Subgroups — #34047631

Order 19072.

588 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23749518	C₂	1	2	yes	maximal, complemented
2	#23749511	C₂	1	2	yes	maximal, complemented
3	#23749505	C₂	1	2	yes	maximal
4	#23749493	C₂	1	1	yes	maximal, characteristic, complemented
5	#23749488	C₂	1	1	yes	maximal, characteristic, complemented
6	#23749485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×Dic₁₄₉	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#23749471	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#18011818	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₁₄₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
14	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
19	#18011818	C₂²	1	1	yes	characteristic
20	#18011818	C₂²	1	1	yes	characteristic
21	#18011818	C₂²	1	2	yes	—
22	#18011818	C₂²	1	1	yes	characteristic
23	#18011818	C₂²	1	2	yes	—
24	#18011818	C₂²	1	1	yes	characteristic
25	#18011818	C₂²	1	2	yes	—
26	#18011818	C₂²	1	2	yes	—
27	#18011818	C₂²	1	2	yes	—
28	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
30	C₂³×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₄₉	C₄	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
36	C₂²×Dic₁₄₉	—	2	4	no	—
37	C₂²×Dic₁₄₉	—	2	4	no	—
38	C₂²×Dic₁₄₉	—	4	2	no	—
39	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
40	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
41	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
42	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
43	C₄⋊Dic₁₄₉	—	4	2	no	—
44	C₂²⋊Dic₁₄₉	—	2	4	no	—
45	C₂²⋊Dic₁₄₉	—	2	4	no	—
46	C₂²⋊Dic₁₄₉	—	4	2	no	—
47	C₄×Dic₁₄₉	—	2	4	no	—
48	C₄×Dic₁₄₉	—	4	2	no	—
49	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
50	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₁₄₉	—	2	2	no	—
52	C₂×D₄×C₁₄₉	—	2	2	no	—
53	C₂×D₄×C₁₄₉	C₂×C₄	1	4	yes	—
54	C₂²×C₄×C₁₄₉	—	4	2	no	—
55	C₂²×C₄×C₁₄₉	—	4	2	no	—
56	C₂²×C₄×C₁₄₉	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₁₄₉	C₂×C₄	1	4	yes	complemented
58	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₄₉	C₂×C₄	1	4	yes	complemented
64	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	4	yes	—
65	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	4	yes	complemented
66	C₄²×C₁₄₉	C₂×C₄	1	4	yes	complemented
67	C₂×Dic₁₄₉	—	2	2	no	—
68	C₂×Dic₁₄₉	—	2	2	no	—
69	C₂×Dic₁₄₉	—	2	2	no	—
70	C₂×Dic₁₄₉	—	2	2	no	—
71	C₂×Dic₁₄₉	—	2	2	no	—
72	C₂×Dic₁₄₉	—	2	2	no	—
73	C₂×Dic₁₄₉	—	2	2	no	—
74	C₂×Dic₁₄₉	—	2	2	no	—
75	C₂×Dic₁₄₉	—	2	2	no	—
76	C₂×Dic₁₄₉	—	2	2	no	—
77	C₂×Dic₁₄₉	—	2	2	no	—
78	C₂×Dic₁₄₉	—	4	8	no	—
79	C₂×Dic₁₄₉	#41	1	4	yes	—
80	C₂³×C₁₄₉	—	2	2	no	—
81	C₂³×C₁₄₉	—	2	2	no	—
82	C₂³×C₁₄₉	—	2	1	no	—
83	C₂³×C₁₄₉	—	2	1	no	—
84	C₂³×C₁₄₉	—	2	2	no	—
85	C₂³×C₁₄₉	—	2	2	no	—
86	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
