4	no	—
83	#41×C₂₉	—	8	2	no	—
84	C₂×Q₈×C₂₉	—	2	1	no	—
85	C₂×Q₈×C₂₉	—	2	1	no	—
86	C₂×Q₈×C₂₉	—	4	1	no	—
87	C₂×D₄×C₂₉	—	2	2	no	—
88	C₂×D₄×C₂₉	—	4	2	no	—
89	C₂×D₄×C₂₉	—	4	2	no	—
90	C₂×D₄×C₂₉	—	4	1	no	—
91	C₂×D₄×C₂₉	—	4	1	no	—
92	C₂×D₄×C₂₉	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₂₉	—	4	2	no	—
94	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
95	Q₁₆×C₂₉	—	2	1	no	—
96	Q₁₆×C₂₉	—	2	1	no	—
97	Q₁₆×C₂₉	—	2	1	no	—
98	Q₁₆×C₂₉	#41	1	2	yes	—
99	SD₁₆×C₂₉	—	4	4	no	—
100	D₈×C₂₉	—	2	2	no	—
101	D₈×C₂₉	—	2	2	no	—
102	D₈×C₂₉	—	8	1	no	—
103	C₂×C₈×C₂₉	—	2	2	no	—
104	C₂×C₈×C₂₉	—	2	2	no	—
105	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₂₉	—	2	1	no	—
109	C₄⋊C₄×C₂₉	—	2	2	no	—
110	C₄⋊C₄×C₂₉	—	2	1	no	—
111	C₄⋊C₄×C₂₉	—	4	1	no	—
112	C₄⋊C₄×C₂₉	—	4	1	no	—
113	C₄⋊C₄×C₂₉	—	4	2	no	—
114	C₄⋊C₄×C₂₉	—	4	1	no	—
115	C₄⋊C₄×C₂₉	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₂₉	—	4	2	no	—
117	C₂²⋊C₄×C₂₉	—	4	2	no	—
118	C₂²⋊C₄×C₂₉	—	4	2	no	—
119	C₂²⋊C₄×C₂₉	D₈	1	2	yes	—
120	C₄²×C₂₉	—	4	2	no	—
121	C₄²×C₂₉	—	4	1	no	—
122	C₄²×C₂₉	—	4	1	no	—
123	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₂₉	—	2	1	no	—
125	C₁₆×C₂₉	—	2	1	no	—
126	C₁₆×C₂₉	—	2	2	no	—
127	#33871	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₂₉	—	4	2	no	—
129	C₂³×C₂₉	C₂×D₈	1	2	yes	—
130	Q₈×C₂₉	—	2	2	no	—
131	Q₈×C₂₉	—	4	1	no	—
132	Q₈×C₂₉	—	4	1	no	—
133	Q₈×C₂₉	—	4	1	no	—
134	Q₈×C₂₉	—	8	2	no	—
135	D₄×C₂₉	—	2	2	no	—
136	D₄×C₂₉	—	4	2	no	—
137	D₄×C₂₉	—	4	2	no	—
138	D₄×C₂₉	—	8	4	no	—
139	D₄×C₂₉	—	8	2	no	—
140	D₄×C₂₉	—	16	1	no	—
141	D₄×C₂₉	—	16	2	no	—
142	C₂×C₄×C₂₉	—	2	2	no	—
143	C₂×C₄×C₂₉	—	2	2	no	—
144	C₂×C₄×C₂₉	—	4	1	no	—
145	C₂×C₄×C₂₉	—	4	1	no	—
146	C₂×C₄×C₂₉	—	4	2	no	—
147	C₂×C₄×C₂₉	—	4	1	no	—
148	C₂×C₄×C₂₉	—	4	1	no	—
149	C₂×C₄×C₂₉	—	8	4	no	—
150	C₂×C₄×C₂₉	C₂×D₈	1	2	yes	—
151	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
154	C₈×C₂₉	—	2	2	no	—
155	C₈×C₂₉	—	2	1	no	—
156	C₈×C₂₉	C₂×#41	1	1	yes	characteristic
157	C₈×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
158	#3342	C₂×C₂₉	1	1	yes	characteristic, complemented
159	#3287	C₂×C₂₉	1	1	yes	characteristic
160	#2240	C₂×C₂₉	1	1	yes	characteristic, complemented
161	#2234	C₂×C₂₉	1	2	yes	complemented
162	#2225	C₂×C₂₉	1	1	yes	characteristic, complemented
163	#2217	C₂×C₂₉	1	2	yes	complemented
164	#2207	C₂×C₂₉	1	1	yes	characteristic, complemented
165	#2189	C₂×C₂₉	1	2	yes	complemented
166	C₄×Q₃₂	C₂×C₂₉	1	1	yes	characteristic, complemented
167	D₄×C₁₆	C₂×C₂₉	1	1	yes	characteristic, complemented
168	#2138	C₂×C₂₉	1	2	yes	complemented
169	C₂²×C₂₉	—	2	2	no	—
170	C₂²×C₂₉	—	2	2	no	—
171	C₂²×C₂₉	—	2	2	no	—
172	C₂²×C₂₉	—	8	2	no	—
173	C₂²×C₂₉	—	8	2	no	—
