Subgroups — #3211×C₁₄₉ #34042124

Order 19072.

410 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#529×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#124×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#495×C₁₄₉	C₂	1	2	yes	maximal, complemented
4	#488×C₁₄₉	C₂	1	2	yes	maximal, complemented
5	#464×C₁₄₉	C₂	1	2	yes	maximal, complemented
6	#463×C₁₄₉	C₂	1	2	yes	maximal, complemented
7	#451×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#450×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#448×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#447×C₁₄₉	C₂	1	1	yes	maximal, characteristic
11	#431×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₁₄₉	C₂²	1	2	yes	complemented
15	C₂×SD₁₆×C₁₄₉	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₁₄₉	C₂²	1	2	yes	complemented
17	C₂×D₈×C₁₄₉	C₂²	1	2	yes	—
18	#126×C₁₄₉	—	2	1	no	—
19	#125×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
20	#124×C₁₄₉	—	2	1	no	—
21	#124×C₁₄₉	—	2	1	no	—
22	#124×C₁₄₉	C₂²	1	2	yes	—
23	#124×C₁₄₉	C₂²	1	2	yes	—
24	#124×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
25	#123×C₁₄₉	—	2	1	no	—
26	C₂²⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
27	#121×C₁₄₉	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₄×C₁₄₉	—	2	1	no	—
30	C₄×D₄×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
31	#117×C₁₄₉	—	2	1	no	—
32	C₂×C₂²⋊C₄×C₁₄₉	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₁₄₉	—	2	1	no	—
34	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₄₉	C₂²	1	2	yes	complemented
36	#103×C₁₄₉	C₂²	1	2	yes	complemented
37	#103×C₁₄₉	C₂²	1	2	yes	complemented
38	#102×C₁₄₉	C₂²	1	2	yes	—
39	#102×C₁₄₉	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₁₄₉	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₁₄₉	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₄₉	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₄₉	—	2	1	no	—
44	C₂×Q₈×C₁₄₉	—	2	1	no	—
45	C₂×Q₈×C₁₄₉	—	4	2	no	—
46	C₂×Q₈×C₁₄₉	C₂³	1	2	yes	—
47	C₂×Q₈×C₁₄₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₄₉	—	2	1	no	—
49	C₂×D₄×C₁₄₉	—	2	1	no	—
50	C₂×D₄×C₁₄₉	—	4	2	no	—
51	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
52	C₂×D₄×C₁₄₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₄₉	—	2	1	no	—
54	C₂²×C₄×C₁₄₉	—	2	1	no	—
55	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₄₉	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
58	Q₁₆×C₁₄₉	—	4	2	no	—
59	SD₁₆×C₁₄₉	—	4	2	no	—
60	SD₁₆×C₁₄₉	—	4	2	no	—
61	D₈×C₁₄₉	—	4	2	no	—
62	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
63	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₄₉	—	2	1	no	—
65	C₄⋊C₄×C₁₄₉	—	2	1	no	—
66	C₄⋊C₄×C₁₄₉	—	2	1	no	—
67	C₄⋊C₄×C₁₄₉	—	2	1	no	—
68	C₄⋊C₄×C₁₄₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₄₉	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₁₄₉	—	2	1	no	—
71	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
72	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
73	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
74	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
75	C₂²⋊C₄×C₁₄₉	—	2	1	no	—
76	C₂²⋊C₄×C₁₄₉	—	2	1	no	—
77	C₂²⋊C₄×C₁₄₉	—	2	1	no	—
78	C₂²⋊C₄×C₁₄₉	—	2	1	no	—
79	C₄²×C₁₄₉	—	2	1	no	—
80	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
81	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
