Subgroups — #3191×C₂₂₉ #44420252

Order 29312.

424 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#525×C₂₂₉	C₂	1	2	yes	maximal, complemented
2	#485×C₂₂₉	C₂	1	2	yes	maximal, complemented
3	#482×C₂₂₉	C₂	1	2	yes	maximal, complemented
4	#467×C₂₂₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#466×C₂₂₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#465×C₂₂₉	C₂	1	2	yes	maximal, complemented
7	#452×C₂₂₉	C₂	1	2	yes	maximal, complemented
8	#449×C₂₂₉	C₂	1	2	yes	maximal, complemented
9	C₂×C₂²⋊C₈×C₂₂₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₂₉	C₂²	1	2	yes	—
11	C₂³×C₄×C₂₂₉	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₂₂₉	C₂²	1	1	yes	characteristic, complemented
13	C₂×SD₁₆×C₂₂₉	C₂²	1	2	yes	complemented
14	C₂×D₈×C₂₂₉	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₈×C₂₂₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₂₂₉	C₂²	1	1	yes	characteristic
17	#123×C₂₂₉	—	2	2	no	—
18	C₂²⋊Q₈×C₂₂₉	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₂₂₉	C₂²	1	2	yes	—
20	#121×C₂₂₉	C₂²	1	2	yes	—
21	#121×C₂₂₉	C₂²	1	2	yes	—
22	#120×C₂₂₉	—	2	2	no	—
23	C₄×D₄×C₂₂₉	—	2	2	no	—
24	C₄×D₄×C₂₂₉	—	2	2	no	—
25	#117×C₂₂₉	C₂²	1	2	yes	—
26	#107×C₂₂₉	C₂²	1	1	yes	characteristic, complemented
27	#107×C₂₂₉	C₂²	1	1	yes	characteristic, complemented
28	#106×C₂₂₉	C₂²	1	2	yes	complemented
29	#103×C₂₂₉	C₂²	1	2	yes	complemented
30	#103×C₂₂₉	C₂²	1	2	yes	complemented
31	#102×C₂₂₉	C₂²	1	2	yes	complemented
32	#102×C₂₂₉	C₂²	1	2	yes	complemented
33	C₂²⋊C₈×C₂₂₉	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₂₂₉	C₂²	1	2	yes	—
35	C₂⁴×C₂₂₉	D₄	1	1	yes	characteristic
36	#41×C₂₂₉	—	4	2	no	—
37	#41×C₂₂₉	—	4	2	no	—
38	C₂×Q₈×C₂₂₉	C₂³	1	2	yes	—
39	C₂×D₄×C₂₂₉	—	2	2	no	—
40	C₂×D₄×C₂₂₉	—	2	2	no	—
41	C₂×D₄×C₂₂₉	—	2	2	no	—
42	C₂×D₄×C₂₂₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₂₉	—	2	2	no	—
44	C₂²×C₄×C₂₂₉	—	4	1	no	—
45	C₂²×C₄×C₂₂₉	—	4	1	no	—
46	C₂²×C₄×C₂₂₉	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₂₉	D₄	1	2	yes	—
48	C₂²×C₄×C₂₂₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₂₉	C₂³	1	1	yes	characteristic
50	Q₁₆×C₂₂₉	—	4	1	no	—
51	SD₁₆×C₂₂₉	—	4	2	no	—
52	D₈×C₂₂₉	—	4	1	no	—
53	C₂×C₈×C₂₂₉	—	4	1	no	—
54	C₂×C₈×C₂₂₉	—	4	1	no	—
55	C₂×C₈×C₂₂₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₂₉	C₂³	1	2	yes	—
57	C₂×C₈×C₂₂₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₂₉	—	2	2	no	—
59	C₄⋊C₄×C₂₂₉	—	2	2	no	—
60	C₄⋊C₄×C₂₂₉	C₂³	1	2	yes	—
61	C₄⋊C₄×C₂₂₉	C₂³	1	2	yes	—
62	C₂²⋊C₄×C₂₂₉	—	2	2	no	—
63	C₂²⋊C₄×C₂₂₉	—	2	2	no	—
64	C₂²⋊C₄×C₂₂₉	—	2	2	no	—
65	C₂²⋊C₄×C₂₂₉	—	2	2	no	—
66	C₂²⋊C₄×C₂₂₉	—	2	2	no	—
67	C₄²×C₂₂₉	—	2	2	no	—
68	C₂³×C₂₂₉	—	2	2	no	—
69	C₂³×C₂₂₉	—	2	2	no	—
70	C₂³×C₂₂₉	—	4	1	no	—
71	C₂³×C₂₂₉	—	4	1	no	—
72	C₂³×C₂₂₉	C₂×D₄	1	2	yes	—
73	C₂³×C₂₂₉	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₂₂₉	—	2	2	no	—
75	Q₈×C₂₂₉	—	2	2	no	—
76	D₄×C₂₂₉	—	2	2	no	—
77	D₄×C₂₂₉	—	2	2	no	—
78	D₄×C₂₂₉	—	4	4	no	—
