Subgroups — C₁₄₉×#18410 #49197359

Order 38144.

906 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3502×C₁₄₉	C₂	1	1	yes	maximal, characteristic
2	#3229×C₁₄₉	C₂	1	2	yes	maximal
3	#2546×C₁₄₉	C₂	1	1	yes	maximal, characteristic
4	#1611×C₁₄₉	C₂	1	8	yes	maximal
5	#1503×C₁₄₉	C₂	1	2	yes	maximal
6	#1461×C₁₄₉	C₂	1	1	yes	maximal, characteristic
7	Q₈²×C₁₄₉	—	2	2	no	—
8	#557×C₁₄₉	—	2	2	no	—
9	#557×C₁₄₉	—	2	4	no	—
10	#544×C₁₄₉	—	2	1	no	—
11	#544×C₁₄₉	C₂²	1	4	yes	—
12	#533×C₁₄₉	—	2	2	no	—
13	#533×C₁₄₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₁₄₉	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
16	#519×C₁₄₉	—	2	1	no	—
17	C₂×C₄×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
18	#498×C₁₄₉	—	2	2	no	—
19	#487×C₁₄₉	—	2	2	no	—
20	#483×C₁₄₉	C₂²	1	8	yes	—
21	#475×C₁₄₉	—	2	4	no	—
22	#431×C₁₄₉	C₂²	1	2	yes	—
23	#398×C₁₄₉	C₂²	1	4	yes	—
24	#391×C₁₄₉	—	2	1	no	—
25	#389×C₁₄₉	—	2	2	no	—
26	C₄×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
27	#328×C₁₄₉	C₂²	1	8	yes	—
28	#324×C₁₄₉	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₄₉	—	2	1	no	—
30	C₂²×Q₈×C₁₄₉	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
32	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
33	C₄⋊Q₈×C₁₄₉	—	2	4	no	—
34	C₄⋊Q₈×C₁₄₉	—	2	8	no	—
35	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
36	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
37	C₄⋊Q₈×C₁₄₉	—	4	4	no	—
38	C₄⋊Q₈×C₁₄₉	D₄	1	2	yes	—
39	C₄⋊Q₈×C₁₄₉	D₄	1	2	yes	—
40	#125×C₁₄₉	—	2	4	no	—
41	#125×C₁₄₉	—	2	4	no	—
42	#125×C₁₄₉	—	2	8	no	—
43	#125×C₁₄₉	—	2	2	no	—
44	C₂²⋊Q₈×C₁₄₉	—	2	2	no	—
45	C₂²⋊Q₈×C₁₄₉	—	2	4	no	—
46	C₂²⋊Q₈×C₁₄₉	C₂³	1	4	yes	—
47	C₄×Q₈×C₁₄₉	—	2	4	no	—
48	C₄×Q₈×C₁₄₉	—	2	4	no	—
49	C₄×Q₈×C₁₄₉	—	2	4	no	—
50	C₄×Q₈×C₁₄₉	—	2	4	no	—
51	C₄×Q₈×C₁₄₉	—	2	2	no	—
52	C₄×Q₈×C₁₄₉	—	2	2	no	—
53	C₄×Q₈×C₁₄₉	—	2	2	no	—
54	C₄×Q₈×C₁₄₉	D₄	1	4	yes	—
55	#117×C₁₄₉	—	2	4	no	—
56	#117×C₁₄₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₄₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₄₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₄₉	—	2	4	no	—
60	C₂×C₄⋊C₄×C₁₄₉	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₁₄₉	—	2	1	no	—
64	C₂×C₄²×C₁₄₉	D₄	1	2	yes	—
65	C₂×C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
66	#106×C₁₄₉	—	2	8	no	—