82	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
83	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
84	C₂³×C₁₄₉	—	2	1	no	—
85	C₂³×C₁₄₉	—	2	2	no	—
86	C₂³×C₁₄₉	—	4	2	no	—
87	C₂³×C₁₄₉	—	4	2	no	—
88	C₂³×C₁₄₉	—	4	2	no	—
89	C₂³×C₁₄₉	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₁₄₉	—	2	2	no	—
91	Q₈×C₁₄₉	—	2	2	no	—
92	Q₈×C₁₄₉	—	2	2	no	—
93	Q₈×C₁₄₉	—	4	2	no	—
94	D₄×C₁₄₉	—	2	2	no	—
95	D₄×C₁₄₉	—	2	2	no	—
96	D₄×C₁₄₉	—	2	2	no	—
97	D₄×C₁₄₉	—	4	2	no	—
98	D₄×C₁₄₉	—	4	2	no	—
99	C₂×C₄×C₁₄₉	—	2	2	no	—
100	C₂×C₄×C₁₄₉	—	2	2	no	—
101	C₂×C₄×C₁₄₉	—	2	1	no	—
102	C₂×C₄×C₁₄₉	—	2	1	no	—
103	C₂×C₄×C₁₄₉	—	2	2	no	—
104	C₂×C₄×C₁₄₉	—	2	1	no	—
105	C₂×C₄×C₁₄₉	—	4	1	no	—
106	C₂×C₄×C₁₄₉	—	4	2	no	—
107	C₂×C₄×C₁₄₉	—	4	2	no	—
108	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
114	C₈×C₁₄₉	—	2	2	no	—
115	C₈×C₁₄₉	—	2	2	no	—
116	C₂²×C₁₄₉	—	2	1	no	—
117	C₂²×C₁₄₉	—	2	1	no	—
118	C₂²×C₁₄₉	—	4	2	no	—
119	C₂²×C₁₄₉	—	4	1	no	—
120	C₂²×C₁₄₉	—	4	1	no	—
121	C₂²×C₁₄₉	—	4	2	no	—
122	C₂²×C₁₄₉	—	4	1	no	—
123	C₂²×C₁₄₉	—	4	2	no	—
124	C₂²×C₁₄₉	—	8	4	no	—
125	C₂²×C₁₄₉	#135	1	2	yes	—
126	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₄₉	—	2	1	no	—
128	C₄×C₁₄₉	—	2	1	no	—
129	C₄×C₁₄₉	—	2	1	no	—
130	C₄×C₁₄₉	—	2	1	no	—
131	C₄×C₁₄₉	—	2	1	no	—
132	C₄×C₁₄₉	—	4	2	no	—
133	C₄×C₁₄₉	—	4	1	no	—
134	C₄×C₁₄₉	—	4	1	no	—
135	C₄×C₁₄₉	—	4	1	no	—
136	C₄×C₁₄₉	H(5,𝔽₂)	1	2	yes	—
137	C₂×C₁₄₉	—	2	2	no	—
138	C₂×C₁₄₉	—	8	1	no	—
139	C₂×C₁₄₉	—	8	2	no	—
140	C₂×C₁₄₉	#535	1	1	yes	characteristic
141	C₂×C₁₄₉	C₂×#135	1	1	yes	characteristic
142	C₂×C₁₄₉	#575	1	1	yes	characteristic
143	C₁₄₉	#3211	1	1	yes	characteristic, direct product factor, Sylow
144	#3211	C₁₄₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	#529	C₂×C₁₄₉	1	1	yes	characteristic, complemented
146	C₂×#124	C₂×C₁₄₉	1	1	yes	characteristic, complemented
147	#495	C₂×C₁₄₉	1	2	yes	complemented
148	#488	C₂×C₁₄₉	1	2	yes	complemented
149	#464	C₂×C₁₄₉	1	2	yes	complemented
150	#463	C₂×C₁₄₉	1	2	yes	complemented
151	#451	C₂×C₁₄₉	1	1	yes	characteristic, complemented
152	#450	C₂×C₁₄₉	1	1	yes	characteristic, complemented
153	#448	C₂×C₁₄₉	1	1	yes	characteristic, complemented
154	#447	C₂×C₁₄₉	1	1	yes	characteristic
155	#431	C₂×C₁₄₉	1	1	yes	characteristic, complemented
156	C₂²×Q₈	C₂²×C₁₄₉	1	1	yes	characteristic
157	C₂²×D₄	C₂²×C₁₄₉	1	1	yes	characteristic
158	C₂×Q₁₆	C₂²×C₁₄₉	1	2	yes	complemented
159	C₂×SD₁₆	C₂²×C₁₄₉	1	2	yes	—
160	C₂×SD₁₆	C₂²×C₁₄₉	1	2	yes	complemented
161	C₂×D₈	C₂²×C₁₄₉	1	2	yes	—
162	#126	—	2	1	no	—
163	#125	C₂²×C₁₄₉	1	1	yes	characteristic, complemented
164	#124	—	2	1	no	—
165	#124	—	2	1	no	—
166	#124	C₂²×C₁₄₉	1	2	yes	—
167	#124	C₂²×C₁₄₉	1	2	yes	—
168	#124	C₂²×C₁₄₉	1	1	yes	characteristic, complemented
169	#123	—	2	1	no	—
170	C₂²⋊Q₈	C₂²×C₁₄₉	1	1	yes	characteristic
171	#121	C₂²×C₁₄₉	1	1	yes	characteristic
172	C₄×Q₈	C₂²×C₁₄₉	1	1	yes	characteristic, complemented
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	C₂²×C₁₄₉	1	1	yes	characteristic, complemented
175	#117	—	2	1	no	—
176	C₂×C₂²⋊C₄	—	2	1	no	—
177	C₂×C₂²⋊C₄	—	2	1	no	—
178	C₂×C₄²	C₂²×C₁₄₉	1	1	yes	characteristic
179	C₄⋊C₈	C₂²×C₁₄₉	1	2	yes	complemented
180	#103	C₂²×C₁₄₉	1	2	yes	complemented
181	#103	C₂²×C₁₄₉	1	2	yes	complemented
182	#102	C₂²×C₁₄₉	1	2	yes	—
183	#102	C₂²×C₁₄₉	1	2	yes	—
184	C₂²⋊C₈	C₂²×C₁₄₉	1	1	yes	characteristic
185	C₂²⋊C₈	C₂²×C₁₄₉	1	1	yes	characteristic