79	D₄×C₂₂₉	—	4	4	no	—
80	D₄×C₂₂₉	—	4	4	no	—
81	C₂×C₄×C₂₂₉	—	2	1	no	—
82	C₂×C₄×C₂₂₉	—	2	1	no	—
83	C₂×C₄×C₂₂₉	—	2	2	no	—
84	C₂×C₄×C₂₂₉	—	2	2	no	—
85	C₂×C₄×C₂₂₉	—	2	2	no	—
86	C₂×C₄×C₂₂₉	—	2	2	no	—
87	C₂×C₄×C₂₂₉	—	2	2	no	—
88	C₂×C₄×C₂₂₉	—	4	1	no	—
89	C₂×C₄×C₂₂₉	—	4	2	no	—
90	C₂×C₄×C₂₂₉	—	4	4	no	—
91	C₂×C₄×C₂₂₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₂₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₂₉	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₂₉	—	2	1	no	—
95	C₈×C₂₂₉	—	2	2	no	—
96	C₈×C₂₂₉	—	2	1	no	—
97	C₂²×C₂₂₉	—	2	1	no	—
98	C₂²×C₂₂₉	—	2	1	no	—
99	C₂²×C₂₂₉	—	2	2	no	—
100	C₂²×C₂₂₉	—	2	2	no	—
101	C₂²×C₂₂₉	—	2	2	no	—
102	C₂²×C₂₂₉	—	4	4	no	—
103	C₂²×C₂₂₉	—	4	2	no	—
104	C₂²×C₂₂₉	—	4	2	no	—
105	C₂²×C₂₂₉	—	4	2	no	—
106	C₂²×C₂₂₉	#135	1	2	yes	—
107	C₂²×C₂₂₉	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₂₂₉	—	2	2	no	—
109	C₄×C₂₂₉	—	2	1	no	—
110	C₄×C₂₂₉	—	4	2	no	—
111	C₄×C₂₂₉	—	4	2	no	—
112	C₄×C₂₂₉	—	4	2	no	—
113	C₄×C₂₂₉	H(5,𝔽₂)	1	1	yes	characteristic
114	C₄×C₂₂₉	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₂₂₉	—	2	2	no	—
116	C₂×C₂₂₉	—	4	2	no	—
117	C₂×C₂₂₉	—	8	2	no	—
118	C₂×C₂₂₉	#534	1	1	yes	characteristic
119	C₂×C₂₂₉	C₂×#135	1	1	yes	characteristic
120	C₂×C₂₂₉	#575	1	1	yes	characteristic
121	C₂₂₉	#3191	1	1	yes	characteristic, direct product factor, Sylow
122	#3191	C₂₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	#525	C₂×C₂₂₉	1	2	yes	complemented
124	#485	C₂×C₂₂₉	1	2	yes	complemented
125	#482	C₂×C₂₂₉	1	2	yes	complemented
126	#467	C₂×C₂₂₉	1	1	yes	characteristic, complemented
127	#466	C₂×C₂₂₉	1	1	yes	characteristic, complemented
128	#465	C₂×C₂₂₉	1	2	yes	complemented
129	#452	C₂×C₂₂₉	1	2	yes	complemented
130	#449	C₂×C₂₂₉	1	2	yes	complemented
131	C₂×C₂²⋊C₈	C₂×C₂₂₉	1	1	yes	characteristic, complemented
132	C₂×#41	C₂²×C₂₂₉	1	2	yes	—
133	C₂³×C₄	C₂²×C₂₂₉	1	1	yes	characteristic
134	C₂×Q₁₆	C₂²×C₂₂₉	1	1	yes	characteristic, complemented
135	C₂×SD₁₆	C₂²×C₂₂₉	1	2	yes	complemented
136	C₂×D₈	C₂²×C₂₂₉	1	1	yes	characteristic, complemented
137	C₂²×C₈	C₂²×C₂₂₉	1	1	yes	characteristic
138	C₂²×C₈	C₂²×C₂₂₉	1	1	yes	characteristic
139	#123	—	2	2	no	—
140	C₂²⋊Q₈	C₂²×C₂₂₉	1	2	yes	—
141	C₂²⋊Q₈	C₂²×C₂₂₉	1	2	yes	—
142	#121	C₂²×C₂₂₉	1	2	yes	—
143	#121	C₂²×C₂₂₉	1	2	yes	—
144	#120	—	2	2	no	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	#117	C₂²×C₂₂₉	1	2	yes	—
148	#107	C₂²×C₂₂₉	1	1	yes	characteristic, complemented
149	#107	C₂²×C₂₂₉	1	1	yes	characteristic, complemented
150	#106	C₂²×C₂₂₉	1	2	yes	complemented
151	#103	C₂²×C₂₂₉	1	2	yes	complemented
152	#103	C₂²×C₂₂₉	1	2	yes	complemented
153	#102	C₂²×C₂₂₉	1	2	yes	complemented
154	#102	C₂²×C₂₂₉	1	2	yes	complemented
155	C₂²⋊C₈	C₂²×C₂₂₉	1	2	yes	—
156	C₂²⋊C₈	C₂²×C₂₂₉	1	2	yes	—
157	C₂⁴	D₄×C₂₂₉	1	1	yes	characteristic
158	#41	—	4	2	no	—
159	#41	—	4	2	no	—
160	C₂×Q₈	C₂³×C₂₂₉	1	2	yes	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	2	2	no	—
