Subgroups — C₂×#1525×C₁₄₉ #49196585

Order 38144.

1402 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₁₄₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₁₄₉	C₂	1	2	yes	maximal, complemented
3	#1525×C₁₄₉	C₂	1	8	yes	maximal, direct product factor
4	C₂×#329×C₁₄₉	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₁₄₉	C₄	1	2	yes	—
6	C₂²×M₁₆×C₁₄₉	—	2	2	no	—
7	#532×C₁₄₉	—	2	4	no	—
8	#532×C₁₄₉	C₄	1	4	yes	—
9	#532×C₁₄₉	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₁₄₉	C₄	1	1	yes	characteristic
11	C₂×#127×C₁₄₉	C₄	1	1	yes	characteristic
12	C₂×#124×C₁₄₉	C₂²	1	2	yes	—
13	C₂×#121×C₁₄₉	—	2	2	no	—
14	C₂×C₄×D₄×C₁₄₉	—	2	2	no	—
15	C₂²×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₁₄₉	—	2	4	no	—
17	C₄×M₁₆×C₁₄₉	C₂²	1	8	yes	complemented
18	C₂×#97×C₁₄₉	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₁₄₉	—	2	2	no	—
20	#329×C₁₄₉	C₂²	1	16	yes	complemented
21	C₂×#41×C₁₄₉	—	2	4	no	—
22	C₂×#41×C₁₄₉	—	4	8	no	—
23	C₂×#41×C₁₄₉	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₁₄₉	C₂×C₄	1	2	yes	—
25	C₂²×D₄×C₁₄₉	—	2	2	no	—
26	C₂²×D₄×C₁₄₉	—	2	2	no	—
27	C₂²×D₄×C₁₄₉	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₄₉	—	2	2	no	—
29	C₂³×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₄₉	D₄	1	2	yes	—
31	C₂×M₁₆×C₁₄₉	—	2	16	no	—
32	C₂×M₁₆×C₁₄₉	—	2	8	no	—
33	C₂²×C₈×C₁₄₉	—	2	4	no	—
34	C₄⋊Q₈×C₁₄₉	—	2	2	no	—
35	C₄⋊Q₈×C₁₄₉	C₂×C₄	1	4	yes	—
36	#127×C₁₄₉	—	2	2	no	—
37	#127×C₁₄₉	C₂×C₄	1	4	yes	—
38	#124×C₁₄₉	—	2	4	no	—
39	#124×C₁₄₉	C₂×C₄	1	4	yes	—
40	#124×C₁₄₉	C₂³	1	4	yes	—
41	#121×C₁₄₉	—	2	8	no	—
42	#121×C₁₄₉	—	2	8	no	—
43	C₄×D₄×C₁₄₉	—	2	8	no	—
44	C₄×D₄×C₁₄₉	—	2	8	no	—
45	C₂×C₄⋊C₄×C₁₄₉	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₁₄₉	—	2	4	no	—
47	C₂×C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₄₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₄₉	D₄	1	4	yes	—
50	C₂×C₄²×C₁₄₉	D₄	1	2	yes	—
51	C₂×C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₄₉	C₂³	1	2	yes	—
53	#97×C₁₄₉	—	2	4	no	—
54	#97×C₁₄₉	C₂³	1	8	yes	—
55	C₄×C₈×C₁₄₉	—	2	8	no	—
56	C₂⁴×C₁₄₉	—	2	2	no	—
57	C₂⁴×C₁₄₉	C₂²⋊C₄	1	1	yes	characteristic
58	#41×C₁₄₉	—	4	16	no	—
59	#41×C₁₄₉	—	4	16	no	—
60	C₂×Q₈×C₁₄₉	—	2	2	no	—
61	C₂×Q₈×C₁₄₉	—	4	4	no	—
62	C₂×Q₈×C₁₄₉	C₂²×C₄	1	4	yes	—
63	C₂×D₄×C₁₄₉	—	2	8	no	—
64	C₂×D₄×C₁₄₉	—	2	8	no	—
65	C₂×D₄×C₁₄₉	—	2	2	no	—
66	C₂×D₄×C₁₄₉	—	4	4	no	—
67	C₂×D₄×C₁₄₉	—	4	8	no	—
68	C₂×D₄×C₁₄₉	—	4	8	no	—
69	C₂×D₄×C₁₄₉	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₄₉	—	2	4	no	—
71	C₂²×C₄×C₁₄₉	—	2	8	no	—
