Subgroups — C₂₉×#12823 #20307723

Order 7424.

430 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₉×#360	C₂	1	2	yes	maximal
4	C₂₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#1383	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₉×#135	C₂²	1	1	yes	characteristic
7	C₂×C₂₉×#131	—	2	1	no	—
8	C₂×C₂₉×M₃₂	C₂²	1	2	yes	—
9	C₂₉×#443	—	2	2	no	—
10	C₂₉×#443	C₄	1	4	yes	complemented
11	C₂×C₂₉×#108	C₂²	1	1	yes	characteristic
12	C₂×C₂₉×#106	—	2	1	no	—
13	C₂×C₂₉×#106	C₂²	1	2	yes	—
14	C₂×C₂₉×#104	—	2	1	no	—
15	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
16	C₂₉×#384	—	2	1	no	—
17	C₂₉×#364	—	2	2	no	—
18	C₂₉×#360	—	2	4	no	—
19	C₂×#41×C₂₉	—	2	1	no	—
20	C₂₉×#135	—	2	1	no	—
21	C₂₉×#135	—	2	1	no	—
22	C₂₉×#135	C₂×C₄	1	2	yes	—
23	C₂₉×#135	D₄	1	2	yes	—
24	C₂×Q₁₆×C₂₉	Q₈	1	1	yes	characteristic
25	C₂×SD₁₆×C₂₉	—	2	1	no	—
26	C₂×D₈×C₂₉	Q₈	1	1	yes	characteristic
27	C₂₉×#131	—	2	4	no	—
28	C₂₉×#131	—	4	2	no	—
29	C₂×M₁₆×C₂₉	—	2	1	no	—
30	C₂×M₁₆×C₂₉	—	4	1	no	—
31	C₂²×C₈×C₂₉	—	4	1	no	—
32	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₉	—	4	2	no	—
36	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
37	C₂₉×M₃₂	—	2	2	no	—
38	C₂₉×M₃₂	—	2	2	no	—
39	C₂×C₁₆×C₂₉	—	2	2	no	—
40	C₂₉×#108	—	2	1	no	—
41	C₂₉×#108	C₂×C₄	1	2	yes	—
42	C₂₉×#106	—	2	2	no	—
43	C₂₉×#106	—	2	2	no	—
44	C₂₉×#106	—	4	2	no	—
45	C₂₉×#104	—	2	4	no	—
46	C₄×C₈×C₂₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₂₉	D₄	1	2	yes	—
48	#41×C₂₉	—	2	2	no	—
49	#41×C₂₉	—	2	2	no	—
50	#41×C₂₉	—	4	1	no	—
51	#41×C₂₉	—	4	1	no	—
52	C₂×Q₈×C₂₉	—	2	1	no	—
53	C₂×D₄×C₂₉	—	2	1	no	—
54	C₂×D₄×C₂₉	—	4	1	no	—
55	C₂²×C₄×C₂₉	—	2	1	no	—
56	C₂²×C₄×C₂₉	—	4	1	no	—
57	C₂²×C₄×C₂₉	—	4	2	no	—
58	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₂₉	—	2	1	no	—
60	Q₁₆×C₂₉	C₄⋊C₄	1	2	yes	complemented
61	SD₁₆×C₂₉	—	2	2	no	—
62	SD₁₆×C₂₉	—	2	2	no	—
63	D₈×C₂₉	—	2	1	no	—
64	D₈×C₂₉	C₄⋊C₄	1	2	yes	complemented
65	M₁₆×C₂₉	—	2	2	no	—
66	M₁₆×C₂₉	—	4	1	no	—
67	M₁₆×C₂₉	—	4	4	no	—
68	C₂×C₈×C₂₉	—	4	1	no	—
69	C₂×C₈×C₂₉	—	4	4	no	—
70	C₂×C₈×C₂₉	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₂₉	D₈	1	2	yes	—
72	C₂×C₈×C₂₉	SD₁₆	1	1	yes	characteristic
73	C₂×C₈×C₂₉	SD₁₆	1	1	yes	characteristic
74	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₂₉	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
79	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
80	C₄⋊C₄×C₂₉	—	4	2	no	—
81	C₄⋊C₄×C₂₉	—	4	2	no	—
82	C₄⋊C₄×C₂₉	—	8	4	no	—
83	C₄²×C₂₉	—	2	1	no	—
84	C₄²×C₂₉	C₂²⋊C₄	1	2	yes	—
85	C₁₆×C₂₉	—	4	2	no	—
86	#12823	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₉	—	4	1	no	—
88	C₂³×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
89	Q₈×C₂₉	—	2	2	no	—
90	Q₈×C₂₉	—	4	1	no	—
91	D₄×C₂₉	—	2	2	no	—
92	D₄×C₂₉	—	4	2	no	—
