Subgroups — C₂×#1675×C₁₄₉ #49198053

Order 38144.

726 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#476×C₁₄₉	C₂	1	2	yes	maximal
3	C₂×C₈×D₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic
4	#1675×C₁₄₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×#335×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#333×C₁₄₉	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₁₄₉	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₁₄₉	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
10	#500×C₁₄₉	—	2	2	no	—
11	#500×C₁₄₉	C₂²	1	4	yes	complemented
12	#476×C₁₄₉	—	2	4	no	—
13	#476×C₁₄₉	C₂²	1	8	yes	—
14	C₈×D₄×C₁₄₉	—	2	2	no	—
15	C₈×D₄×C₁₄₉	C₂²	1	4	yes	—
16	C₂×#107×C₁₄₉	—	2	1	no	—
17	C₂×#107×C₁₄₉	—	2	1	no	—
18	C₂×#106×C₁₄₉	—	2	2	no	—
19	C₂×C₄⋊C₈×C₁₄₉	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₄₉	C₂²	1	1	yes	characteristic
21	C₂×#102×C₁₄₉	—	2	2	no	—
22	C₂×C₂²⋊C₈×C₁₄₉	—	2	1	no	—
23	C₂×C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
24	#335×C₁₄₉	C₂²	1	4	yes	complemented
25	#333×C₁₄₉	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₁₄₉	—	2	2	no	—
27	C₂²×D₄×C₁₄₉	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₄₉	—	2	1	no	—
29	C₂²×C₈×C₁₄₉	—	2	1	no	—
30	C₂²×C₈×C₁₄₉	—	2	2	no	—
31	C₂²×C₈×C₁₄₉	—	2	4	no	—
32	C₂²×C₈×C₁₄₉	—	4	4	no	—
33	C₂²×C₈×C₁₄₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₄₉	D₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
36	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
37	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
38	C₄⋊Q₈×C₁₄₉	C₂³	1	4	yes	—
39	C₄×D₄×C₁₄₉	—	2	1	no	—
40	C₄×D₄×C₁₄₉	—	2	1	no	—
41	C₄×D₄×C₁₄₉	D₄	1	2	yes	—
42	C₄×D₄×C₁₄₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₄₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₄₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₄₉	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₄₉	—	2	1	no	—
47	C₂×C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
48	#107×C₁₄₉	—	2	4	no	—
49	#107×C₁₄₉	—	2	4	no	—
50	#106×C₁₄₉	—	2	8	no	—
51	C₄⋊C₈×C₁₄₉	—	2	1	no	—
52	C₄⋊C₈×C₁₄₉	—	2	2	no	—
53	C₄⋊C₈×C₁₄₉	C₂³	1	2	yes	—
54	C₄⋊C₈×C₁₄₉	C₂³	1	4	yes	—
55	#102×C₁₄₉	—	2	8	no	—
56	C₂²⋊C₈×C₁₄₉	—	2	4	no	—
57	C₄×C₈×C₁₄₉	D₄	1	2	yes	—
58	C₄×C₈×C₁₄₉	C₂³	1	2	yes	—
59	C₂⁴×C₁₄₉	—	2	1	no	—
60	C₂×Q₈×C₁₄₉	—	2	4	no	—
61	C₂×Q₈×C₁₄₉	—	4	2	no	—
62	C₂×Q₈×C₁₄₉	—	8	4	no	—
63	C₂×D₄×C₁₄₉	—	2	1	no	—
64	C₂×D₄×C₁₄₉	—	2	2	no	—
65	C₂×D₄×C₁₄₉	Q₁₆	1	4	yes	—
66	C₂×D₄×C₁₄₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₄₉	—	2	1	no	—
68	C₂²×C₄×C₁₄₉	—	2	2	no	—
69	C₂²×C₄×C₁₄₉	—	2	2	no	—
