Subgroups — C₂₉×#12998 #20307898

Order 7424.

500 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#2143	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₉×#349	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#1389	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₉×#135	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₉	—	2	1	no	—
8	C₂²×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
9	C₂₉×#525	—	2	1	no	—
10	C₂×C₂₉×M₃₂	C₂²	1	2	yes	—
11	C₂₉×#470	C₄	1	2	yes	—
12	C₂₉×#469	C₄	1	2	yes	—
13	C₂₉×#468	—	2	2	no	—
14	C₂₉×#430	—	2	4	no	—
15	C₂×C₂₉×#108	—	2	1	no	—
16	C₂×C₂₉×#108	C₂²	1	2	yes	—
17	C₂₉×#427	C₂²	1	1	yes	characteristic
18	C₂₉×#416	—	2	1	no	—
19	C₂₉×#359	—	2	4	no	—
20	C₂₉×#349	—	2	2	no	—
21	C₂×#41×C₂₉	—	2	1	no	—
22	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
23	C₂₉×#135	—	2	2	no	—
24	C₂₉×#135	—	2	2	no	—
25	C₂×Q₁₆×C₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂×SD₁₆×C₂₉	—	2	1	no	—
27	C₂×D₈×C₂₉	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₂₉	—	2	2	no	—
29	C₂×M₁₆×C₂₉	—	2	2	no	—
30	C₂×M₁₆×C₂₉	—	2	4	no	—
31	C₂×M₁₆×C₂₉	—	2	2	no	—
32	C₂×M₁₆×C₂₉	—	4	1	no	—
33	C₂×M₁₆×C₂₉	—	4	2	no	—
34	C₂×M₁₆×C₂₉	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₂₉	D₄	1	2	yes	—
36	C₂²×C₈×C₂₉	—	4	1	no	—
37	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
39	C₂₉×#123	—	4	1	no	—
40	C₂₉×C₂²⋊Q₈	—	2	2	no	—
41	C₂₉×#121	—	2	2	no	—
42	C₂₉×#120	—	4	1	no	—
43	C₄×D₄×C₂₉	—	4	2	no	—
44	C₂₉×#117	—	2	1	no	—
45	C₂₉×M₃₂	—	2	2	no	—
46	C₂₉×M₃₂	—	2	2	no	—
47	C₂×C₁₆×C₂₉	—	2	2	no	—
48	C₂₉×#108	—	2	4	no	—
49	C₂₉×#108	—	2	2	no	—
50	C₂₉×#108	D₄	1	4	yes	complemented
51	C₂₉×#107	—	2	1	no	—
52	C₂₉×#106	C₂×C₄	1	2	yes	—
53	C₂₉×#103	—	2	2	no	—
54	C₂₉×#102	—	2	2	no	—
55	C₂⁴×C₂₉	SD₁₆	1	1	yes	characteristic
56	#41×C₂₉	—	4	1	no	—
57	#41×C₂₉	—	4	1	no	—
58	#41×C₂₉	—	4	1	no	—
59	#41×C₂₉	—	4	1	no	—
60	C₂×Q₈×C₂₉	—	2	1	no	—
61	C₂×D₄×C₂₉	—	2	1	no	—
62	C₂×D₄×C₂₉	—	4	2	no	—
63	C₂×D₄×C₂₉	—	4	1	no	—
64	C₂²×C₄×C₂₉	—	2	1	no	—
65	C₂²×C₄×C₂₉	—	2	1	no	—
66	C₂²×C₄×C₂₉	—	2	1	no	—
67	C₂²×C₄×C₂₉	—	2	1	no	—
68	C₂²×C₄×C₂₉	—	4	1	no	—
69	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₂₉	D₈	1	2	yes	—
71	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₂₉	—	2	2	no	—
74	SD₁₆×C₂₉	—	4	2	no	—
75	D₈×C₂₉	—	2	2	no	—
76	M₁₆×C₂₉	—	2	4	no	—
77	M₁₆×C₂₉	—	2	4	no	—
78	M₁₆×C₂₉	—	2	4	no	—
79	M₁₆×C₂₉	—	4	4	no	—
80	M₁₆×C₂₉	—	4	2	no	—
81	C₂×C₈×C₂₉	—	2	1	no	—
82	C₂×C₈×C₂₉	—	2	1	no	—
83	C₂×C₈×C₂₉	—	4	4	no	—
84	C₂×C₈×C₂₉	—	4	2	no	—
85	C₂×C₈×C₂₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₂₉	#41	1	2	yes	—
90	C₄⋊C₄×C₂₉	—	2	2	no	—
91	C₄⋊C₄×C₂₉	—	4	2	no	—
92	C₂²⋊C₄×C₂₉	—	4	2	no	—
93	C₂²⋊C₄×C₂₉	—	4	1	no	—
94	C₂²⋊C₄×C₂₉	—	4	2	no	—
95	C₄²×C₂₉	—	4	1	no	—
96	C₁₆×C₂₉	—	4	2	no	—
97	#12998	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₂₉	—	2	1	no	—
99	C₂³×C₂₉	—	2	1	no	—
100	C₂³×C₂₉	—	4	1	no	—
101	C₂³×C₂₉	—	8	1	no	—