93	D₄×C₂₉	—	4	2	no	—
94	D₄×C₂₉	—	4	1	no	—
95	C₂×C₄×C₂₉	—	2	4	no	—
96	C₂×C₄×C₂₉	—	2	2	no	—
97	C₂×C₄×C₂₉	—	4	1	no	—
98	C₂×C₄×C₂₉	—	4	2	no	—
99	C₂×C₄×C₂₉	—	4	2	no	—
100	C₂×C₄×C₂₉	—	8	4	no	—
101	C₂×C₄×C₂₉	—	8	2	no	—
102	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₂₉	#137	1	1	yes	characteristic
108	C₈×C₂₉	—	4	2	no	—
109	C₈×C₂₉	#102	1	2	yes	—
110	C₈×C₂₉	#102	1	2	yes	—
111	C₈×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₂₉	#117	1	1	yes	characteristic
113	C₈×C₂₉	#121	1	1	yes	characteristic
114	C₈×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×#443	C₂×C₂₉	1	1	yes	characteristic
116	C₂×#364	C₂×C₂₉	1	1	yes	characteristic, complemented
117	C₂×#360	C₂×C₂₉	1	2	yes	—
118	#1844	C₂×C₂₉	1	1	yes	characteristic, complemented
119	#1383	C₂×C₂₉	1	2	yes	complemented
120	C₂²×C₂₉	—	2	2	no	—
121	C₂²×C₂₉	—	4	2	no	—
122	C₂²×C₂₉	—	8	2	no	—
123	C₂²×C₂₉	#417	1	1	yes	characteristic
124	C₂²×C₂₉	#459	1	1	yes	characteristic
125	C₂²×C₂₉	#476	1	1	yes	characteristic
126	C₄×C₂₉	—	2	4	no	—
127	C₄×C₂₉	—	4	2	no	—
128	C₄×C₂₉	—	16	2	no	—
129	C₄×C₂₉	#380	1	1	yes	characteristic
130	C₄×C₂₉	C₂×#102	1	1	yes	characteristic
131	C₄×C₂₉	#461	1	1	yes	characteristic
132	C₄×C₂₉	#474	1	1	yes	characteristic
133	C₂×#135	C₂²×C₂₉	1	1	yes	characteristic
134	C₂×#131	—	2	1	no	—
135	C₂×M₃₂	C₂²×C₂₉	1	2	yes	—
136	#443	—	2	2	no	—
137	#443	C₄×C₂₉	1	4	yes	complemented
138	C₂×#108	C₂²×C₂₉	1	1	yes	characteristic
139	C₂×#106	—	2	1	no	—
140	C₂×#106	C₂²×C₂₉	1	2	yes	—
141	C₂×#104	—	2	1	no	—
142	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
143	#384	—	2	1	no	—
144	#364	—	2	2	no	—
145	#360	—	2	4	no	—
146	C₂×C₂₉	—	2	1	no	—
147	C₂×C₂₉	—	2	1	no	—
148	C₂×C₂₉	—	8	2	no	—
149	C₂×C₂₉	#1802	1	1	yes	characteristic
150	C₂×C₂₉	#2230	1	2	yes	—
151	C₂×#41	—	2	1	no	—
152	#135	—	2	1	no	—
153	#135	—	2	1	no	—
154	#135	C₂×C₄×C₂₉	1	2	yes	—
155	#135	D₄×C₂₉	1	2	yes	—
156	C₂×Q₁₆	Q₈×C₂₉	1	1	yes	characteristic
157	C₂×SD₁₆	—	2	1	no	—
158	C₂×D₈	Q₈×C₂₉	1	1	yes	characteristic
159	#131	—	2	4	no	—
160	#131	—	4	2	no	—
161	C₂×M₁₆	—	2	1	no	—
162	C₂×M₁₆	—	4	1	no	—
163	C₂²×C₈	—	4	1	no	—
164	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	4	2	no	—
168	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	#108	—	2	1	no	—
173	#108	C₂×C₄×C₂₉	1	2	yes	—
174	#106	—	2	2	no	—
175	#106	—	2	2	no	—
176	#106	—	4	2	no	—
177	#104	—	2	4	no	—
178	C₄×C₈	C₂×C₄×C₂₉	1	2	yes	—
179	C₄×C₈	D₄×C₂₉	1	2	yes	—
180	C₂₉	#12823	1	1	yes	characteristic, direct product factor, Sylow
181	#41	—	2	2	no	—
182	#41	—	2	2	no	—
183	#41	—	4	1	no	—
184	#41	—	4	1	no	—
185	C₂×Q₈	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
192	Q₁₆	—	2	1	no	—
193	Q₁₆	C₄⋊C₄×C₂₉	1	2	yes	complemented
194	SD₁₆	—	2	2	no	—
195	SD₁₆	—	2	2	no	—
196	D₈	—	2	1	no	—
