Subgroups — C₂₉×#19954 #20314854

Order 7424.

618 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3354	C₂	1	2	yes	maximal, complemented
2	C₂₉×#3303	C₂	1	2	yes	maximal, complemented
3	C₂₉×#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#1691	C₂	1	2	yes	maximal, complemented
5	C₂₉×#1683	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1647	C₂	1	2	yes	maximal, complemented
7	C₂₉×#1634	C₂	1	2	yes	maximal, complemented
8	C₂₉×#1589	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₉×#555	C₂²	1	2	yes	—
11	C₂₉×#547	C₂²	1	2	yes	—
12	C₂₉×#500	C₂²	1	1	yes	characteristic
13	C₂₉×#498	C₂²	1	2	yes	—
14	C₂₉×#494	C₂²	1	1	yes	characteristic
15	C₂₉×#487	C₂²	1	2	yes	—
16	C₂₉×#485	—	2	2	no	—
17	C₂₉×#482	—	2	2	no	—
18	C₂₉×#481	—	2	2	no	—
19	C₂₉×#476	C₂²	1	2	yes	—
20	C₂₉×#476	C₂²	1	2	yes	—
21	C₂₉×#474	C₂²	1	2	yes	—
22	C₂₉×#469	—	2	2	no	—
23	C₂₉×#465	—	2	2	no	—
24	C₂₉×#461	C₂²	1	2	yes	—
25	C₂₉×#449	—	2	2	no	—
26	C₄×D₈×C₂₉	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₂₉	C₂²	1	1	yes	characteristic
28	C₂₉×#436	C₂²	1	2	yes	—
29	C₈×D₄×C₂₉	C₂²	1	2	yes	—
30	C₂₉×#428	—	2	2	no	—
31	C₂₉×#416	—	2	2	no	—
32	C₂₉×#335	C₂²	1	1	yes	characteristic
33	C₂₉×#335	C₂²	1	1	yes	characteristic
34	C₂₉×#333	C₂²	1	2	yes	—
35	C₂₉×#332	C₂²	1	2	yes	—
36	C₂₉×D₄⋊C₈	C₂²	1	2	yes	—
37	C₂₉×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₂₉	C₂²	1	1	yes	characteristic
39	C₂×#41×C₂₉	—	2	2	no	—
40	C₂×Q₁₆×C₂₉	—	4	1	no	—
41	C₂×SD₁₆×C₂₉	—	2	2	no	—
42	C₂×D₈×C₂₉	D₄	1	1	yes	characteristic
43	C₂×D₈×C₂₉	D₄	1	1	yes	characteristic
44	C₂×M₁₆×C₂₉	—	2	2	no	—
45	C₂²×C₈×C₂₉	—	2	2	no	—
46	C₂₉×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₉×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₉×#124	—	4	2	no	—
49	C₂₉×#123	—	4	2	no	—
50	C₂₉×C₂²⋊Q₈	—	2	2	no	—
51	C₂₉×#121	—	2	2	no	—
52	C₂₉×#121	—	2	2	no	—
53	C₂₉×#121	—	2	2	no	—
54	C₄×D₄×C₂₉	C₂³	1	2	yes	—
55	C₄×D₄×C₂₉	C₂³	1	2	yes	—
56	C₂₉×#117	—	2	2	no	—
57	C₂₉×#117	—	2	2	no	—
58	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
59	C₂₉×#107	—	2	2	no	—
60	C₂₉×#107	—	4	1	no	—
61	C₂₉×#107	D₄	1	1	yes	characteristic
62	C₂₉×#107	D₄	1	1	yes	characteristic
63	C₂₉×#106	—	2	2	no	—
64	C₂₉×#106	—	2	2	no	—
65	C₂₉×#106	—	2	2	no	—
66	C₂₉×#106	—	2	2	no	—
67	C₂₉×C₄⋊C₈	C₂³	1	2	yes	—
68	C₂₉×C₄⋊C₈	C₂³	1	2	yes	—
69	C₂₉×#103	—	2	2	no	—
70	C₂₉×#103	—	2	2	no	—
71	C₂₉×#103	—	2	2	no	—
72	C₂₉×#102	—	2	2	no	—