72	C₂²×C₄×C₁₄₉	—	2	2	no	—
73	C₂²×C₄×C₁₄₉	—	2	8	no	—
74	C₂²×C₄×C₁₄₉	—	4	8	no	—
75	C₂²×C₄×C₁₄₉	C₂²⋊C₄	1	4	yes	—
76	C₂²×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₁₄₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₄₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₄₉	C₂×D₄	1	4	yes	—
83	M₁₆×C₁₄₉	—	2	32	no	—
84	C₂×C₈×C₁₄₉	—	2	16	no	—
85	C₂×C₈×C₁₄₉	—	2	8	no	—
86	C₄⋊C₄×C₁₄₉	—	2	8	no	—
87	C₂²⋊C₄×C₁₄₉	—	2	8	no	—
88	C₂²⋊C₄×C₁₄₉	—	2	8	no	—
89	C₄²×C₁₄₉	C₂²⋊C₄	1	4	yes	—
90	C₄²×C₁₄₉	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₄₉	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₄₉	C₂×D₄	1	4	yes	—
94	C₄²×C₁₄₉	C₂×D₄	1	2	yes	—
95	C₄²×C₁₄₉	C₂⁴	1	1	yes	characteristic
96	C₂³×C₁₄₉	—	2	2	no	—
97	C₂³×C₁₄₉	—	2	4	no	—
98	C₂³×C₁₄₉	—	2	4	no	—
99	C₂³×C₁₄₉	—	4	4	no	—
100	C₂³×C₁₄₉	—	4	8	no	—
101	C₂³×C₁₄₉	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂³×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	Q₈×C₁₄₉	—	4	4	no	—
104	Q₈×C₁₄₉	—	4	4	no	—
105	D₄×C₁₄₉	—	4	4	no	—
106	D₄×C₁₄₉	—	4	4	no	—
107	D₄×C₁₄₉	—	4	16	no	—
108	D₄×C₁₄₉	—	4	16	no	—
109	C₂×C₄×C₁₄₉	—	2	2	no	—
110	C₂×C₄×C₁₄₉	—	2	4	no	—
111	C₂×C₄×C₁₄₉	—	2	4	no	—
112	C₂×C₄×C₁₄₉	—	2	4	no	—
113	C₂×C₄×C₁₄₉	—	2	8	no	—
114	C₂×C₄×C₁₄₉	—	2	8	no	—
115	C₂×C₄×C₁₄₉	—	2	4	no	—
116	C₂×C₄×C₁₄₉	—	4	4	no	—
117	C₂×C₄×C₁₄₉	—	4	16	no	—
118	C₂×C₄×C₁₄₉	#100	1	4	yes	—
119	C₂×C₄×C₁₄₉	#104	1	8	yes	—
120	C₂×C₄×C₁₄₉	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₄₉	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₁₄₉	C₂×C₂²⋊C₄	1	4	yes	—
124	C₂×C₄×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₄₉	C₂²×D₄	1	2	yes	—
126	C₈×C₁₄₉	—	2	16	no	—
127	C₂²×C₁₄₉	—	2	2	no	—
128	C₂²×C₁₄₉	—	2	4	no	—
129	C₂²×C₁₄₉	—	4	4	no	—
130	C₂²×C₁₄₉	—	4	4	no	—
131	C₂²×C₁₄₉	—	4	8	no	—
132	C₂²×C₁₄₉	—	8	8	no	—
133	C₂²×C₁₄₉	C₂×#100	1	2	yes	—
134	C₂²×C₁₄₉	C₂×#104	1	4	yes	—
135	C₂²×C₁₄₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₁₄₉	—	2	1	no	—
137	C₄×C₁₄₉	—	2	1	no	—
138	C₄×C₁₄₉	—	2	2	no	—
139	C₄×C₁₄₉	—	2	2	no	—
140	C₄×C₁₄₉	—	4	4	no	—
141	C₄×C₁₄₉	C₂×#100	1	4	yes	—
142	C₄×C₁₄₉	C₂×#104	1	8	yes	—
143	C₂×C₁₄₉	—	4	2	no	—
144	C₂×C₁₄₉	—	8	4	no	—
145	C₂×C₁₄₉	#1525	1	4	yes	direct product factor
146	C₂×C₁₄₉	C₂²×#100	1	1	yes	characteristic
147	C₂×C₁₄₉	C₂²×#104	1	2	yes	—
148	C₂×#1525	C₁₄₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₁₄₉	C₂×#1525	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×#532	C₂×C₁₄₉	1	1	yes	characteristic
151	C₂×C₄×M₁₆	C₂×C₁₄₉	1	2	yes	complemented
