Subgroups — C₁₄₉×#10728 #49189677

Order 38144.

1374 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#117×C₁₄₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#407×C₁₄₉	C₂	1	2	yes	maximal
3	C₂×#343×C₁₄₉	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₄₉	C₂²	1	4	yes	—
6	C₂×#117×C₁₄₉	—	2	4	no	—
7	C₂×#117×C₁₄₉	—	2	8	no	—
8	C₂²×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
9	C₂²×C₂²⋊C₄×C₁₄₉	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₄₉	C₄	1	2	yes	—
11	#407×C₁₄₉	—	2	4	no	—
12	#407×C₁₄₉	C₄	1	8	yes	complemented
13	C₂×C₂²⋊C₈×C₁₄₉	—	2	4	no	—
14	#343×C₁₄₉	—	2	8	no	—
15	C₂⁵×C₁₄₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₄₉	—	2	4	no	—
17	C₂³×C₄×C₁₄₉	—	2	4	no	—
18	C₂³×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₄₉	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₁₄₉	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₄₉	D₄	1	2	yes	—
22	C₂³×C₄×C₁₄₉	D₄	1	4	yes	—
23	C₂³×C₄×C₁₄₉	Q₈	1	2	yes	—
24	C₂³×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₁₄₉	—	2	4	no	—
26	C₂×M₁₆×C₁₄₉	—	2	8	no	—
27	C₂×M₁₆×C₁₄₉	—	2	8	no	—
28	C₂×M₁₆×C₁₄₉	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₁₄₉	—	2	4	no	—
30	#117×C₁₄₉	—	2	32	no	—
31	C₂×C₄⋊C₄×C₁₄₉	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₄₉	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₄₉	D₄	1	8	yes	—
34	C₂×C₂²⋊C₄×C₁₄₉	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₁₄₉	—	2	8	no	—
36	C₂×C₄²×C₁₄₉	—	2	8	no	—
37	C₂×C₄²×C₁₄₉	—	2	4	no	—
38	C₂²⋊C₈×C₁₄₉	—	2	16	no	—
39	C₂⁴×C₁₄₉	—	2	2	no	—
40	C₂⁴×C₁₄₉	—	2	2	no	—
41	C₂⁴×C₁₄₉	—	4	4	no	—
42	C₂⁴×C₁₄₉	C₄²	1	2	yes	—
43	C₂⁴×C₁₄₉	C₂²⋊C₄	1	2	yes	—
44	C₂⁴×C₁₄₉	C₄⋊C₄	1	2	yes	—
45	C₂⁴×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₄₉	—	2	2	no	—
47	C₂²×C₄×C₁₄₉	—	2	2	no	—
48	C₂²×C₄×C₁₄₉	—	2	16	no	—
49	C₂²×C₄×C₁₄₉	—	2	8	no	—
50	C₂²×C₄×C₁₄₉	—	2	8	no	—
51	C₂²×C₄×C₁₄₉	—	2	8	no	—
52	C₂²×C₄×C₁₄₉	—	2	8	no	—
53	C₂²×C₄×C₁₄₉	—	2	4	no	—
54	C₂²×C₄×C₁₄₉	—	2	4	no	—
55	C₂²×C₄×C₁₄₉	—	4	8	no	—
56	C₂²×C₄×C₁₄₉	C₄²	1	2	yes	—
57	C₂²×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₄₉	C₂²⋊C₄	1	8	yes	—
59	C₂²×C₄×C₁₄₉	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₁₄₉	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₄₉	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₄₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₄₉	#41	1	4	yes	—
68	M₁₆×C₁₄₉	—	2	16	no	—
69	M₁₆×C₁₄₉	—	4	8	no	—
70	C₂×C₈×C₁₄₉	—	2	8	no	—
71	C₂×C₈×C₁₄₉	—	4	4	no	—
72	C₂×C₈×C₁₄₉	—	4	4	no	—
73	C₄⋊C₄×C₁₄₉	—	2	16	no	—
74	C₂²⋊C₄×C₁₄₉	—	2	16	no	—
75	C₄²×C₁₄₉	—	2	16	no	—
76	C₂³×C₁₄₉	—	2	4	no	—
77	C₂³×C₁₄₉	—	2	4	no	—
78	C₂³×C₁₄₉	—	2	2	no	—
79	C₂³×C₁₄₉	—	2	4	no	—
80	C₂³×C₁₄₉	—	2	2	no	—
81	C₂³×C₁₄₉	—	2	8	no	—
82	C₂³×C₁₄₉	—	4	4	no	—
83	C₂³×C₁₄₉	—	4	4	no	—
84	C₂³×C₁₄₉	—	4	8	no	—
85	C₂³×C₁₄₉	—	4	4	no	—
86	C₂³×C₁₄₉	—	4	4	no	—
87	C₂³×C₁₄₉	#95	1	4	yes	—
88	C₂³×C₁₄₉	C₂×C₄²	1	1	yes	characteristic