73	C₂₉×#102	—	2	2	no	—
74	C₂₉×#102	—	2	2	no	—
75	C₂₉×#102	D₄	1	2	yes	—
76	C₂₉×#102	D₄	1	2	yes	—
77	C₂₉×C₂²⋊C₈	—	2	2	no	—
78	C₂₉×C₂²⋊C₈	—	2	2	no	—
79	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
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	2	no	—
85	C₂×Q₈×C₂₉	—	2	2	no	—
86	C₂×D₄×C₂₉	—	4	2	no	—
87	C₂×D₄×C₂₉	—	4	2	no	—
88	C₂×D₄×C₂₉	C₂×D₄	1	2	yes	—
89	C₂×D₄×C₂₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₂₉	—	2	2	no	—
91	C₂²×C₄×C₂₉	—	2	2	no	—
92	C₂²×C₄×C₂₉	—	4	2	no	—
93	Q₁₆×C₂₉	—	8	2	no	—
94	SD₁₆×C₂₉	—	8	2	no	—
95	D₈×C₂₉	—	2	2	no	—
96	D₈×C₂₉	—	4	1	no	—
97	M₁₆×C₂₉	—	4	4	no	—
98	C₂×C₈×C₂₉	—	2	1	no	—
99	C₂×C₈×C₂₉	—	2	2	no	—
100	C₂×C₈×C₂₉	—	2	2	no	—
101	C₂×C₈×C₂₉	—	8	2	no	—
102	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₂₉	—	2	2	no	—
107	C₄⋊C₄×C₂₉	—	2	2	no	—
108	C₄⋊C₄×C₂₉	—	2	2	no	—
109	C₄⋊C₄×C₂₉	—	2	2	no	—
110	C₄⋊C₄×C₂₉	—	4	2	no	—
111	C₄⋊C₄×C₂₉	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₂₉	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₂₉	—	2	2	no	—
114	C₂²⋊C₄×C₂₉	—	2	2	no	—
115	C₂²⋊C₄×C₂₉	—	4	2	no	—
116	C₂²⋊C₄×C₂₉	—	4	2	no	—
117	C₂²⋊C₄×C₂₉	—	4	2	no	—
118	C₄²×C₂₉	—	4	2	no	—
119	C₄²×C₂₉	C₂⁴	1	1	yes	characteristic
120	#19954	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₂₉	—	2	2	no	—
122	C₂³×C₂₉	—	2	2	no	—
123	Q₈×C₂₉	—	4	2	no	—
124	Q₈×C₂₉	—	4	2	no	—
125	Q₈×C₂₉	—	8	2	no	—
126	D₄×C₂₉	—	2	2	no	—
127	D₄×C₂₉	—	2	2	no	—
128	D₄×C₂₉	—	2	2	no	—
129	D₄×C₂₉	—	2	2	no	—
130	D₄×C₂₉	—	8	4	no	—
131	D₄×C₂₉	—	16	2	no	—
132	C₂×C₄×C₂₉	—	2	2	no	—
133	C₂×C₄×C₂₉	—	2	2	no	—
134	C₂×C₄×C₂₉	—	4	2	no	—
135	C₂×C₄×C₂₉	—	4	2	no	—
136	C₂×C₄×C₂₉	—	4	2	no	—
137	C₂×C₄×C₂₉	—	4	2	no	—
138	C₂×C₄×C₂₉	—	4	2	no	—
139	C₂×C₄×C₂₉	—	4	2	no	—
140	C₂×C₄×C₂₉	—	8	4	no	—
141	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₂₉	—	2	2	no	—
145	C₈×C₂₉	—	2	1	no	—
146	C₈×C₂₉	—	2	1	no	—
147	C₈×C₂₉	—	2	1	no	—
148	C₈×C₂₉	—	4	2	no	—
149	C₈×C₂₉	—	4	2	no	—
150	C₈×C₂₉	#135	1	2	yes	—
151	#3354	C₂×C₂₉	1	2	yes	complemented
152	#3303	C₂×C₂₉	1	2	yes	complemented
153	#1707	C₂×C₂₉	1	1	yes	characteristic, complemented
154	#1691	C₂×C₂₉	1	2	yes	complemented
155	#1683	C₂×C₂₉	1	2	yes	complemented
156	#1647	C₂×C₂₉	1	2	yes	complemented
157	#1634	C₂×C₂₉	1	2	yes	complemented
158	#1589	C₂×C₂₉	1	1	yes	characteristic, complemented
159	C₈×D₈	C₂×C₂₉	1	1	yes	characteristic, complemented
160	C₂²×C₂₉	—	4	2	no	—
161	C₂²×C₂₉	—	4	2	no	—
