Subgroups — C₂²×C₂₉×#391 #20328921

Order 7424.

6174 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄²×C₂₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₉×#391	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₂₉×#95	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₂₉	—	2	8	no	—
7	C₂³×Q₈×C₂₉	—	2	8	no	—
8	C₂³×Q₈×C₂₉	C₄	1	16	yes	complemented
9	C₂³×Q₈×C₂₉	C₂²	1	12	yes	—
10	C₂⁴×C₄×C₂₉	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₂₉	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₂₉	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₂₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₂₉	—	2	8	no	—
15	C₂²×C₄⋊C₄×C₂₉	C₂²	1	12	yes	—
16	C₂²×C₄²×C₂₉	—	2	8	no	—
17	C₂²×C₄²×C₂₉	C₂²	1	12	yes	—
18	C₂₉×#391	C₂²	1	64	yes	direct product factor
19	C₂×C₂₉×#95	C₂²	1	48	yes	—
20	C₂⁵×C₂₉	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₂₉	—	2	128	no	—
22	C₂²×Q₈×C₂₉	—	2	48	no	—
23	C₂²×Q₈×C₂₉	—	2	48	no	—
24	C₂²×Q₈×C₂₉	C₂×C₄	1	96	yes	complemented
25	C₂²×Q₈×C₂₉	C₂³	1	16	yes	—
26	C₂³×C₄×C₂₉	—	2	8	no	—
27	C₂³×C₄×C₂₉	—	2	8	no	—
28	C₂³×C₄×C₂₉	—	2	8	no	—
29	C₂³×C₄×C₂₉	—	2	8	no	—
30	C₂³×C₄×C₂₉	—	2	8	no	—
31	C₂³×C₄×C₂₉	—	2	16	no	—
32	C₂³×C₄×C₂₉	C₂×C₄	1	16	yes	—
33	C₂³×C₄×C₂₉	C₂×C₄	1	8	yes	—
34	C₂³×C₄×C₂₉	D₄	1	8	yes	—
35	C₂³×C₄×C₂₉	D₄	1	8	yes	—
36	C₂³×C₄×C₂₉	Q₈	1	16	yes	complemented
37	C₂³×C₄×C₂₉	C₂³	1	12	yes	—
38	C₂³×C₄×C₂₉	C₂³	1	12	yes	—
39	C₂³×C₄×C₂₉	C₂³	1	12	yes	—
40	C₂³×C₄×C₂₉	C₂³	1	6	yes	—
41	C₂×C₄⋊C₄×C₂₉	—	2	48	no	—
42	C₂×C₄⋊C₄×C₂₉	C₂³	1	16	yes	—
43	C₂×C₄²×C₂₉	—	2	48	no	—
44	C₂×C₄²×C₂₉	C₂³	1	16	yes	—
45	C₂₉×#95	C₂³	1	64	yes	—
46	C₂⁴×C₂₉	C₂²×C₄	1	4	yes	—
47	C₂⁴×C₂₉	C₂×D₄	1	4	yes	—
48	C₂⁴×C₂₉	C₂×D₄	1	4	yes	—
49	C₂⁴×C₂₉	C₂×Q₈	1	8	yes	—
50	C₂⁴×C₂₉	#41	1	8	yes	—
51	C₂⁴×C₂₉	C₂⁴	1	3	yes	—
52	C₂×Q₈×C₂₉	—	2	384	no	—
53	C₂×Q₈×C₂₉	—	2	32	no	—
54	C₂×Q₈×C₂₉	—	2	32	no	—
55	C₂×Q₈×C₂₉	C₂²×C₄	1	64	yes	complemented
56	C₂²×C₄×C₂₉	—	2	64	no	—
57	C₂²×C₄×C₂₉	—	2	64	no	—
58	C₂²×C₄×C₂₉	—	2	24	no	—
59	C₂²×C₄×C₂₉	—	2	24	no	—
60	C₂²×C₄×C₂₉	—	2	24	no	—
61	C₂²×C₄×C₂₉	—	2	24	no	—
62	C₂²×C₄×C₂₉	—	2	24	no	—
63	C₂²×C₄×C₂₉	—	2	48	no	—
64	C₂²×C₄×C₂₉	—	4	32	no	—
65	C₂²×C₄×C₂₉	C₂²⋊C₄	1	64	yes	—
66	C₂²×C₄×C₂₉	C₂²×C₄	1	48	yes	—
67	C₂²×C₄×C₂₉	C₂²×C₄	1	24	yes	—
68	C₂²×C₄×C₂₉	C₂×D₄	1	24	yes	—
69	C₂²×C₄×C₂₉	C₂×D₄	1	24	yes	—