67	C₄⋊C₈×C₁₄₉	—	2	2	no	—
68	#103×C₁₄₉	—	2	8	no	—
69	C₂²⋊C₈×C₁₄₉	C₂³	1	4	yes	—
70	C₄×C₈×C₁₄₉	—	2	2	no	—
71	#95×C₁₄₉	—	2	4	no	—
72	#95×C₁₄₉	C₂³	1	2	yes	—
73	C₂×Q₈×C₁₄₉	—	2	4	no	—
74	C₂×Q₈×C₁₄₉	—	2	2	no	—
75	C₂×Q₈×C₁₄₉	—	2	2	no	—
76	C₂×Q₈×C₁₄₉	—	2	2	no	—
77	C₂×Q₈×C₁₄₉	—	4	2	no	—
78	C₂×Q₈×C₁₄₉	—	4	2	no	—
79	C₂×Q₈×C₁₄₉	—	4	2	no	—
80	C₂×Q₈×C₁₄₉	SD₁₆	1	4	yes	—
81	C₂×Q₈×C₁₄₉	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₁₄₉	—	2	2	no	—
83	C₂²×C₄×C₁₄₉	—	2	2	no	—
84	C₂²×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₁₄₉	—	2	4	no	—
90	C₄⋊C₄×C₁₄₉	—	2	4	no	—
91	C₄⋊C₄×C₁₄₉	—	2	4	no	—
92	C₄⋊C₄×C₁₄₉	—	2	4	no	—
93	C₄⋊C₄×C₁₄₉	—	2	4	no	—
94	C₄⋊C₄×C₁₄₉	—	2	8	no	—
95	C₄⋊C₄×C₁₄₉	—	2	2	no	—
96	C₄⋊C₄×C₁₄₉	—	2	2	no	—
97	C₄⋊C₄×C₁₄₉	—	2	2	no	—
98	C₄⋊C₄×C₁₄₉	—	2	2	no	—
99	C₄⋊C₄×C₁₄₉	—	2	4	no	—
100	C₄⋊C₄×C₁₄₉	—	2	4	no	—
101	C₄⋊C₄×C₁₄₉	—	2	4	no	—
102	C₄⋊C₄×C₁₄₉	—	2	4	no	—
103	C₄⋊C₄×C₁₄₉	—	4	1	no	—
104	C₄⋊C₄×C₁₄₉	—	4	1	no	—
105	C₄⋊C₄×C₁₄₉	—	4	8	no	—
106	C₄⋊C₄×C₁₄₉	C₂×D₄	1	4	yes	—
107	C₂²⋊C₄×C₁₄₉	—	2	4	no	—
108	C₄²×C₁₄₉	—	2	8	no	—
109	C₄²×C₁₄₉	—	2	2	no	—
110	C₄²×C₁₄₉	—	2	2	no	—
111	C₄²×C₁₄₉	—	2	4	no	—
112	C₄²×C₁₄₉	—	4	2	no	—
113	C₄²×C₁₄₉	C₂×D₄	1	2	yes	—
114	C₄²×C₁₄₉	C₂×D₄	1	2	yes	—
115	C₂³×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₁₄₉	—	2	8	no	—
117	Q₈×C₁₄₉	—	4	4	no	—
118	Q₈×C₁₄₉	—	4	4	no	—
119	Q₈×C₁₄₉	—	8	4	no	—
120	Q₈×C₁₄₉	—	8	4	no	—
121	C₂×C₄×C₁₄₉	—	2	2	no	—
122	C₂×C₄×C₁₄₉	—	2	2	no	—
123	C₂×C₄×C₁₄₉	—	2	2	no	—
124	C₂×C₄×C₁₄₉	—	2	4	no	—
125	C₂×C₄×C₁₄₉	—	2	4	no	—
126	C₂×C₄×C₁₄₉	—	2	4	no	—
127	C₂×C₄×C₁₄₉	—	2	2	no	—
128	C₂×C₄×C₁₄₉	—	4	1	no	—
129	C₂×C₄×C₁₄₉	—	4	2	no	—
130	C₂×C₄×C₁₄₉	—	4	2	no	—
131	C₂×C₄×C₁₄₉	—	4	2	no	—
132	C₂×C₄×C₁₄₉	—	4	2	no	—
133	C₂×C₄×C₁₄₉	—	4	2	no	—
134	C₂×C₄×C₁₄₉	—	4	2	no	—
135	C₂×C₄×C₁₄₉	#120	1	1	yes	characteristic
136	C₂×C₄×C₁₄₉	#120	1	2	yes	—
137	C₂×C₄×C₁₄₉	#120	1	1	yes	characteristic
138	C₂×C₄×C₁₄₉	C₂×SD₁₆	1	4	yes	—
139	C₂×C₄×C₁₄₉	C₂×SD₁₆	1	2	yes	—
140	C₂×C₄×C₁₄₉	#137	1	2	yes	—
141	C₂×C₄×C₁₄₉	C₂²×D₄	1	2	yes	—
142	C₈×C₁₄₉	—	4	4	no	—
143	C₂²×C₁₄₉	—	2	1	no	—
144	C₂²×C₁₄₉	—	2	1	no	—
145	C₂²×C₁₄₉	C₂×#120	1	1	yes	characteristic
146	C₂²×C₁₄₉	C₂²×SD₁₆	1	1	yes	characteristic
147	C₂²×C₁₄₉	C₂×#137	1	1	yes	characteristic
148	C₄×C₁₄₉	—	2	4	no	—
149	C₄×C₁₄₉	—	2	1	no	—
150	C₄×C₁₄₉	—	2	2	no	—
151	C₄×C₁₄₉	—	4	4	no	—