102	C₂³×C₂₉	#102	1	2	yes	—
103	C₂³×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
104	Q₈×C₂₉	—	4	1	no	—
105	Q₈×C₂₉	—	4	1	no	—
106	D₄×C₂₉	—	4	1	no	—
107	D₄×C₂₉	—	4	1	no	—
108	D₄×C₂₉	—	8	4	no	—
109	D₄×C₂₉	—	8	1	no	—
110	D₄×C₂₉	—	8	1	no	—
111	C₂×C₄×C₂₉	—	2	1	no	—
112	C₂×C₄×C₂₉	—	2	1	no	—
113	C₂×C₄×C₂₉	—	2	2	no	—
114	C₂×C₄×C₂₉	—	2	2	no	—
115	C₂×C₄×C₂₉	—	2	2	no	—
116	C₂×C₄×C₂₉	—	2	2	no	—
117	C₂×C₄×C₂₉	—	4	2	no	—
118	C₂×C₄×C₂₉	—	4	1	no	—
119	C₂×C₄×C₂₉	—	8	1	no	—
120	C₂×C₄×C₂₉	—	8	1	no	—
121	C₂×C₄×C₂₉	#102	1	2	yes	—
122	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
124	C₂×C₄×C₂₉	#123	1	1	yes	characteristic
125	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₂₉	#137	1	1	yes	characteristic
128	C₈×C₂₉	—	2	2	no	—
129	C₈×C₂₉	—	4	4	no	—
130	C₈×C₂₉	C₄×D₄	1	2	yes	—
131	C₈×C₂₉	#121	1	2	yes	—
132	#3054	C₂×C₂₉	1	1	yes	characteristic
133	C₂×#430	C₂×C₂₉	1	1	yes	characteristic, complemented
134	#2143	C₂×C₂₉	1	2	yes	complemented
135	C₂×#349	C₂×C₂₉	1	1	yes	characteristic, complemented
136	#1389	C₂×C₂₉	1	2	yes	complemented
137	C₂²×C₂₉	—	2	2	no	—
138	C₂²×C₂₉	—	4	1	no	—
139	C₂²×C₂₉	—	4	2	no	—
140	C₂²×C₂₉	—	4	2	no	—
141	C₂²×C₂₉	—	8	1	no	—
142	C₂²×C₂₉	—	8	2	no	—
143	C₂²×C₂₉	—	8	1	no	—
144	C₂²×C₂₉	C₂×#102	1	1	yes	characteristic
145	C₂²×C₂₉	#448	1	1	yes	characteristic
146	C₂²×C₂₉	#481	1	1	yes	characteristic
147	C₄×C₂₉	—	2	2	no	—
148	C₄×C₂₉	—	8	2	no	—
149	C₄×C₂₉	—	8	1	no	—
150	C₄×C₂₉	#386	1	1	yes	characteristic
151	C₄×C₂₉	#417	1	1	yes	characteristic
152	C₄×C₂₉	#447	1	1	yes	characteristic
153	C₄×C₂₉	#480	1	1	yes	characteristic
154	C₂×#135	C₂²×C₂₉	1	1	yes	characteristic
155	C₂²×M₁₆	—	2	1	no	—
156	C₂²×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
157	#525	—	2	1	no	—
158	C₂×M₃₂	C₂²×C₂₉	1	2	yes	—
159	#470	C₄×C₂₉	1	2	yes	—
160	#469	C₄×C₂₉	1	2	yes	—
161	#468	—	2	2	no	—
162	#430	—	2	4	no	—
163	C₂×#108	—	2	1	no	—
164	C₂×#108	C₂²×C₂₉	1	2	yes	—
165	#427	C₂²×C₂₉	1	1	yes	characteristic
166	#416	—	2	1	no	—
167	#359	—	2	4	no	—
168	#349	—	2	2	no	—
169	C₂×C₂₉	—	2	1	no	—
170	C₂×C₂₉	—	2	1	no	—
171	C₂×C₂₉	—	2	1	no	—
172	C₂×C₂₉	—	4	2	no	—
173	C₂×C₂₉	—	16	1	no	—
174	C₂×C₂₉	#1874	1	1	yes	characteristic
175	C₂×#41	—	2	1	no	—
176	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
177	#135	—	2	2	no	—
178	#135	—	2	2	no	—
179	C₂×Q₁₆	C₂×C₄×C₂₉	1	1	yes	characteristic
180	C₂×SD₁₆	—	2	1	no	—
181	C₂×D₈	C₂×C₄×C₂₉	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	4	1	no	—
187	C₂×M₁₆	—	4	2	no	—
188	C₂×M₁₆	C₂×C₄×C₂₉	1	2	yes	—
189	C₂×M₁₆	D₄×C₂₉	1	2	yes	—
190	C₂²×C₈	—	4	1	no	—
191	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
193	#123	—	4	1	no	—
194	C₂²⋊Q₈	—	2	2	no	—
195	#121	—	2	2	no	—
196	#120	—	4	1	no	—
197	C₄×D₄	—	4	2	no	—
198	#117	—	2	1	no	—
199	M₃₂	—	2	2	no	—
200	M₃₂	—	2	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	#108	—	2	4	no	—
203	#108	—	2	2	no	—
204	#108	D₄×C₂₉	1	4	yes	complemented
205	#107	—	2	1	no	—