186	C₄×C₈	C₂²×C₁₄₉	1	2	yes	complemented
187	C₂⁴	—	2	1	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	C₂³×C₁₄₉	1	2	yes	—
191	C₂×Q₈	C₂³×C₁₄₉	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	C₂³×C₁₄₉	1	2	yes	—
196	C₂×D₄	C₂³×C₁₄₉	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	D₄×C₁₄₉	1	1	yes	characteristic
200	C₂²×C₄	D₄×C₁₄₉	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₁₄₉	1	1	yes	characteristic
202	Q₁₆	—	4	2	no	—
203	SD₁₆	—	4	2	no	—
204	SD₁₆	—	4	2	no	—
205	D₈	—	4	2	no	—
206	C₂×C₈	C₂³×C₁₄₉	1	2	yes	—
207	C₂×C₈	C₂³×C₁₄₉	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	C₂³×C₁₄₉	1	1	yes	characteristic
213	C₄⋊C₄	C₂³×C₁₄₉	1	1	yes	characteristic
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	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	C₄²	—	2	1	no	—
224	C₄²	D₄×C₁₄₉	1	1	yes	characteristic
225	C₄²	D₄×C₁₄₉	1	1	yes	characteristic
226	C₄²	C₂³×C₁₄₉	1	1	yes	characteristic
227	C₄²	C₂³×C₁₄₉	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×D₄×C₁₄₉	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	4	2	no	—
238	D₄	—	2	2	no	—
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
254	C₂×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
257	C₂×C₄	C₂⁴×C₁₄₉	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	8	4	no	—
269	C₂²	#135×C₁₄₉	1	2	yes	—
270	C₂²	C₂²×D₄×C₁₄₉	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	H(5,𝔽₂)×C₁₄₉	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	—	8	1	no	—
284	C₂	#535×C₁₄₉	1	1	yes	characteristic
285	C₂	C₂×#135×C₁₄₉	1	1	yes	characteristic
286	C₂	#575×C₁₄₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₄₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₄₉
prime radical	#447×C₁₄₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #529×C₁₄₉	C₂	semidirect
2. C₂×#124×C₁₄₉	C₂	semidirect
3. #495×C₁₄₉	C₂	semidirect
4. #488×C₁₄₉	C₂	semidirect
5. #464×C₁₄₉	C₂	semidirect
6. #463×C₁₄₉	C₂	semidirect
7. #451×C₁₄₉	C₂	semidirect
8. #450×C₁₄₉	C₂	semidirect
9. #448×C₁₄₉	C₂	semidirect
11. #431×C₁₄₉	C₂	semidirect
14. C₂×Q₁₆×C₁₄₉	C₂²	semidirect
16. C₂×SD₁₆×C₁₄₉	C₂²	semidirect
19. #125×C₁₄₉	C₂²	semidirect
24. #124×C₁₄₉	C₂²	semidirect
28. C₄×Q₈×C₁₄₉	C₂²	semidirect
30. C₄×D₄×C₁₄₉	C₂²	semidirect
35. C₄⋊C₈×C₁₄₉	C₂²	semidirect
36. #103×C₁₄₉	C₂²	semidirect
37. #103×C₁₄₉	C₂²	semidirect
42. C₄×C₈×C₁₄₉	C₂²	semidirect
143. C₁₄₉	#3211	direct (N × H)
144. #3211	C₁₄₉	direct (N × H)
145. #529	C₂×C₁₄₉	semidirect
146. C₂×#124	C₂×C₁₄₉	semidirect
147. #495	C₂×C₁₄₉	semidirect
148. #488	C₂×C₁₄₉	semidirect
149. #464	C₂×C₁₄₉	semidirect
150. #463	C₂×C₁₄₉	semidirect
151. #451	C₂×C₁₄₉	semidirect
152. #450	C₂×C₁₄₉	semidirect
153. #448	C₂×C₁₄₉	semidirect
155. #431	C₂×C₁₄₉	semidirect
158. C₂×Q₁₆	C₂²×C₁₄₉	semidirect
160. C₂×SD₁₆	C₂²×C₁₄₉	semidirect
163. #125	C₂²×C₁₄₉	semidirect
168. #124	C₂²×C₁₄₉	semidirect
172. C₄×Q₈	C₂²×C₁₄₉	semidirect
174. C₄×D₄	C₂²×C₁₄₉	semidirect
179. C₄⋊C₈	C₂²×C₁₄₉	semidirect
180. #103	C₂²×C₁₄₉	semidirect
181. #103	C₂²×C₁₄₉	semidirect
186. C₄×C₈	C₂²×C₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#529×C₁₄₉	2
C₂×#124×C₁₄₉	2
#495×C₁₄₉	2
#488×C₁₄₉	2
#464×C₁₄₉	2
#463×C₁₄₉	2
#451×C₁₄₉	2
#450×C₁₄₉	2
#448×C₁₄₉	2
#447×C₁₄₉	2
#431×C₁₄₉	2
#3211	149

Subgroups — #3211×C149 #34042124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3211×C₁₄₉ #34042124