152	C₄×C₁₄₉	—	4	4	no	—
153	C₄×C₁₄₉	—	4	2	no	—
154	C₄×C₁₄₉	—	4	4	no	—
155	C₄×C₁₄₉	—	8	2	no	—
156	C₄×C₁₄₉	#448	1	4	yes	—
157	C₄×C₁₄₉	#456	1	2	yes	—
158	C₂×C₁₄₉	—	2	1	no	—
159	C₂×C₁₄₉	—	2	1	no	—
160	C₂×C₁₄₉	C₂×#448	1	1	yes	characteristic
161	C₂×C₁₄₉	C₂×#456	1	1	yes	characteristic
162	C₂×C₁₄₉	#3028	1	1	yes	characteristic
163	#18410	C₁₄₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₁₄₉	#18410	1	1	yes	characteristic, direct product factor, Sylow
165	#3502	C₂×C₁₄₉	1	1	yes	characteristic
166	#3229	C₂×C₁₄₉	1	2	yes	—
167	#2546	C₂×C₁₄₉	1	1	yes	characteristic
168	#1611	C₂×C₁₄₉	1	8	yes	—
169	#1503	C₂×C₁₄₉	1	2	yes	—
170	#1461	C₂×C₁₄₉	1	1	yes	characteristic
171	Q₈²	—	2	2	no	—
172	#557	—	2	2	no	—
173	#557	—	2	4	no	—
174	#544	—	2	1	no	—
175	#544	C₂²×C₁₄₉	1	4	yes	—
176	#533	—	2	2	no	—
177	#533	C₂²×C₁₄₉	1	2	yes	—
178	C₂×C₄⋊Q₈	—	2	1	no	—
179	C₂×C₄⋊Q₈	C₂²×C₁₄₉	1	1	yes	characteristic
180	#519	—	2	1	no	—
181	C₂×C₄×Q₈	C₂²×C₁₄₉	1	1	yes	characteristic
182	#498	—	2	2	no	—
183	#487	—	2	2	no	—
184	#483	C₂²×C₁₄₉	1	8	yes	—
185	#475	—	2	4	no	—
186	#431	C₂²×C₁₄₉	1	2	yes	—
187	#398	C₂²×C₁₄₉	1	4	yes	—
188	#391	—	2	1	no	—
189	#389	—	2	2	no	—
190	C₄×C₄⋊C₄	C₂²×C₁₄₉	1	1	yes	characteristic
191	#328	C₂²×C₁₄₉	1	8	yes	—
192	#324	C₂²×C₁₄₉	1	4	yes	—
193	C₂²×Q₈	—	2	1	no	—
194	C₂²×Q₈	D₄×C₁₄₉	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	2	4	no	—
198	C₄⋊Q₈	—	2	8	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	4	4	no	—
202	C₄⋊Q₈	D₄×C₁₄₉	1	2	yes	—
203	C₄⋊Q₈	D₄×C₁₄₉	1	2	yes	—
204	#125	—	2	4	no	—
205	#125	—	2	4	no	—
206	#125	—	2	8	no	—
207	#125	—	2	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	2	4	no	—
210	C₂²⋊Q₈	C₂³×C₁₄₉	1	4	yes	—
211	C₄×Q₈	—	2	4	no	—
212	C₄×Q₈	—	2	4	no	—
213	C₄×Q₈	—	2	4	no	—
214	C₄×Q₈	—	2	4	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	D₄×C₁₄₉	1	4	yes	—
219	#117	—	2	4	no	—
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	4	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	D₄×C₁₄₉	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂³×C₁₄₉	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₁₄₉	1	2	yes	—
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	D₄×C₁₄₉	1	2	yes	—
229	C₂×C₄²	C₂³×C₁₄₉	1	1	yes	characteristic
230	#106	—	2	8	no	—
231	C₄⋊C₈	—	2	2	no	—
232	#103	—	2	8	no	—