70	C₂²×C₄×C₂₉	C₂×Q₈	1	48	yes	complemented
71	C₂²×C₄×C₂₉	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₂₉	C₂⁴	1	8	yes	—
73	C₂²×C₄×C₂₉	C₂⁴	1	8	yes	—
74	C₂²×C₄×C₂₉	C₂⁴	1	4	yes	—
75	C₄⋊C₄×C₂₉	—	2	32	no	—
76	C₄²×C₂₉	—	2	32	no	—
77	C₂²×#391	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₉	C₂×C₂²⋊C₄	1	16	yes	—
79	C₂³×C₂₉	C₄×Q₈	1	32	yes	—
80	C₂³×C₂₉	C₂²⋊Q₈	1	32	yes	—
81	C₂³×C₂₉	#124	1	16	yes	—
82	C₂³×C₂₉	C₄⋊Q₈	1	16	yes	—
83	C₂³×C₂₉	C₂³×C₄	1	6	yes	—
84	C₂³×C₂₉	C₂²×D₄	1	6	yes	—
85	C₂³×C₂₉	C₂²×D₄	1	6	yes	—
86	C₂³×C₂₉	C₂²×Q₈	1	12	yes	—
87	C₂³×C₂₉	C₂×#41	1	12	yes	—
88	C₂³×C₂₉	C₂⁵	1	1	yes	characteristic
89	Q₈×C₂₉	—	2	128	no	—
90	C₂×C₄×C₂₉	—	2	96	no	—
91	C₂×C₄×C₂₉	—	2	96	no	—
92	C₂×C₄×C₂₉	—	2	8	no	—
93	C₂×C₄×C₂₉	—	2	8	no	—
94	C₂×C₄×C₂₉	—	2	8	no	—
95	C₂×C₄×C₂₉	—	2	8	no	—
96	C₂×C₄×C₂₉	—	2	8	no	—
97	C₂×C₄×C₂₉	—	2	16	no	—
98	C₂×C₄×C₂₉	—	4	48	no	—
99	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	96	yes	—
100	C₂×C₄×C₂₉	C₂³×C₄	1	16	yes	—
101	C₂×C₄×C₂₉	C₂³×C₄	1	8	yes	—
102	C₂×C₄×C₂₉	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₂₉	C₂²×D₄	1	8	yes	—
104	C₂×C₄×C₂₉	C₂²×Q₈	1	16	yes	complemented
105	C₂⁴×Q₈	C₂×C₂₉	1	1	yes	characteristic
106	C₂³×C₄⋊C₄	C₂×C₂₉	1	1	yes	characteristic
107	C₂³×C₄²	C₂×C₂₉	1	1	yes	characteristic
108	C₂×#391	C₂×C₂₉	1	24	yes	direct product factor
109	C₂²×#95	C₂×C₂₉	1	4	yes	—
110	C₂²×C₂₉	#391	1	64	yes	direct product factor
111	C₂²×C₂₉	C₂²×C₂²⋊C₄	1	12	yes	—
112	C₂²×C₂₉	C₂×C₄×Q₈	1	24	yes	—
113	C₂²×C₂₉	C₂×C₂²⋊Q₈	1	24	yes	—
114	C₂²×C₂₉	C₂×#124	1	12	yes	—
115	C₂²×C₂₉	C₂×C₄⋊Q₈	1	12	yes	—
116	C₂²×C₂₉	C₂⁴×C₄	1	1	yes	characteristic
117	C₂²×C₂₉	C₂³×D₄	1	1	yes	characteristic
118	C₂²×C₂₉	C₂³×D₄	1	1	yes	characteristic
119	C₂²×C₂₉	C₂³×Q₈	1	2	yes	—
120	C₂²×C₂₉	C₂²×#41	1	2	yes	—
121	C₄×C₂₉	—	2	16	no	—
122	C₄×C₂₉	—	2	16	no	—
123	C₄×C₂₉	—	4	8	no	—
124	C₄×C₂₉	C₂²×C₂²⋊C₄	1	16	yes	—
125	C₂³×Q₈	—	2	8	no	—
126	C₂³×Q₈	—	2	8	no	—
127	C₂³×Q₈	C₄×C₂₉	1	16	yes	complemented
128	C₂³×Q₈	C₂²×C₂₉	1	12	yes	—
129	C₂⁴×C₄	C₂²×C₂₉	1	2	yes	—
130	C₂⁴×C₄	C₂²×C₂₉	1	2	yes	—
131	C₂⁴×C₄	C₂²×C₂₉	1	2	yes	—
132	C₂⁴×C₄	C₂²×C₂₉	1	1	yes	characteristic
133	C₂²×C₄⋊C₄	—	2	8	no	—
134	C₂²×C₄⋊C₄	C₂²×C₂₉	1	12	yes	—
135	C₂²×C₄²	—	2	8	no	—
136	C₂²×C₄²	C₂²×C₂₉	1	12	yes	—
137	#391	C₂²×C₂₉	1	64	yes	direct product factor
138	C₂×#95	C₂²×C₂₉	1	48	yes	—
