Subgroups — C₂₉×#33638 #20328538

Order 7424.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₉×#2957	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2231	C₂	1	4	yes	maximal, complemented
4	C₂₉×#2230	C₂	1	4	yes	maximal, complemented
5	C₂₉×#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₉×#365	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#2145	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₉×#360	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₉×#135	C₂²	1	1	yes	characteristic
11	C₂₉×#568	—	2	1	no	—
12	C₂₉×#544	—	2	1	no	—
13	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
15	C₂₉×#501	—	2	2	no	—
16	C₂₉×#499	C₂²	1	2	yes	complemented
17	C₂₉×#498	C₂²	1	2	yes	complemented
18	C₂₉×#444	—	2	2	no	—
19	C₂₉×#443	C₂²	1	4	yes	—
20	C₂×C₂₉×#108	C₂²	1	1	yes	characteristic
21	C₂₉×#428	—	2	1	no	—
22	C₂₉×#427	C₂²	1	1	yes	characteristic
23	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
24	C₂₉×#421	—	2	1	no	—
25	C₂₉×#405	C₂²	1	1	yes	characteristic
26	C₂₉×#365	C₂²	1	2	yes	complemented
27	C₂₉×#365	C₂²	1	4	yes	complemented
28	C₂₉×#365	C₂²	1	2	yes	complemented
29	C₂₉×#364	C₂²	1	4	yes	complemented
30	C₂₉×#360	C₂²	1	2	yes	—
31	C₂₉×#360	C₂²	1	4	yes	—
32	C₂₉×#360	C₂²	1	2	yes	—
33	C₂×#41×C₂₉	—	2	1	no	—
34	C₂₉×#135	—	2	1	no	—
35	C₂₉×#135	—	2	1	no	—
36	C₂₉×#135	Q₈	1	2	yes	complemented
37	C₂₉×#135	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₂₉	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₂₉	—	2	1	no	—
40	C₂×D₈×C₂₉	Q₈	1	1	yes	characteristic, complemented
41	C₂₉×#131	—	2	4	no	—
42	C₂₉×#131	—	4	2	no	—
43	C₂×M₁₆×C₂₉	—	2	1	no	—
44	C₂×M₁₆×C₂₉	—	4	2	no	—
45	C₂×M₁₆×C₂₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
47	C₂₉×C₄⋊Q₈	—	2	2	no	—
48	C₂₉×C₄⋊Q₈	—	4	1	no	—
49	C₂₉×#125	—	2	2	no	—
50	C₂₉×#125	—	4	1	no	—
51	C₂₉×C₂²⋊Q₈	—	4	2	no	—
52	C₂₉×#117	—	2	1	no	—
53	C₂₉×#117	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
55	C₂₉×M₃₂	—	2	1	no	—
56	C₂₉×M₃₂	—	2	1	no	—
57	C₂₉×M₃₂	C₂³	1	2	yes	—
58	C₂₉×M₃₂	C₂³	1	2	yes	—
59	C₂×C₁₆×C₂₉	—	2	1	no	—
60	C₂×C₁₆×C₂₉	—	2	1	no	—
61	C₂₉×#108	—	2	1	no	—
62	C₂₉×#108	C₂³	1	2	yes	—
63	C₂₉×#107	—	2	1	no	—
64	C₂₉×#107	—	2	2	no	—
65	C₂₉×#106	—	2	1	no	—
66	C₂₉×#106	—	2	2	no	—
67	C₂₉×#106	C₂³	1	2	yes	—
68	C₂₉×#106	C₂³	1	1	yes	characteristic
69	C₂₉×#106	C₂³	1	1	yes	characteristic
70	C₂₉×#104	—	2	4	no	—
71	C₂₉×#97	D₄	1	2	yes	—
72	C₄×C₈×C₂₉	C₂³	1	2	yes	—
73	#41×C₂₉	—	2	2	no	—
74	#41×C₂₉	—	2	2	no	—
75	#41×C₂₉	—	4	1	no	—
76	#41×C₂₉	—	4	1	no	—
77	C₂×Q₈×C₂₉	—	2	1	no	—
78	C₂×Q₈×C₂₉	—	4	2	no	—
79	C₂×D₄×C₂₉	—	2	1	no	—