233	C₂²⋊C₈	C₂³×C₁₄₉	1	4	yes	—
234	C₄×C₈	—	2	2	no	—
235	#95	—	2	4	no	—
236	#95	C₂³×C₁₄₉	1	2	yes	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	SD₁₆×C₁₄₉	1	4	yes	—
245	C₂×Q₈	C₂×D₄×C₁₄₉	1	4	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₄₉	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₄₉	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
252	C₂²×C₄	C₂⁴×C₁₄₉	1	1	yes	characteristic
253	C₂×C₈	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	8	no	—
270	C₄⋊C₄	C₂×D₄×C₁₄₉	1	4	yes	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₄²	—	2	8	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	4	no	—
276	C₄²	—	4	2	no	—
277	C₄²	C₂×D₄×C₁₄₉	1	2	yes	—
278	C₄²	C₂×D₄×C₁₄₉	1	2	yes	—
279	C₂³	C₂²×D₄×C₁₄₉	1	1	yes	characteristic
280	Q₈	—	2	8	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	4	4	no	—
283	Q₈	—	8	4	no	—
284	Q₈	—	8	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	#120×C₁₄₉	1	1	yes	characteristic
300	C₂×C₄	#120×C₁₄₉	1	2	yes	—
301	C₂×C₄	#120×C₁₄₉	1	1	yes	characteristic
302	C₂×C₄	C₂×SD₁₆×C₁₄₉	1	4	yes	—
303	C₂×C₄	C₂×SD₁₆×C₁₄₉	1	2	yes	—
304	C₂×C₄	#137×C₁₄₉	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₁₄₉	1	2	yes	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	C₂×#120×C₁₄₉	1	1	yes	characteristic
310	C₂²	C₂²×SD₁₆×C₁₄₉	1	1	yes	characteristic
311	C₂²	C₂×#137×C₁₄₉	1	1	yes	characteristic
312	C₄	—	2	4	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	4	no	—
319	C₄	—	8	2	no	—
320	C₄	#448×C₁₄₉	1	4	yes	—
321	C₄	#456×C₁₄₉	1	2	yes	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	C₂×#448×C₁₄₉	1	1	yes	characteristic
325	C₂	C₂×#456×C₁₄₉	1	1	yes	characteristic
326	C₂	#3028×C₁₄₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₄₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₄₉
prime radical	C₂³×C₁₄₉	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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
163. #18410	C₁₄₉	direct (N × H)
164. C₁₄₉	#18410	direct (N × H)

Maximal subgroups

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

subgroup	index
#3502×C₁₄₉	2
#3229×C₁₄₉	2
#2546×C₁₄₉	2
#1611×C₁₄₉	2
#1503×C₁₄₉	2
#1461×C₁₄₉	2
#18410	149

Subgroups — C149×#18410 #49197359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₄₉×#18410 #49197359