89	C₂³×C₁₄₉	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂³×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₁₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₄₉	#117	1	2	yes	—
93	C₂×C₄×C₁₄₉	—	2	4	no	—
94	C₂×C₄×C₁₄₉	—	2	2	no	—
95	C₂×C₄×C₁₄₉	—	2	4	no	—
96	C₂×C₄×C₁₄₉	—	2	2	no	—
97	C₂×C₄×C₁₄₉	—	2	16	no	—
98	C₂×C₄×C₁₄₉	—	2	16	no	—
99	C₂×C₄×C₁₄₉	—	2	8	no	—
100	C₂×C₄×C₁₄₉	—	4	8	no	—
101	C₂×C₄×C₁₄₉	—	4	8	no	—
102	C₂×C₄×C₁₄₉	—	4	8	no	—
103	C₂×C₄×C₁₄₉	#95	1	4	yes	—
104	C₂×C₄×C₁₄₉	C₂×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₄₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₄₉	C₄×D₄	1	8	yes	—
107	C₂×C₄×C₁₄₉	#120	1	4	yes	—
108	C₂×C₄×C₁₄₉	#121	1	4	yes	—
109	C₂×C₄×C₁₄₉	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₁₄₉	#123	1	4	yes	—
111	C₈×C₁₄₉	—	4	8	no	—
112	C₂²×C₁₄₉	—	2	4	no	—
113	C₂²×C₁₄₉	—	2	2	no	—
114	C₂²×C₁₄₉	—	2	2	no	—
115	C₂²×C₁₄₉	—	2	2	no	—
116	C₂²×C₁₄₉	—	2	2	no	—
117	C₂²×C₁₄₉	—	2	4	no	—
118	C₂²×C₁₄₉	—	2	8	no	—
119	C₂²×C₁₄₉	—	4	8	no	—
120	C₂²×C₁₄₉	—	4	8	no	—
121	C₂²×C₁₄₉	—	4	4	no	—
122	C₂²×C₁₄₉	—	4	4	no	—
123	C₂²×C₁₄₉	C₂×#95	1	1	yes	characteristic
124	C₂²×C₁₄₉	C₄×C₂²⋊C₄	1	2	yes	—
125	C₂²×C₁₄₉	#379	1	1	yes	characteristic
126	C₂²×C₁₄₉	#380	1	2	yes	—
127	C₂²×C₁₄₉	#381	1	1	yes	characteristic
128	C₂²×C₁₄₉	#413	1	4	yes	—
129	C₄×C₁₄₉	—	2	4	no	—
130	C₄×C₁₄₉	—	4	8	no	—
131	C₄×C₁₄₉	#385	1	4	yes	—
132	C₄×C₁₄₉	#386	1	4	yes	—
133	C₂×C₁₄₉	—	2	2	no	—
134	C₂×C₁₄₉	—	2	2	no	—
135	C₂×C₁₄₉	—	4	4	no	—
136	C₂×C₁₄₉	C₂²⋊#95	1	1	yes	characteristic
137	C₂×C₁₄₉	#1742	1	2	yes	—
138	C₂×C₁₄₉	#1785	1	2	yes	—
139	C₂×C₁₄₉	#1822	1	2	yes	—
140	#10728	C₁₄₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₄₉	#10728	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×#117	C₂×C₁₄₉	1	1	yes	characteristic
143	C₂×#407	C₂×C₁₄₉	1	2	yes	—
144	C₂×#343	C₂×C₁₄₉	1	4	yes	complemented
145	C₂⁴×C₄	C₂²×C₁₄₉	1	1	yes	characteristic
146	C₂²×M₁₆	C₂²×C₁₄₉	1	4	yes	—
147	C₂×#117	—	2	4	no	—
148	C₂×#117	—	2	8	no	—
149	C₂²×C₄⋊C₄	C₂²×C₁₄₉	1	2	yes	—
150	C₂²×C₂²⋊C₄	C₄×C₁₄₉	1	2	yes	—
151	C₂²×C₄²	C₄×C₁₄₉	1	2	yes	—
152	#407	—	2	4	no	—
153	#407	C₄×C₁₄₉	1	8	yes	complemented
154	C₂×C₂²⋊C₈	—	2	4	no	—
155	#343	—	2	8	no	—
156	C₂⁵	C₂×C₄×C₁₄₉	1	1	yes	characteristic
157	C₂³×C₄	—	2	4	no	—
158	C₂³×C₄	—	2	4	no	—
159	C₂³×C₄	C₂×C₄×C₁₄₉	1	1	yes	characteristic
160	C₂³×C₄	C₂×C₄×C₁₄₉	1	4	yes	—
161	C₂³×C₄	C₂×C₄×C₁₄₉	1	4	yes	—
162	C₂³×C₄	D₄×C₁₄₉	1	2	yes	—
163	C₂³×C₄	D₄×C₁₄₉	1	4	yes	—
164	C₂³×C₄	Q₈×C₁₄₉	1	2	yes	—
165	C₂³×C₄	C₂³×C₁₄₉	1	1	yes	characteristic
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	8	no	—
168	C₂×M₁₆	—	2	8	no	—
169	C₂×M₁₆	C₂×C₄×C₁₄₉	1	8	yes	—
170	C₂²×C₈	—	2	4	no	—
171	#117	—	2	32	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	D₄×C₁₄₉	1	8	yes	—
175	C₂×C₂²⋊C₄	—	2	4	no	—
176	C₂×C₂²⋊C₄	—	2	8	no	—
177	C₂×C₄²	—	2	8	no	—
178	C₂×C₄²	—	2	4	no	—
179	C₂²⋊C₈	—	2	16	no	—
180	C₂⁴	—	2	2	no	—
181	C₂⁴	—	2	2	no	—
182	C₂⁴	—	4	4	no	—
183	C₂⁴	C₄²×C₁₄₉	1	2	yes	—
184	C₂⁴	C₂²⋊C₄×C₁₄₉	1	2	yes	—