164	C₂×D₄	C₂³×C₂₂₉	1	2	yes	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	D₄×C₂₂₉	1	1	yes	characteristic
169	C₂²×C₄	D₄×C₂₂₉	1	2	yes	—
170	C₂²×C₄	C₂³×C₂₂₉	1	2	yes	—
171	C₂²×C₄	C₂³×C₂₂₉	1	1	yes	characteristic
172	Q₁₆	—	4	1	no	—
173	SD₁₆	—	4	2	no	—
174	D₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	C₂³×C₂₂₉	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₂₂₉	1	2	yes	—
179	C₂×C₈	C₂³×C₂₂₉	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	C₂³×C₂₂₉	1	2	yes	—
183	C₄⋊C₄	C₂³×C₂₂₉	1	2	yes	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×D₄×C₂₂₉	1	2	yes	—
195	C₂³	C₂×D₄×C₂₂₉	1	1	yes	characteristic
196	Q₈	—	2	2	no	—
197	Q₈	—	2	2	no	—
198	D₄	—	2	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	C₂×D₄×C₂₂₉	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₂₂₉	1	1	yes	characteristic
215	C₂×C₄	C₂⁴×C₂₂₉	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	#135×C₂₂₉	1	2	yes	—
229	C₂²	C₂²×D₄×C₂₂₉	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	H(5,𝔽₂)×C₂₂₉	1	1	yes	characteristic
236	C₄	H(5,𝔽₂)×C₂₂₉	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	4	2	no	—
239	C₂	—	8	2	no	—
240	C₂	#534×C₂₂₉	1	1	yes	characteristic
241	C₂	C₂×#135×C₂₂₉	1	1	yes	characteristic
242	C₂	#575×C₂₂₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₉ ◁ C₂³×C₄×C₂₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #525×C₂₂₉	C₂	semidirect
2. #485×C₂₂₉	C₂	semidirect
3. #482×C₂₂₉	C₂	semidirect
4. #467×C₂₂₉	C₂	semidirect
5. #466×C₂₂₉	C₂	semidirect
6. #465×C₂₂₉	C₂	semidirect
7. #452×C₂₂₉	C₂	semidirect
8. #449×C₂₂₉	C₂	semidirect
9. C₂×C₂²⋊C₈×C₂₂₉	C₂	semidirect
12. C₂×Q₁₆×C₂₂₉	C₂²	semidirect
13. C₂×SD₁₆×C₂₂₉	C₂²	semidirect
14. C₂×D₈×C₂₂₉	C₂²	semidirect
26. #107×C₂₂₉	C₂²	semidirect
27. #107×C₂₂₉	C₂²	semidirect
28. #106×C₂₂₉	C₂²	semidirect
29. #103×C₂₂₉	C₂²	semidirect
30. #103×C₂₂₉	C₂²	semidirect
31. #102×C₂₂₉	C₂²	semidirect
32. #102×C₂₂₉	C₂²	semidirect
121. C₂₂₉	#3191	direct (N × H)
122. #3191	C₂₂₉	direct (N × H)
123. #525	C₂×C₂₂₉	semidirect
124. #485	C₂×C₂₂₉	semidirect
125. #482	C₂×C₂₂₉	semidirect
126. #467	C₂×C₂₂₉	semidirect
127. #466	C₂×C₂₂₉	semidirect
128. #465	C₂×C₂₂₉	semidirect
129. #452	C₂×C₂₂₉	semidirect
130. #449	C₂×C₂₂₉	semidirect
131. C₂×C₂²⋊C₈	C₂×C₂₂₉	semidirect
134. C₂×Q₁₆	C₂²×C₂₂₉	semidirect
135. C₂×SD₁₆	C₂²×C₂₂₉	semidirect
136. C₂×D₈	C₂²×C₂₂₉	semidirect
148. #107	C₂²×C₂₂₉	semidirect
149. #107	C₂²×C₂₂₉	semidirect
150. #106	C₂²×C₂₂₉	semidirect
151. #103	C₂²×C₂₂₉	semidirect
152. #103	C₂²×C₂₂₉	semidirect
153. #102	C₂²×C₂₂₉	semidirect
154. #102	C₂²×C₂₂₉	semidirect

Maximal subgroups

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

subgroup	index
#525×C₂₂₉	2
#485×C₂₂₉	2
#482×C₂₂₉	2
#467×C₂₂₉	2
#466×C₂₂₉	2
#465×C₂₂₉	2
#452×C₂₂₉	2
#449×C₂₂₉	2
C₂×C₂²⋊C₈×C₂₂₉	2
#3191	229

Subgroups — #3191×C229 #44420252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3191×C₂₂₉ #44420252