80	C₂×D₄×C₂₉	—	4	1	no	—
81	C₂²×C₄×C₂₉	—	4	1	no	—
82	C₂²×C₄×C₂₉	—	4	1	no	—
83	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₂₉	—	2	1	no	—
85	Q₁₆×C₂₉	C₂×Q₈	1	2	yes	—
86	SD₁₆×C₂₉	—	2	2	no	—
87	SD₁₆×C₂₉	—	2	2	no	—
88	D₈×C₂₉	—	2	1	no	—
89	D₈×C₂₉	C₂×Q₈	1	2	yes	—
90	M₁₆×C₂₉	—	2	2	no	—
91	M₁₆×C₂₉	—	4	1	no	—
92	M₁₆×C₂₉	—	4	4	no	—
93	M₁₆×C₂₉	SD₁₆	1	2	yes	—
94	M₁₆×C₂₉	SD₁₆	1	2	yes	—
95	C₂×C₈×C₂₉	—	4	1	no	—
96	C₂×C₈×C₂₉	—	4	4	no	—
97	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₂₉	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₂₉	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
103	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₂₉	—	2	1	no	—
105	C₄⋊C₄×C₂₉	—	2	2	no	—
106	C₄⋊C₄×C₂₉	—	2	1	no	—
107	C₄⋊C₄×C₂₉	—	2	1	no	—
108	C₄⋊C₄×C₂₉	—	4	2	no	—
109	C₄⋊C₄×C₂₉	—	4	2	no	—
110	C₄⋊C₄×C₂₉	—	4	2	no	—
111	C₄⋊C₄×C₂₉	—	4	1	no	—
112	C₂²⋊C₄×C₂₉	—	2	1	no	—
113	C₂²⋊C₄×C₂₉	—	4	1	no	—
114	C₄²×C₂₉	—	4	1	no	—
115	C₄²×C₂₉	C₂×D₄	1	2	yes	—
116	C₁₆×C₂₉	—	2	2	no	—
117	C₁₆×C₂₉	—	2	2	no	—
118	#33638	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₂₉	—	4	1	no	—
120	C₂³×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	Q₈×C₂₉	—	2	2	no	—
122	Q₈×C₂₉	—	4	1	no	—
123	Q₈×C₂₉	—	16	2	no	—
124	D₄×C₂₉	—	2	2	no	—
125	D₄×C₂₉	—	4	2	no	—
126	D₄×C₂₉	—	4	2	no	—
127	D₄×C₂₉	—	4	1	no	—
128	C₂×C₄×C₂₉	—	2	2	no	—
129	C₂×C₄×C₂₉	—	4	1	no	—
130	C₂×C₄×C₂₉	—	4	2	no	—
131	C₂×C₄×C₂₉	—	4	2	no	—
132	C₂×C₄×C₂₉	—	4	1	no	—
133	C₂×C₄×C₂₉	—	4	1	no	—
134	C₂×C₄×C₂₉	—	4	2	no	—
135	C₂×C₄×C₂₉	—	16	1	no	—
136	C₂×C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₂₉	—	4	2	no	—
143	C₈×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₂₉	C₂×SD₁₆	1	4	yes	—
146	C₈×C₂₉	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₂₉	C₂×#41	1	1	yes	characteristic
148	#3163	C₂×C₂₉	1	1	yes	characteristic, complemented
149	#2957	C₂×C₂₉	1	1	yes	characteristic
150	#2231	C₂×C₂₉	1	4	yes	complemented
151	#2230	C₂×C₂₉	1	4	yes	complemented
152	#2155	C₂×C₂₉	1	1	yes	characteristic, complemented
153	C₂×#365	C₂×C₂₉	1	1	yes	characteristic, complemented
154	#2153	C₂×C₂₉	1	1	yes	characteristic, complemented
155	#2145	C₂×C₂₉	1	1	yes	characteristic
156	C₂×#360	C₂×C₂₉	1	1	yes	characteristic
157	C₂²×C₂₉	—	4	1	no	—
158	C₂²×C₂₉	—	4	2	no	—
159	C₂²×C₂₉	—	16	1	no	—
160	C₂²×C₂₉	#476	1	1	yes	characteristic
161	C₂²×C₂₉	#476	1	1	yes	characteristic
162	C₂²×C₂₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₄×C₂₉	—	4	2	no	—
164	C₄×C₂₉	—	4	2	no	—
165	C₄×C₂₉	—	8	1	no	—
166	C₄×C₂₉	—	8	2	no	—
167	C₄×C₂₉	—	8	1	no	—