139	C₂×C₂₉	C₂×#391	1	24	yes	direct product factor
140	C₂×C₂₉	C₂³×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₂₉	C₂²×C₄×Q₈	1	2	yes	—
142	C₂×C₂₉	C₂²×C₂²⋊Q₈	1	2	yes	—
143	C₂×C₂₉	C₂²×#124	1	1	yes	characteristic
144	C₂×C₂₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
145	C₂⁵	C₂³×C₂₉	1	1	yes	characteristic
146	C₂²×Q₈	—	2	128	no	—
147	C₂²×Q₈	—	2	48	no	—
148	C₂²×Q₈	—	2	48	no	—
149	C₂²×Q₈	C₂×C₄×C₂₉	1	96	yes	complemented
150	C₂²×Q₈	C₂³×C₂₉	1	16	yes	—
151	C₂³×C₄	—	2	8	no	—
152	C₂³×C₄	—	2	8	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	—	2	16	no	—
157	C₂³×C₄	C₂×C₄×C₂₉	1	16	yes	—
158	C₂³×C₄	C₂×C₄×C₂₉	1	8	yes	—
159	C₂³×C₄	D₄×C₂₉	1	8	yes	—
160	C₂³×C₄	D₄×C₂₉	1	8	yes	—
161	C₂³×C₄	Q₈×C₂₉	1	16	yes	complemented
162	C₂³×C₄	C₂³×C₂₉	1	12	yes	—
163	C₂³×C₄	C₂³×C₂₉	1	12	yes	—
164	C₂³×C₄	C₂³×C₂₉	1	12	yes	—
165	C₂³×C₄	C₂³×C₂₉	1	6	yes	—
166	C₂×C₄⋊C₄	—	2	48	no	—
167	C₂×C₄⋊C₄	C₂³×C₂₉	1	16	yes	—
168	C₂×C₄²	—	2	48	no	—
169	C₂×C₄²	C₂³×C₂₉	1	16	yes	—
170	#95	C₂³×C₂₉	1	64	yes	—
171	C₂₉	C₂²×#391	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	C₂²×C₄×C₂₉	1	4	yes	—
173	C₂⁴	C₂×D₄×C₂₉	1	4	yes	—
174	C₂⁴	C₂×D₄×C₂₉	1	4	yes	—
175	C₂⁴	C₂×Q₈×C₂₉	1	8	yes	—
176	C₂⁴	#41×C₂₉	1	8	yes	—
177	C₂⁴	C₂⁴×C₂₉	1	3	yes	—
178	C₂×Q₈	—	2	384	no	—
179	C₂×Q₈	—	2	32	no	—
180	C₂×Q₈	—	2	32	no	—
181	C₂×Q₈	C₂²×C₄×C₂₉	1	64	yes	complemented
182	C₂²×C₄	—	2	64	no	—
183	C₂²×C₄	—	2	64	no	—
184	C₂²×C₄	—	2	24	no	—
185	C₂²×C₄	—	2	24	no	—
186	C₂²×C₄	—	2	24	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	—	2	24	no	—
189	C₂²×C₄	—	2	48	no	—
190	C₂²×C₄	—	4	32	no	—
191	C₂²×C₄	C₂²⋊C₄×C₂₉	1	64	yes	—
192	C₂²×C₄	C₂²×C₄×C₂₉	1	48	yes	—
193	C₂²×C₄	C₂²×C₄×C₂₉	1	24	yes	—
194	C₂²×C₄	C₂×D₄×C₂₉	1	24	yes	—
195	C₂²×C₄	C₂×D₄×C₂₉	1	24	yes	—
196	C₂²×C₄	C₂×Q₈×C₂₉	1	48	yes	complemented
197	C₂²×C₄	C₂⁴×C₂₉	1	8	yes	—
198	C₂²×C₄	C₂⁴×C₂₉	1	8	yes	—
199	C₂²×C₄	C₂⁴×C₂₉	1	8	yes	—
200	C₂²×C₄	C₂⁴×C₂₉	1	4	yes	—
201	C₄⋊C₄	—	2	32	no	—
202	C₄²	—	2	32	no	—
203	C₂³	C₂×C₂²⋊C₄×C₂₉	1	16	yes	—
204	C₂³	C₄×Q₈×C₂₉	1	32	yes	—
205	C₂³	C₂₉×C₂²⋊Q₈	1	32	yes	—
206	C₂³	C₂₉×#124	1	16	yes	—
207	C₂³	C₂₉×C₄⋊Q₈	1	16	yes	—
208	C₂³	C₂³×C₄×C₂₉	1	6	yes	—
209	C₂³	C₂²×D₄×C₂₉	1	6	yes	—
210	C₂³	C₂²×D₄×C₂₉	1	6	yes	—
211	C₂³	C₂²×Q₈×C₂₉	1	12	yes	—
212	C₂³	C₂×#41×C₂₉	1	12	yes	—
213	C₂³	C₂⁵×C₂₉	1	1	yes	characteristic