152	#1525	C₂×C₁₄₉	1	8	yes	direct product factor
153	C₂×#329	C₂×C₁₄₉	1	4	yes	complemented
154	C₂²×#41	C₄×C₁₄₉	1	2	yes	—
155	C₂²×M₁₆	—	2	2	no	—
156	#532	—	2	4	no	—
157	#532	C₄×C₁₄₉	1	4	yes	—
158	#532	C₂²×C₁₄₉	1	4	yes	—
159	C₂×C₄⋊Q₈	C₄×C₁₄₉	1	1	yes	characteristic
160	C₂×#127	C₄×C₁₄₉	1	1	yes	characteristic
161	C₂×#124	C₂²×C₁₄₉	1	2	yes	—
162	C₂×#121	—	2	2	no	—
163	C₂×C₄×D₄	—	2	2	no	—
164	C₂²×C₄²	C₂²×C₁₄₉	1	1	yes	characteristic
165	C₄×M₁₆	—	2	4	no	—
166	C₄×M₁₆	C₂²×C₁₄₉	1	8	yes	complemented
167	C₂×#97	C₂²×C₁₄₉	1	4	yes	—
168	C₂×C₄×C₈	—	2	2	no	—
169	#329	C₂²×C₁₄₉	1	16	yes	complemented
170	C₂×#41	—	2	4	no	—
171	C₂×#41	—	4	8	no	—
172	C₂×#41	C₂×C₄×C₁₄₉	1	8	yes	—
173	C₂²×Q₈	C₂×C₄×C₁₄₉	1	2	yes	—
174	C₂²×D₄	—	2	2	no	—
175	C₂²×D₄	—	2	2	no	—
176	C₂²×D₄	C₂×C₄×C₁₄₉	1	2	yes	—
177	C₂³×C₄	—	2	2	no	—
178	C₂³×C₄	C₂×C₄×C₁₄₉	1	1	yes	characteristic
179	C₂³×C₄	D₄×C₁₄₉	1	2	yes	—
180	C₂×M₁₆	—	2	16	no	—
181	C₂×M₁₆	—	2	8	no	—
182	C₂²×C₈	—	2	4	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	C₂×C₄×C₁₄₉	1	4	yes	—
185	#127	—	2	2	no	—
186	#127	C₂×C₄×C₁₄₉	1	4	yes	—
187	#124	—	2	4	no	—
188	#124	C₂×C₄×C₁₄₉	1	4	yes	—
189	#124	C₂³×C₁₄₉	1	4	yes	—
190	#121	—	2	8	no	—
191	#121	—	2	8	no	—
192	C₄×D₄	—	2	8	no	—
193	C₄×D₄	—	2	8	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	4	no	—
196	C₂×C₄²	C₂×C₄×C₁₄₉	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₁₄₉	1	2	yes	—
198	C₂×C₄²	D₄×C₁₄₉	1	2	yes	—
199	C₂×C₄²	D₄×C₁₄₉	1	4	yes	—
200	C₂×C₄²	C₂³×C₁₄₉	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₁₄₉	1	2	yes	—
202	#97	—	2	4	no	—
203	#97	C₂³×C₁₄₉	1	8	yes	—
204	C₄×C₈	—	2	8	no	—
205	C₂⁴	—	2	2	no	—
206	C₂⁴	C₂²⋊C₄×C₁₄₉	1	1	yes	characteristic
207	#41	—	4	16	no	—
208	#41	—	4	16	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	C₂²×C₄×C₁₄₉	1	4	yes	—
212	C₂×D₄	—	2	8	no	—
213	C₂×D₄	—	2	8	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	—	4	8	no	—
217	C₂×D₄	—	4	8	no	—
218	C₂×D₄	C₂²×C₄×C₁₄₉	1	4	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	4	8	no	—
224	C₂²×C₄	C₂²⋊C₄×C₁₄₉	1	4	yes	—
225	C₂²×C₄	C₂²⋊C₄×C₁₄₉	1	2	yes	—
226	C₂²×C₄	C₂²⋊C₄×C₁₄₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊C₄×C₁₄₉	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₄₉	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₁₄₉	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₄₉	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₁₄₉	1	4	yes	—
232	M₁₆	—	2	32	no	—
233	C₂×C₈	—	2	16	no	—