162	C₂²×C₂₉	—	4	2	no	—
163	C₂²×C₂₉	—	4	2	no	—
164	C₂²×C₂₉	—	4	2	no	—
165	C₂²×C₂₉	—	4	2	no	—
166	C₂²×C₂₉	D₄²	1	1	yes	characteristic
167	C₄×C₂₉	—	2	1	no	—
168	C₄×C₂₉	—	4	2	no	—
169	C₄×C₂₉	—	4	2	no	—
170	C₄×C₂₉	—	8	2	no	—
171	C₄×C₂₉	—	8	2	no	—
172	C₄×C₂₉	C₂×#135	1	1	yes	characteristic
173	C₄×C₂₉	#575	1	1	yes	characteristic
174	C₄×C₂₉	#576	1	1	yes	characteristic
175	C₄×C₂₉	#578	1	1	yes	characteristic
176	#555	C₂²×C₂₉	1	2	yes	—
177	#547	C₂²×C₂₉	1	2	yes	—
178	#500	C₂²×C₂₉	1	1	yes	characteristic
179	#498	C₂²×C₂₉	1	2	yes	—
180	#494	C₂²×C₂₉	1	1	yes	characteristic
181	#487	C₂²×C₂₉	1	2	yes	—
182	#485	—	2	2	no	—
183	#482	—	2	2	no	—
184	#481	—	2	2	no	—
185	#476	C₂²×C₂₉	1	2	yes	—
186	#476	C₂²×C₂₉	1	2	yes	—
187	#474	C₂²×C₂₉	1	2	yes	—
188	#469	—	2	2	no	—
189	#465	—	2	2	no	—
190	#461	C₂²×C₂₉	1	2	yes	—
191	#449	—	2	2	no	—
192	C₄×D₈	C₂²×C₂₉	1	1	yes	characteristic
193	C₄×D₈	C₂²×C₂₉	1	1	yes	characteristic
194	#436	C₂²×C₂₉	1	2	yes	—
195	C₈×D₄	C₂²×C₂₉	1	2	yes	—
196	#428	—	2	2	no	—
197	#416	—	2	2	no	—
198	#335	C₂²×C₂₉	1	1	yes	characteristic
199	#335	C₂²×C₂₉	1	1	yes	characteristic
200	#333	C₂²×C₂₉	1	2	yes	—
201	#332	C₂²×C₂₉	1	2	yes	—
202	D₄⋊C₈	C₂²×C₂₉	1	2	yes	—
203	D₄⋊C₈	C₂²×C₂₉	1	2	yes	—
204	C₈²	C₂²×C₂₉	1	1	yes	characteristic
205	C₂×C₂₉	—	8	2	no	—
206	C₂×C₂₉	—	8	2	no	—
207	C₂×C₂₉	#3273	1	1	yes	characteristic
208	C₂×C₂₉	#3283	1	1	yes	characteristic
209	C₂×C₂₉	#3291	1	1	yes	characteristic
210	C₂×#41	—	2	2	no	—
211	C₂×Q₁₆	—	4	1	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	D₄×C₂₉	1	1	yes	characteristic
214	C₂×D₈	D₄×C₂₉	1	1	yes	characteristic
215	C₂×M₁₆	—	2	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₄⋊Q₈	C₂³×C₂₉	1	2	yes	—
218	C₄⋊Q₈	C₂³×C₂₉	1	2	yes	—
219	#124	—	4	2	no	—
220	#123	—	4	2	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	#121	—	2	2	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	C₄×D₄	C₂³×C₂₉	1	2	yes	—
226	C₄×D₄	C₂³×C₂₉	1	2	yes	—
227	#117	—	2	2	no	—
228	#117	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	#107	—	2	2	no	—
231	#107	—	4	1	no	—
232	#107	D₄×C₂₉	1	1	yes	characteristic
233	#107	D₄×C₂₉	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	#106	—	2	2	no	—
236	#106	—	2	2	no	—
237	#106	—	2	2	no	—
238	C₄⋊C₈	C₂³×C₂₉	1	2	yes	—
239	C₄⋊C₈	C₂³×C₂₉	1	2	yes	—
240	#103	—	2	2	no	—
241	#103	—	2	2	no	—
242	#103	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	D₄×C₂₉	1	2	yes	—