214	Q₈	—	2	128	no	—
215	C₂×C₄	—	2	96	no	—
216	C₂×C₄	—	2	96	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	8	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	4	48	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	96	yes	—
225	C₂×C₄	C₂³×C₄×C₂₉	1	16	yes	—
226	C₂×C₄	C₂³×C₄×C₂₉	1	8	yes	—
227	C₂×C₄	C₂²×D₄×C₂₉	1	8	yes	—
228	C₂×C₄	C₂²×D₄×C₂₉	1	8	yes	—
229	C₂×C₄	C₂²×Q₈×C₂₉	1	16	yes	complemented
230	C₂²	C₂₉×#391	1	64	yes	direct product factor
231	C₂²	C₂²×C₂²⋊C₄×C₂₉	1	12	yes	—
232	C₂²	C₂×C₄×Q₈×C₂₉	1	24	yes	—
233	C₂²	C₂×C₂₉×C₂²⋊Q₈	1	24	yes	—
234	C₂²	C₂×C₂₉×#124	1	12	yes	—
235	C₂²	C₂×C₂₉×C₄⋊Q₈	1	12	yes	—
236	C₂²	C₂⁴×C₄×C₂₉	1	1	yes	characteristic
237	C₂²	C₂³×D₄×C₂₉	1	1	yes	characteristic
238	C₂²	C₂³×D₄×C₂₉	1	1	yes	characteristic
239	C₂²	C₂³×Q₈×C₂₉	1	2	yes	—
240	C₂²	C₂²×#41×C₂₉	1	2	yes	—
241	C₄	—	2	16	no	—
242	C₄	—	2	16	no	—
243	C₄	—	4	8	no	—
244	C₄	C₂²×C₂²⋊C₄×C₂₉	1	16	yes	—
245	C₂	C₂×C₂₉×#391	1	24	yes	direct product factor
246	C₂	C₂³×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
247	C₂	C₂²×C₄×Q₈×C₂₉	1	2	yes	—
248	C₂	C₂²×C₂₉×C₂²⋊Q₈	1	2	yes	—
249	C₂	C₂²×C₂₉×#124	1	1	yes	characteristic
250	C₂	C₂²×C₂₉×C₄⋊Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₂₉	C₂³
commutator	C₂²	C₂⁴×C₄×C₂₉
Frattini, cosocle	C₂³	C₂⁵×C₂₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₂₉×#391	C₂	direct (N × H)
8. C₂³×Q₈×C₂₉	C₄	semidirect
18. C₂₉×#391	C₂²	direct (N × H)
24. C₂²×Q₈×C₂₉	C₂×C₄	semidirect
36. C₂³×C₄×C₂₉	Q₈	semidirect
55. C₂×Q₈×C₂₉	C₂²×C₄	semidirect
70. C₂²×C₄×C₂₉	C₂×Q₈	semidirect
77. C₂²×#391	C₂₉	direct (N × H)
104. C₂×C₄×C₂₉	C₂²×Q₈	semidirect
108. C₂×#391	C₂×C₂₉	direct (N × H)
110. C₂²×C₂₉	#391	direct (N × H)
127. C₂³×Q₈	C₄×C₂₉	semidirect
137. #391	C₂²×C₂₉	direct (N × H)
139. C₂×C₂₉	C₂×#391	direct (N × H)
149. C₂²×Q₈	C₂×C₄×C₂₉	semidirect
161. C₂³×C₄	Q₈×C₂₉	semidirect
171. C₂₉	C₂²×#391	direct (N × H)
181. C₂×Q₈	C₂²×C₄×C₂₉	semidirect
196. C₂²×C₄	C₂×Q₈×C₂₉	semidirect
229. C₂×C₄	C₂²×Q₈×C₂₉	semidirect
230. C₂²	C₂₉×#391	direct (N × H)
245. C₂	C₂×C₂₉×#391	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₂₉	2
C₂³×C₄⋊C₄×C₂₉	2
C₂³×C₄²×C₂₉	2
C₂×C₂₉×#391	2
C₂²×C₂₉×#95	2
C₂²×#391	29

Subgroups — C22×C29×#391 #20328921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₉×#391 #20328921