206	#106	C₂×C₄×C₂₉	1	2	yes	—
207	#103	—	2	2	no	—
208	#102	—	2	2	no	—
209	C₂₉	#12998	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	SD₁₆×C₂₉	1	1	yes	characteristic
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	#41	—	4	1	no	—
214	#41	—	4	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
225	C₂²×C₄	D₈×C₂₉	1	2	yes	—
226	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
228	Q₁₆	—	2	2	no	—
229	SD₁₆	—	4	2	no	—
230	D₈	—	2	2	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	4	no	—
235	M₁₆	—	4	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	4	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	C₂²⋊C₄×C₂₉	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
244	C₂×C₈	#41×C₂₉	1	2	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	—	4	1	no	—
251	C₁₆	—	4	2	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	4	1	no	—
255	C₂³	—	8	1	no	—
256	C₂³	C₂₉×#102	1	2	yes	—
257	C₂³	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
258	Q₈	—	4	1	no	—
259	Q₈	—	4	1	no	—
260	D₄	—	4	1	no	—
261	D₄	—	4	1	no	—
262	D₄	—	8	4	no	—
263	D₄	—	8	1	no	—
264	D₄	—	8	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	C₂₉×#102	1	2	yes	—
276	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
277	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
278	C₂×C₄	C₂₉×#123	1	1	yes	characteristic
279	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
280	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₂×C₄	C₂₉×#137	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	4	4	no	—
284	C₈	C₄×D₄×C₂₉	1	2	yes	—
285	C₈	C₂₉×#121	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	1	no	—
293	C₂²	C₂×C₂₉×#102	1	1	yes	characteristic
294	C₂²	C₂₉×#448	1	1	yes	characteristic
295	C₂²	C₂₉×#481	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	1	no	—
299	C₄	C₂₉×#386	1	1	yes	characteristic
300	C₄	C₂₉×#417	1	1	yes	characteristic
301	C₄	C₂₉×#447	1	1	yes	characteristic
302	C₄	C₂₉×#480	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	4	2	no	—
307	C₂	—	16	1	no	—
308	C₂	C₂₉×#1874	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₉	#447
commutator	C₂×C₈	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₈	C₂³×C₂₉
socle	C₂×C₂₉	#1874
prime radical	C₂₉×#3054	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂²×M₁₆×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₉×#430	C₂	semidirect
3. C₂₉×#2143	C₂	semidirect
4. C₂×C₂₉×#349	C₂	semidirect
5. C₂₉×#1389	C₂	semidirect
50. C₂₉×#108	D₄	semidirect
97. #12998	C₂₉	direct (N × H)
133. C₂×#430	C₂×C₂₉	semidirect
134. #2143	C₂×C₂₉	semidirect
135. C₂×#349	C₂×C₂₉	semidirect
136. #1389	C₂×C₂₉	semidirect
204. #108	D₄×C₂₉	semidirect
209. C₂₉	#12998	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×#3054	2
C₂×C₂₉×#430	2
C₂₉×#2143	2
C₂×C₂₉×#349	2
C₂₉×#1389	2
#12998	29

Subgroups — C29×#12998 #20307898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#12998 #20307898