185	C₂⁴	C₄⋊C₄×C₁₄₉	1	2	yes	—
186	C₂⁴	C₂²×C₄×C₁₄₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	16	no	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	C₄²×C₁₄₉	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₁₄₉	1	2	yes	—
199	C₂²×C₄	C₂²⋊C₄×C₁₄₉	1	8	yes	—
200	C₂²×C₄	C₂²⋊C₄×C₁₄₉	1	8	yes	—
201	C₂²×C₄	C₄⋊C₄×C₁₄₉	1	2	yes	—
202	C₂²×C₄	C₄⋊C₄×C₁₄₉	1	8	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₄₉	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₁₄₉	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₄₉	1	4	yes	—
207	C₂²×C₄	C₂×Q₈×C₁₄₉	1	1	yes	characteristic
208	C₂²×C₄	#41×C₁₄₉	1	4	yes	—
209	M₁₆	—	2	16	no	—
210	M₁₆	—	4	8	no	—
211	C₂×C₈	—	2	8	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₄⋊C₄	—	2	16	no	—
215	C₂²⋊C₄	—	2	16	no	—
216	C₄²	—	2	16	no	—
217	C₂³	—	2	4	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	8	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	8	no	—
228	C₂³	#95×C₁₄₉	1	4	yes	—
229	C₂³	C₂×C₄²×C₁₄₉	1	1	yes	characteristic
230	C₂³	C₂×C₂²⋊C₄×C₁₄₉	1	2	yes	—
231	C₂³	C₂×C₂²⋊C₄×C₁₄₉	1	1	yes	characteristic
232	C₂³	C₂×C₄⋊C₄×C₁₄₉	1	1	yes	characteristic
233	C₂³	#117×C₁₄₉	1	2	yes	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	16	no	—
239	C₂×C₄	—	2	16	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	#95×C₁₄₉	1	4	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₁₄₉	1	2	yes	—
246	C₂×C₄	C₂×C₄⋊C₄×C₁₄₉	1	2	yes	—
247	C₂×C₄	C₄×D₄×C₁₄₉	1	8	yes	—
248	C₂×C₄	#120×C₁₄₉	1	4	yes	—
249	C₂×C₄	#121×C₁₄₉	1	4	yes	—
250	C₂×C₄	C₂²⋊Q₈×C₁₄₉	1	4	yes	—
251	C₂×C₄	#123×C₁₄₉	1	4	yes	—
252	C₈	—	4	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	4	8	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	C₂×#95×C₁₄₉	1	1	yes	characteristic
265	C₂²	C₄×C₂²⋊C₄×C₁₄₉	1	2	yes	—
266	C₂²	#379×C₁₄₉	1	1	yes	characteristic
267	C₂²	#380×C₁₄₉	1	2	yes	—
268	C₂²	#381×C₁₄₉	1	1	yes	characteristic
269	C₂²	#413×C₁₄₉	1	4	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	4	8	no	—
272	C₄	#385×C₁₄₉	1	4	yes	—
273	C₄	#386×C₁₄₉	1	4	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	2	2	no	—
276	C₂	—	4	4	no	—
277	C₂	C₂²⋊#95×C₁₄₉	1	1	yes	characteristic
278	C₂	#1742×C₁₄₉	1	2	yes	—
279	C₂	#1785×C₁₄₉	1	2	yes	—
280	C₂	#1822×C₁₄₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₄₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₁₄₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₄₉
prime radical	C₂⁵×C₁₄₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₄₉ ◁ C₂⁴×C₄×C₁₄₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#343×C₁₄₉	C₂	semidirect
12. #407×C₁₄₉	C₄	semidirect
140. #10728	C₁₄₉	direct (N × H)
141. C₁₄₉	#10728	direct (N × H)
144. C₂×#343	C₂×C₁₄₉	semidirect
153. #407	C₄×C₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#117×C₁₄₉	2
C₂×#407×C₁₄₉	2
C₂×#343×C₁₄₉	2
#10728	149

Subgroups — C149×#10728 #49189677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₄₉×#10728 #49189677