70	C₂²×C₄×C₁₄₉	—	4	2	no	—
71	C₂²×C₄×C₁₄₉	—	4	2	no	—
72	C₂²×C₄×C₁₄₉	—	4	2	no	—
73	C₂²×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₄₉	#41	1	1	yes	characteristic
76	C₂×C₈×C₁₄₉	—	2	2	no	—
77	C₂×C₈×C₁₄₉	—	2	2	no	—
78	C₂×C₈×C₁₄₉	—	2	4	no	—
79	C₂×C₈×C₁₄₉	—	4	2	no	—
80	C₂×C₈×C₁₄₉	—	4	4	no	—
81	C₂×C₈×C₁₄₉	—	4	8	no	—
82	C₂×C₈×C₁₄₉	SD₁₆	1	4	yes	—
83	C₂×C₈×C₁₄₉	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₄₉	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₁₄₉	—	2	1	no	—
86	C₄⋊C₄×C₁₄₉	—	2	4	no	—
87	C₄⋊C₄×C₁₄₉	—	2	4	no	—
88	C₄⋊C₄×C₁₄₉	—	4	2	no	—
89	C₄⋊C₄×C₁₄₉	—	4	2	no	—
90	C₄⋊C₄×C₁₄₉	C₂×D₄	1	2	yes	—
91	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
92	C₂²⋊C₄×C₁₄₉	—	2	2	no	—
93	C₄²×C₁₄₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₁₄₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₄₉	#41	1	1	yes	characteristic
96	C₄²×C₁₄₉	C₂⁴	1	1	yes	characteristic
97	C₂³×C₁₄₉	—	2	2	no	—
98	C₂³×C₁₄₉	—	2	4	no	—
99	C₂³×C₁₄₉	—	4	2	no	—
100	C₂³×C₁₄₉	—	4	2	no	—
101	C₂³×C₁₄₉	#121	1	1	yes	characteristic
102	Q₈×C₁₄₉	—	8	4	no	—
103	Q₈×C₁₄₉	—	8	4	no	—
104	D₄×C₁₄₉	—	2	2	no	—
105	D₄×C₁₄₉	—	2	4	no	—
106	D₄×C₁₄₉	C₂×Q₁₆	1	4	yes	—
107	C₂×C₄×C₁₄₉	—	2	2	no	—
108	C₂×C₄×C₁₄₉	—	2	2	no	—
109	C₂×C₄×C₁₄₉	—	4	2	no	—
110	C₂×C₄×C₁₄₉	—	4	2	no	—
111	C₂×C₄×C₁₄₉	—	4	2	no	—
112	C₂×C₄×C₁₄₉	—	4	4	no	—
113	C₂×C₄×C₁₄₉	—	4	4	no	—
114	C₂×C₄×C₁₄₉	—	8	4	no	—
115	C₂×C₄×C₁₄₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₄₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₄₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₄₉	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₁₄₉	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₁₄₉	#135	1	2	yes	—
121	C₂×C₄×C₁₄₉	#137	1	2	yes	—
122	C₂×C₄×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₄₉	C₂×#41	1	1	yes	characteristic
125	C₈×C₁₄₉	—	2	2	no	—
126	C₈×C₁₄₉	—	4	2	no	—
127	C₈×C₁₄₉	—	4	4	no	—
128	C₈×C₁₄₉	C₂×SD₁₆	1	4	yes	—
129	C₂²×C₁₄₉	—	2	4	no	—
130	C₂²×C₁₄₉	—	4	8	no	—
131	C₂²×C₁₄₉	—	4	2	no	—
132	C₂²×C₁₄₉	—	4	2	no	—
133	C₂²×C₁₄₉	#461	1	2	yes	—
134	C₂²×C₁₄₉	#467	1	2	yes	—
135	C₂²×C₁₄₉	#472	1	2	yes	—
136	C₂²×C₁₄₉	C₂×#121	1	1	yes	characteristic
137	C₄×C₁₄₉	—	2	1	no	—
138	C₄×C₁₄₉	—	2	1	no	—
139	C₄×C₁₄₉	—	4	2	no	—
140	C₄×C₁₄₉	—	8	4	no	—
141	C₄×C₁₄₉	#461	1	2	yes	—
142	C₄×C₁₄₉	#467	1	2	yes	—
143	C₄×C₁₄₉	C₂²×SD₁₆	1	1	yes	characteristic
144	C₄×C₁₄₉	C₂²×Q₁₆	1	1	yes	characteristic
145	C₄×C₁₄₉	C₂×#135	1	1	yes	characteristic
146	C₄×C₁₄₉	C₂×#137	1	1	yes	characteristic
147	C₂×C₁₄₉	—	4	4	no	—
148	C₂×C₁₄₉	#1675	1	4	yes	direct product factor
149	C₂×C₁₄₉	C₂×#461	1	1	yes	characteristic