168	C₄×C₂₉	#476	1	1	yes	characteristic
169	C₄×C₂₉	#476	1	1	yes	characteristic
170	C₄×C₂₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
171	C₄×C₂₉	C₂²×SD₁₆	1	1	yes	characteristic
172	C₂×#135	C₂²×C₂₉	1	1	yes	characteristic
173	#568	—	2	1	no	—
174	#544	—	2	1	no	—
175	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
176	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
177	#501	—	2	2	no	—
178	#499	C₂²×C₂₉	1	2	yes	complemented
179	#498	C₂²×C₂₉	1	2	yes	complemented
180	#444	—	2	2	no	—
181	#443	C₂²×C₂₉	1	4	yes	—
182	C₂×#108	C₂²×C₂₉	1	1	yes	characteristic
183	#428	—	2	1	no	—
184	#427	C₂²×C₂₉	1	1	yes	characteristic
185	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
186	#421	—	2	1	no	—
187	#405	C₂²×C₂₉	1	1	yes	characteristic
188	#365	C₂²×C₂₉	1	4	yes	complemented
189	#365	C₂²×C₂₉	1	2	yes	complemented
190	#365	C₂²×C₂₉	1	2	yes	complemented
191	#364	C₂²×C₂₉	1	4	yes	complemented
192	#360	C₂²×C₂₉	1	4	yes	—
193	#360	C₂²×C₂₉	1	2	yes	—
194	#360	C₂²×C₂₉	1	2	yes	—
195	C₂×C₂₉	—	2	1	no	—
196	C₂×C₂₉	—	2	1	no	—
197	C₂×C₂₉	—	2	1	no	—
198	C₂×C₂₉	—	8	2	no	—
199	C₂×C₂₉	C₂×#476	1	1	yes	characteristic
200	C₂×#41	—	2	1	no	—
201	#135	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	Q₈×C₂₉	1	2	yes	complemented
204	#135	C₂³×C₂₉	1	2	yes	—
205	C₂×Q₁₆	Q₈×C₂₉	1	1	yes	characteristic, complemented
206	C₂×SD₁₆	—	2	1	no	—
207	C₂×D₈	Q₈×C₂₉	1	1	yes	characteristic, complemented
208	#131	—	2	4	no	—
209	#131	—	4	2	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	4	2	no	—
212	C₂×M₁₆	D₄×C₂₉	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	—	4	1	no	—
216	#125	—	2	2	no	—
217	#125	—	4	1	no	—
218	C₂²⋊Q₈	—	4	2	no	—
219	#117	—	2	1	no	—
220	#117	D₄×C₂₉	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	C₂³×C₂₉	1	2	yes	—
225	M₃₂	C₂³×C₂₉	1	2	yes	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	#108	—	2	1	no	—
229	#108	C₂³×C₂₉	1	2	yes	—
230	#107	—	2	1	no	—
231	#107	—	2	2	no	—
232	#106	—	2	1	no	—
233	#106	—	2	2	no	—
234	#106	C₂³×C₂₉	1	2	yes	—
235	#106	C₂³×C₂₉	1	1	yes	characteristic
236	#106	C₂³×C₂₉	1	1	yes	characteristic
237	#104	—	2	4	no	—
238	#97	D₄×C₂₉	1	2	yes	—
239	C₄×C₈	C₂³×C₂₉	1	2	yes	—
240	C₂₉	#33638	1	1	yes	characteristic, direct product factor, Sylow
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
252	Q₁₆	—	2	1	no	—
253	Q₁₆	C₂×Q₈×C₂₉	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Q₈×C₂₉	1	2	yes	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	4	no	—
261	M₁₆	SD₁₆×C₂₉	1	2	yes	—
262	M₁₆	SD₁₆×C₂₉	1	2	yes	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