174	C₂²×C₂₉	—	8	2	no	—
175	C₂²×C₂₉	C₂²×D₈	1	1	yes	characteristic
176	C₄×C₂₉	—	2	1	no	—
177	C₄×C₂₉	—	2	2	no	—
178	C₄×C₂₉	—	4	2	no	—
179	C₄×C₂₉	—	8	2	no	—
180	C₄×C₂₉	—	8	1	no	—
181	C₄×C₂₉	—	8	1	no	—
182	C₄×C₂₉	—	8	1	no	—
183	C₄×C₂₉	#508	1	2	yes	—
184	C₄×C₂₉	#546	1	1	yes	characteristic
185	C₄×C₂₉	#577	1	1	yes	characteristic
186	C₂×#135	C₂²×C₂₉	1	2	yes	—
187	#556	—	2	1	no	—
188	#550	—	2	1	no	—
189	C₂×Q₃₂	C₂²×C₂₉	1	1	yes	characteristic
190	C₂×SD₃₂	C₂²×C₂₉	1	2	yes	—
191	C₂²×C₁₆	C₂²×C₂₉	1	2	yes	—
192	#500	C₂²×C₂₉	1	1	yes	characteristic
193	#493	C₂²×C₂₉	1	1	yes	characteristic
194	#485	—	2	2	no	—
195	#475	—	2	1	no	—
196	#467	C₂²×C₂₉	1	2	yes	—
197	#466	C₂²×C₂₉	1	2	yes	—
198	#460	—	2	1	no	—
199	#449	—	2	2	no	—
200	C₄×Q₁₆	C₂²×C₂₉	1	1	yes	characteristic
201	C₄×Q₁₆	C₂²×C₂₉	1	1	yes	characteristic
202	C₈×D₄	C₂²×C₂₉	1	1	yes	characteristic
203	#427	C₂²×C₂₉	1	2	yes	—
204	#367	C₂²×C₂₉	1	2	yes	—
205	#366	C₂²×C₂₉	1	1	yes	characteristic
206	C₄⋊C₁₆	C₂²×C₂₉	1	1	yes	characteristic
207	#358	C₂²×C₂₉	1	1	yes	characteristic
208	#358	C₂²×C₂₉	1	2	yes	—
209	#358	C₂²×C₂₉	1	1	yes	characteristic
210	#357	C₂²×C₂₉	1	2	yes	—
211	#357	C₂²×C₂₉	1	2	yes	—
212	#357	C₂²×C₂₉	1	2	yes	—
213	C₂²⋊C₁₆	C₂²×C₂₉	1	2	yes	—
214	C₄×C₁₆	C₂²×C₂₉	1	1	yes	characteristic
215	C₂×C₂₉	—	4	2	no	—
216	C₂×C₂₉	—	16	2	no	—
217	C₂×C₂₉	#3294	1	1	yes	characteristic
218	C₂×C₂₉	C₂×#508	1	1	yes	characteristic
219	C₂×C₂₉	#3416	1	1	yes	characteristic
220	C₂×#41	—	2	2	no	—
221	#135	—	2	4	no	—
222	#135	—	4	2	no	—
223	C₂×Q₁₆	C₂³×C₂₉	1	1	yes	characteristic
224	C₂×Q₁₆	C₂³×C₂₉	1	1	yes	characteristic
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	—	2	1	no	—
227	C₂×D₈	C₂³×C₂₉	1	2	yes	—
228	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
229	C₄⋊Q₈	—	2	1	no	—
230	#127	—	2	1	no	—
231	#127	—	4	1	no	—
232	#126	—	4	2	no	—
233	#124	—	4	1	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	#121	—	2	2	no	—
236	#121	—	4	2	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	4	1	no	—
240	C₄×D₄	—	4	1	no	—
241	C₄×D₄	D₄×C₂₉	1	1	yes	characteristic
242	#117	—	2	2	no	—
243	Q₃₂	—	2	2	no	—
244	SD₃₂	—	4	2	no	—
245	C₂×C₁₆	—	4	2	no	—
246	C₂×C₁₆	C₂³×C₂₉	1	1	yes	characteristic
247	C₂×C₁₆	C₂³×C₂₉	1	1	yes	characteristic
248	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
249	#107	—	2	1	no	—
250	#107	C₂³×C₂₉	1	1	yes	characteristic
251	#107	C₂³×C₂₉	1	2	yes	—
252	#107	C₂³×C₂₉	1	1	yes	characteristic
253	#106	—	2	2	no	—
254	C₄⋊C₈	D₄×C₂₉	1	1	yes	characteristic
255	#103	—	2	1	no	—
256	#103	—	2	1	no	—
257	#103	—	2	2	no	—
258	#102	—	2	2	no	—
259	C₂²⋊C₈	D₄×C₂₉	1	2	yes	—
260	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