197	D₈	C₄⋊C₄×C₂₉	1	2	yes	complemented
198	M₁₆	—	2	2	no	—
199	M₁₆	—	4	1	no	—
200	M₁₆	—	4	4	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	C₂²⋊C₄×C₂₉	1	2	yes	—
204	C₂×C₈	D₈×C₂₉	1	2	yes	—
205	C₂×C₈	SD₁₆×C₂₉	1	1	yes	characteristic
206	C₂×C₈	SD₁₆×C₂₉	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
210	C₂×C₈	C₂×Q₈×C₂₉	1	1	yes	characteristic
211	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
212	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	8	4	no	—
216	C₄²	—	2	1	no	—
217	C₄²	C₂²⋊C₄×C₂₉	1	2	yes	—
218	C₁₆	—	4	2	no	—
219	C₂³	—	4	1	no	—
220	C₂³	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
221	Q₈	—	2	2	no	—
222	Q₈	—	4	1	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	1	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
235	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
236	C₂×C₄	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
237	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
238	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
239	C₂×C₄	C₂₉×#137	1	1	yes	characteristic
240	C₈	—	4	2	no	—
241	C₈	C₂₉×#102	1	2	yes	—
242	C₈	C₂₉×#102	1	2	yes	—
243	C₈	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
244	C₈	C₂₉×#117	1	1	yes	characteristic
245	C₈	C₂₉×#121	1	1	yes	characteristic
246	C₈	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂²	—	2	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₂₉×#417	1	1	yes	characteristic
251	C₂²	C₂₉×#459	1	1	yes	characteristic
252	C₂²	C₂₉×#476	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	4	2	no	—
255	C₄	—	16	2	no	—
256	C₄	C₂₉×#380	1	1	yes	characteristic
257	C₄	C₂×C₂₉×#102	1	1	yes	characteristic
258	C₄	C₂₉×#461	1	1	yes	characteristic
259	C₄	C₂₉×#474	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	8	2	no	—
263	C₂	C₂₉×#1802	1	1	yes	characteristic
264	C₂	C₂₉×#2230	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₉	#476
commutator	C₂×C₈	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₈	C₂³×C₂₉
prime radical	C₂×C₂₉×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂×C₄×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₉×#364	C₂	semidirect
4. C₂₉×#1844	C₂	semidirect
5. C₂₉×#1383	C₂	semidirect
10. C₂₉×#443	C₄	semidirect
60. Q₁₆×C₂₉	C₄⋊C₄	semidirect
64. D₈×C₂₉	C₄⋊C₄	semidirect
86. #12823	C₂₉	direct (N × H)
116. C₂×#364	C₂×C₂₉	semidirect
118. #1844	C₂×C₂₉	semidirect
119. #1383	C₂×C₂₉	semidirect
137. #443	C₄×C₂₉	semidirect
180. C₂₉	#12823	direct (N × H)
193. Q₁₆	C₄⋊C₄×C₂₉	semidirect
197. D₈	C₄⋊C₄×C₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#443	2
C₂×C₂₉×#364	2
C₂×C₂₉×#360	2
C₂₉×#1844	2
C₂₉×#1383	2
#12823	29

Subgroups — C29×#12823 #20307723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#12823 #20307723