150	C₂×C₁₄₉	C₂×#467	1	1	yes	characteristic
151	C₂×C₁₄₉	C₂×#472	1	1	yes	characteristic
152	C₂×#1675	C₁₄₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₁₄₉	C₂×#1675	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×#500	C₂×C₁₄₉	1	1	yes	characteristic, complemented
155	C₂×#476	C₂×C₁₄₉	1	2	yes	—
156	C₂×C₈×D₄	C₂×C₁₄₉	1	1	yes	characteristic
157	#1675	C₂×C₁₄₉	1	8	yes	direct product factor
158	C₂×#335	C₂×C₁₄₉	1	1	yes	characteristic, complemented
159	C₂×#333	C₂×C₁₄₉	1	2	yes	complemented
160	C₂³×C₈	—	2	1	no	—
161	C₂×C₄⋊Q₈	C₂²×C₁₄₉	1	2	yes	—
162	C₂×C₄×D₄	C₂²×C₁₄₉	1	1	yes	characteristic
163	#500	—	2	2	no	—
164	#500	C₂²×C₁₄₉	1	4	yes	complemented
165	#476	—	2	4	no	—
166	#476	C₂²×C₁₄₉	1	8	yes	—
167	C₈×D₄	—	2	2	no	—
168	C₈×D₄	C₂²×C₁₄₉	1	4	yes	—
169	C₂×#107	—	2	1	no	—
170	C₂×#107	—	2	1	no	—
171	C₂×#106	—	2	2	no	—
172	C₂×C₄⋊C₈	C₂²×C₁₄₉	1	1	yes	characteristic
173	C₂×C₄⋊C₈	C₂²×C₁₄₉	1	2	yes	—
174	C₂×#102	—	2	2	no	—
175	C₂×C₂²⋊C₈	—	2	1	no	—
176	C₂×C₄×C₈	C₂²×C₁₄₉	1	1	yes	characteristic
177	#335	C₂²×C₁₄₉	1	4	yes	complemented
178	#333	C₂²×C₁₄₉	1	8	yes	complemented
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×D₄	D₄×C₁₄₉	1	1	yes	characteristic
181	C₂³×C₄	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	—	4	4	no	—
186	C₂²×C₈	D₄×C₁₄₉	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₁₄₉	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	C₂³×C₁₄₉	1	4	yes	—
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	D₄×C₁₄₉	1	2	yes	—
195	C₄×D₄	C₂³×C₁₄₉	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	D₄×C₁₄₉	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₄²	C₂³×C₁₄₉	1	1	yes	characteristic
201	#107	—	2	4	no	—
202	#107	—	2	4	no	—
203	#106	—	2	8	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	C₂³×C₁₄₉	1	2	yes	—
207	C₄⋊C₈	C₂³×C₁₄₉	1	4	yes	—
208	#102	—	2	8	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₄×C₈	D₄×C₁₄₉	1	2	yes	—
211	C₄×C₈	C₂³×C₁₄₉	1	2	yes	—
212	C₂⁴	—	2	1	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	Q₁₆×C₁₄₉	1	4	yes	—
219	C₂×D₄	C₂×D₄×C₁₄₉	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₄₉	1	1	yes	characteristic
228	C₂²×C₄	#41×C₁₄₉	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	8	no	—
235	C₂×C₈	SD₁₆×C₁₄₉	1	4	yes	—
236	C₂×C₈	C₂×D₄×C₁₄₉	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₁₄₉	1	2	yes	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	C₂×D₄×C₁₄₉	1	2	yes	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₄²	C₂×D₄×C₁₄₉	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₁₄₉	1	1	yes	characteristic