247	#102	D₄×C₂₉	1	2	yes	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
253	C₂₉	#19954	1	1	yes	characteristic, direct product factor, Sylow
254	#41	—	4	4	no	—
255	#41	—	8	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	C₂×D₄×C₂₉	1	2	yes	—
261	C₂×D₄	C₂×D₄×C₂₉	1	2	yes	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	4	2	no	—
265	Q₁₆	—	8	2	no	—
266	SD₁₆	—	8	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	4	1	no	—
269	M₁₆	—	4	4	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	8	2	no	—
274	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	C₂×D₄×C₂₉	1	2	yes	—
284	C₄⋊C₄	C₂×D₄×C₂₉	1	2	yes	—
285	C₂²⋊C₄	—	2	2	no	—
286	C₂²⋊C₄	—	2	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	C₂⁴×C₂₉	1	1	yes	characteristic
292	C₂³	—	2	2	no	—
293	C₂³	—	2	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	2	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	16	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
314	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	4	2	no	—
320	C₈	—	4	2	no	—
321	C₈	C₂₉×#135	1	2	yes	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	D₄²×C₂₉	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	2	no	—
334	C₄	C₂×C₂₉×#135	1	1	yes	characteristic
335	C₄	C₂₉×#575	1	1	yes	characteristic
336	C₄	C₂₉×#576	1	1	yes	characteristic
337	C₄	C₂₉×#578	1	1	yes	characteristic
338	C₂	—	8	2	no	—
339	C₂	—	8	2	no	—
340	C₂	C₂₉×#3273	1	1	yes	characteristic
341	C₂	C₂₉×#3283	1	1	yes	characteristic
342	C₂	C₂₉×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₉×#3354	C₂	semidirect
2. C₂₉×#3303	C₂	semidirect
3. C₂₉×#1707	C₂	semidirect
4. C₂₉×#1691	C₂	semidirect
5. C₂₉×#1683	C₂	semidirect
6. C₂₉×#1647	C₂	semidirect
7. C₂₉×#1634	C₂	semidirect
8. C₂₉×#1589	C₂	semidirect
9. C₈×D₈×C₂₉	C₂	semidirect
120. #19954	C₂₉	direct (N × H)
151. #3354	C₂×C₂₉	semidirect
152. #3303	C₂×C₂₉	semidirect
153. #1707	C₂×C₂₉	semidirect
154. #1691	C₂×C₂₉	semidirect
155. #1683	C₂×C₂₉	semidirect
156. #1647	C₂×C₂₉	semidirect
157. #1634	C₂×C₂₉	semidirect
158. #1589	C₂×C₂₉	semidirect
159. C₈×D₈	C₂×C₂₉	semidirect
253. C₂₉	#19954	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×#3354	2
C₂₉×#3303	2
C₂₉×#1707	2
C₂₉×#1691	2
C₂₉×#1683	2
C₂₉×#1647	2
C₂₉×#1634	2
C₂₉×#1589	2
C₈×D₈×C₂₉	2
#19954	29

Subgroups — C29×#19954 #20314854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#19954 #20314854