267	C₂×C₈	C₂×Q₈×C₂₉	1	1	yes	characteristic
268	C₂×C₈	C₂×Q₈×C₂₉	1	1	yes	characteristic
269	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
270	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
271	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	C₂×D₄×C₂₉	1	2	yes	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	16	1	no	—
303	C₂×C₄	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
305	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
306	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
307	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
311	C₈	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
312	C₈	C₂×SD₁₆×C₂₉	1	4	yes	—
313	C₈	C₂²×Q₈×C₂₉	1	1	yes	characteristic
314	C₈	C₂×#41×C₂₉	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	C₂₉×#476	1	1	yes	characteristic
319	C₂²	C₂₉×#476	1	1	yes	characteristic
320	C₂²	C₂×C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	C₂₉×#476	1	1	yes	characteristic
327	C₄	C₂₉×#476	1	1	yes	characteristic
328	C₄	C₂×C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
329	C₄	C₂²×SD₁₆×C₂₉	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₂₉×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₉	C₂×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
prime radical	C₂×C₂₉×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₉×#3163	C₂	semidirect
3. C₂₉×#2231	C₂	semidirect
4. C₂₉×#2230	C₂	semidirect
5. C₂₉×#2155	C₂	semidirect
6. C₂×C₂₉×#365	C₂	semidirect
7. C₂₉×#2153	C₂	semidirect
16. C₂₉×#499	C₂²	semidirect
17. C₂₉×#498	C₂²	semidirect
26. C₂₉×#365	C₂²	semidirect
27. C₂₉×#365	C₂²	semidirect
28. C₂₉×#365	C₂²	semidirect
29. C₂₉×#364	C₂²	semidirect
36. C₂₉×#135	Q₈	semidirect
38. C₂×Q₁₆×C₂₉	Q₈	semidirect
40. C₂×D₈×C₂₉	Q₈	semidirect
118. #33638	C₂₉	direct (N × H)
148. #3163	C₂×C₂₉	semidirect
150. #2231	C₂×C₂₉	semidirect
151. #2230	C₂×C₂₉	semidirect
152. #2155	C₂×C₂₉	semidirect
153. C₂×#365	C₂×C₂₉	semidirect
154. #2153	C₂×C₂₉	semidirect
178. #499	C₂²×C₂₉	semidirect
179. #498	C₂²×C₂₉	semidirect
188. #365	C₂²×C₂₉	semidirect
189. #365	C₂²×C₂₉	semidirect
190. #365	C₂²×C₂₉	semidirect
191. #364	C₂²×C₂₉	semidirect
203. #135	Q₈×C₂₉	semidirect
205. C₂×Q₁₆	Q₈×C₂₉	semidirect
207. C₂×D₈	Q₈×C₂₉	semidirect
240. C₂₉	#33638	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×#3163	2
C₂₉×#2957	2
C₂₉×#2231	2
C₂₉×#2230	2
C₂₉×#2155	2
C₂×C₂₉×#365	2
C₂₉×#2153	2
C₂₉×#2145	2
C₂×C₂₉×#360	2
#33638	29

Subgroups — C29×#33638 #20328538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#33638 #20328538