248	C₄²	#41×C₁₄₉	1	1	yes	characteristic
249	C₄²	C₂⁴×C₁₄₉	1	1	yes	characteristic
250	C₂³	—	2	4	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	#121×C₁₄₉	1	1	yes	characteristic
255	Q₈	—	8	4	no	—
256	Q₈	—	8	4	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	2	no	—
259	D₄	C₂×Q₁₆×C₁₄₉	1	4	yes	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	#121×C₁₄₉	1	1	yes	characteristic
269	C₂×C₄	#121×C₁₄₉	1	1	yes	characteristic
270	C₂×C₄	#121×C₁₄₉	1	1	yes	characteristic
271	C₂×C₄	C₂×SD₁₆×C₁₄₉	1	2	yes	—
272	C₂×C₄	C₂×Q₁₆×C₁₄₉	1	2	yes	—
273	C₂×C₄	#135×C₁₄₉	1	2	yes	—
274	C₂×C₄	#137×C₁₄₉	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₁₄₉	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₁₄₉	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₁₄₉	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	4	4	no	—
281	C₈	C₂×SD₁₆×C₁₄₉	1	4	yes	—
282	C₂²	—	2	4	no	—
283	C₂²	—	4	8	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#461×C₁₄₉	1	2	yes	—
287	C₂²	#467×C₁₄₉	1	2	yes	—
288	C₂²	#472×C₁₄₉	1	2	yes	—
289	C₂²	C₂×#121×C₁₄₉	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	4	no	—
294	C₄	#461×C₁₄₉	1	2	yes	—
295	C₄	#467×C₁₄₉	1	2	yes	—
296	C₄	C₂²×SD₁₆×C₁₄₉	1	1	yes	characteristic
297	C₄	C₂²×Q₁₆×C₁₄₉	1	1	yes	characteristic
298	C₄	C₂×#135×C₁₄₉	1	1	yes	characteristic
299	C₄	C₂×#137×C₁₄₉	1	1	yes	characteristic
300	C₂	—	4	4	no	—
301	C₂	#1675×C₁₄₉	1	4	yes	direct product factor
302	C₂	C₂×#461×C₁₄₉	1	1	yes	characteristic
303	C₂	C₂×#467×C₁₄₉	1	1	yes	characteristic
304	C₂	C₂×#472×C₁₄₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₄₉	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₄₉
prime radical	C₂²×D₄×C₁₄₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₄₉ ◁ C₂×C₄²×C₁₄₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₄₉	C₂	semidirect
4. #1675×C₁₄₉	C₂	direct (N × H)
5. C₂×#335×C₁₄₉	C₂	semidirect
6. C₂×#333×C₁₄₉	C₂	semidirect
11. #500×C₁₄₉	C₂²	semidirect
24. #335×C₁₄₉	C₂²	semidirect
25. #333×C₁₄₉	C₂²	semidirect
148. C₂×C₁₄₉	#1675	direct (N × H)
152. C₂×#1675	C₁₄₉	direct (N × H)
153. C₁₄₉	C₂×#1675	direct (N × H)
154. C₂×#500	C₂×C₁₄₉	semidirect
157. #1675	C₂×C₁₄₉	direct (N × H)
158. C₂×#335	C₂×C₁₄₉	semidirect
159. C₂×#333	C₂×C₁₄₉	semidirect
164. #500	C₂²×C₁₄₉	semidirect
177. #335	C₂²×C₁₄₉	semidirect
178. #333	C₂²×C₁₄₉	semidirect
301. C₂	#1675×C₁₄₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₄₉	2
C₂×#476×C₁₄₉	2
C₂×C₈×D₄×C₁₄₉	2
#1675×C₁₄₉	2
C₂×#335×C₁₄₉	2
C₂×#333×C₁₄₉	2
C₂×#1675	149

Subgroups — C2×#1675×C149 #49198053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1675×C₁₄₉ #49198053