234	C₂×C₈	—	2	8	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₄²	C₂²⋊C₄×C₁₄₉	1	4	yes	—
239	C₄²	C₂²⋊C₄×C₁₄₉	1	2	yes	—
240	C₄²	C₂²×C₄×C₁₄₉	1	1	yes	characteristic
241	C₄²	C₂²×C₄×C₁₄₉	1	2	yes	—
242	C₄²	C₂×D₄×C₁₄₉	1	4	yes	—
243	C₄²	C₂×D₄×C₁₄₉	1	2	yes	—
244	C₄²	C₂⁴×C₁₄₉	1	1	yes	characteristic
245	C₂³	—	2	2	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	4	4	no	—
249	C₂³	—	4	8	no	—
250	C₂³	C₂×C₂²⋊C₄×C₁₄₉	1	2	yes	—
251	C₂³	C₂×C₂²⋊C₄×C₁₄₉	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	Q₈	—	4	4	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	16	no	—
257	D₄	—	4	16	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	16	no	—
267	C₂×C₄	#100×C₁₄₉	1	4	yes	—
268	C₂×C₄	#104×C₁₄₉	1	8	yes	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₁₄₉	1	2	yes	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₁₄₉	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊C₄×C₁₄₉	1	2	yes	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₁₄₉	1	4	yes	—
273	C₂×C₄	C₂³×C₄×C₁₄₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₁₄₉	1	2	yes	—
275	C₈	—	2	16	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	8	8	no	—
282	C₂²	C₂×#100×C₁₄₉	1	2	yes	—
283	C₂²	C₂×#104×C₁₄₉	1	4	yes	—
284	C₂²	C₂²×C₂²⋊C₄×C₁₄₉	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	4	no	—
290	C₄	C₂×#100×C₁₄₉	1	4	yes	—
291	C₄	C₂×#104×C₁₄₉	1	8	yes	—
292	C₂	—	4	2	no	—
293	C₂	—	8	4	no	—
294	C₂	#1525×C₁₄₉	1	4	yes	direct product factor
295	C₂	C₂²×#100×C₁₄₉	1	1	yes	characteristic
296	C₂	C₂²×#104×C₁₄₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₄₉	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₄₉
Frattini, cosocle	C₄²	C₂⁴×C₁₄₉
socle	C₂³×C₁₄₉	C₂×C₂²⋊C₄
prime radical	C₂×#532×C₁₄₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₄₉ ◁ C₂²×C₄²×C₁₄₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×M₁₆×C₁₄₉	C₂	semidirect
3. #1525×C₁₄₉	C₂	direct (N × H)
4. C₂×#329×C₁₄₉	C₂	semidirect
17. C₄×M₁₆×C₁₄₉	C₂²	semidirect
20. #329×C₁₄₉	C₂²	semidirect
145. C₂×C₁₄₉	#1525	direct (N × H)
148. C₂×#1525	C₁₄₉	direct (N × H)
149. C₁₄₉	C₂×#1525	direct (N × H)
151. C₂×C₄×M₁₆	C₂×C₁₄₉	semidirect
152. #1525	C₂×C₁₄₉	direct (N × H)
153. C₂×#329	C₂×C₁₄₉	semidirect
166. C₄×M₁₆	C₂²×C₁₄₉	semidirect
169. #329	C₂²×C₁₄₉	semidirect
294. C₂	#1525×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×#532×C₁₄₉	2
C₂×C₄×M₁₆×C₁₄₉	2
#1525×C₁₄₉	2
C₂×#329×C₁₄₉	2
C₂×#1525	149

Subgroups — C2×#1525×C149 #49196585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1525×C₁₄₉ #49196585