261	C₂₉	#33871	1	1	yes	characteristic, direct product factor, Sylow
262	#41	—	4	4	no	—
263	#41	—	8	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	D₈×C₂₉	1	1	yes	characteristic
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	—	2	1	no	—
278	Q₁₆	#41×C₂₉	1	2	yes	—
279	SD₁₆	—	4	4	no	—
280	D₈	—	2	2	no	—
281	D₈	—	2	2	no	—
282	D₈	—	8	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
287	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	D₈×C₂₉	1	1	yes	characteristic
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	D₈×C₂₉	1	2	yes	—
300	C₄²	—	4	2	no	—
301	C₄²	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
304	C₁₆	—	2	1	no	—
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	C₂×D₈×C₂₉	1	2	yes	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	8	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	2	no	—
319	D₄	—	16	1	no	—
320	D₄	—	16	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	8	4	no	—
329	C₂×C₄	C₂×D₈×C₂₉	1	2	yes	—
330	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
331	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	1	no	—
335	C₈	C₂×#41×C₂₉	1	1	yes	characteristic
336	C₈	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	C₂²×D₈×C₂₉	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	C₂₉×#508	1	2	yes	—
352	C₄	C₂₉×#546	1	1	yes	characteristic
353	C₄	C₂₉×#577	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	16	2	no	—
356	C₂	C₂₉×#3294	1	1	yes	characteristic
357	C₂	C₂×C₂₉×#508	1	1	yes	characteristic
358	C₂	C₂₉×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
prime radical	C₂₉×#3287	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄×D₄×C₂₉ ◁ C₈×D₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₉×#3342	C₂	semidirect
3. C₂₉×#2240	C₂	semidirect
4. C₂₉×#2234	C₂	semidirect
5. C₂₉×#2225	C₂	semidirect
6. C₂₉×#2217	C₂	semidirect
7. C₂₉×#2207	C₂	semidirect
8. C₂₉×#2189	C₂	semidirect
9. C₄×C₂₉×Q₃₂	C₂	semidirect
10. D₄×C₁₆×C₂₉	C₂	semidirect
11. C₂₉×#2138	C₂	semidirect
127. #33871	C₂₉	direct (N × H)
158. #3342	C₂×C₂₉	semidirect
160. #2240	C₂×C₂₉	semidirect
161. #2234	C₂×C₂₉	semidirect
162. #2225	C₂×C₂₉	semidirect
163. #2217	C₂×C₂₉	semidirect
164. #2207	C₂×C₂₉	semidirect
165. #2189	C₂×C₂₉	semidirect
166. C₄×Q₃₂	C₂×C₂₉	semidirect
167. D₄×C₁₆	C₂×C₂₉	semidirect
168. #2138	C₂×C₂₉	semidirect
261. C₂₉	#33871	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₉×#3342	2
C₂₉×#3287	2
C₂₉×#2240	2
C₂₉×#2234	2
C₂₉×#2225	2
C₂₉×#2217	2
C₂₉×#2207	2
C₂₉×#2189	2
C₄×C₂₉×Q₃₂	2
D₄×C₁₆×C₂₉	2
C₂₉×#2138	2
#33871	29

Subgroups — C29×#33871 #20328771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#33871 #20328771