87	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
88	C₂³×C₁₄₉	#41	1	2	yes	—
89	C₂³×C₁₄₉	#41	1	2	yes	—
90	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
91	D₄×C₁₄₉	—	2	8	no	—
92	C₂×C₄×C₁₄₉	—	2	1	no	—
93	C₂×C₄×C₁₄₉	—	2	1	no	—
94	C₂×C₄×C₁₄₉	—	2	2	no	—
95	C₂×C₄×C₁₄₉	—	2	2	no	—
96	C₂×C₄×C₁₄₉	—	2	2	no	—
97	C₂×C₄×C₁₄₉	—	2	2	no	—
98	C₂×C₄×C₁₄₉	—	4	4	no	—
99	C₂×C₄×C₁₄₉	—	4	4	no	—
100	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
105	C₂²×C₁₄₉	—	2	2	no	—
106	C₂²×C₁₄₉	—	2	2	no	—
107	C₂²×C₁₄₉	—	2	4	no	—
108	C₂²×C₁₄₉	—	2	4	no	—
109	C₂²×C₁₄₉	—	2	2	no	—
110	C₂²×C₁₄₉	—	2	2	no	—
111	C₂²×C₁₄₉	—	2	2	no	—
112	C₂²×C₁₄₉	—	4	2	no	—
113	C₂²×C₁₄₉	—	4	2	no	—
114	C₂²×C₁₄₉	#117	1	4	yes	—
115	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
117	C₂²×C₁₄₉	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₄₉	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₄₉	H(5,𝔽₂)	1	2	yes	—
120	C₄×C₁₄₉	—	2	2	no	—
121	C₄×C₁₄₉	—	2	2	no	—
122	C₄×C₁₄₉	—	2	2	no	—
123	C₄×C₁₄₉	—	2	2	no	—
124	C₄×C₁₄₉	—	2	2	no	—
125	Dic₁₄₉	—	2	4	no	—
126	Dic₁₄₉	—	4	2	no	—
127	Dic₁₄₉	—	4	2	no	—
128	Dic₁₄₉	—	4	2	no	—
129	C₂×C₁₄₉	—	2	4	no	—
130	C₂×C₁₄₉	—	4	2	no	—
131	C₂×C₁₄₉	C₂×#117	1	1	yes	characteristic
132	C₂×C₁₄₉	C₄×#41	1	1	yes	characteristic
133	C₂×C₁₄₉	#518	1	1	yes	characteristic
134	C₂×C₁₄₉	#551	1	2	yes	—
135	C₂×C₁₄₉	#553	1	2	yes	—
136	C₁₄₉	#2353	1	1	yes	characteristic, complemented, Sylow
137	#2353	—	149	1	no	maximal, Sylow
138	C₂×C₄×D₄	D₁₄₉	1	1	yes	characteristic
139	#388	—	149	2	no	—
140	#387	—	149	2	no	—
141	#386	—	149	2	no	—
142	#383	—	149	1	no	—
143	#381	—	149	1	no	—
144	#380	—	149	1	no	—
145	C₄×C₂²⋊C₄	—	149	1	no	—
146	C₄×C₂²⋊C₄	—	149	2	no	—
147	#376	—	149	1	no	—
148	C₂×#95	—	149	1	no	—
149	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
150	C₂³×C₄	—	149	2	no	—
151	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
152	C₂³×C₄	C₂×D₁₄₉	1	1	yes	characteristic
153	C₄×D₄	Dic₁₄₉	1	8	yes	complemented
154	C₂×C₄⋊C₄	—	149	2	no	—
155	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	149	2	no	—
157	C₂×C₂²⋊C₄	—	149	2	no	—
158	C₂×C₂²⋊C₄	—	149	2	no	—
159	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
161	C₂×C₄²	—	149	2	no	—
162	C₂×C₄²	—	149	2	no	—
163	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
164	#95	—	149	1	no	—
165	#95	—	149	1	no	—
166	#95	—	149	2	no	—
167	#95	—	149	1	no	—
168	#95	—	149	2	no	—
169	#95	—	149	1	no	—
170	#95	—	149	2	no	—
171	#95	—	149	2	no	—
172	#95	—	149	2	no	—
173	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
174	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	C₂×Dic₁₄₉	1	4	yes	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	298	4	no	—
181	C₂²×C₄	—	298	4	no	—
182	C₂²×C₄	—	596	2	no	—
183	C₂²×C₄	—	149	2	no	—
184	C₂²×C₄	—	149	2	no	—
185	C₂²×C₄	—	149	2	no	—
186	C₂²×C₄	—	149	2	no	—
187	C₂²×C₄	C₂×Dic₁₄₉	1	4	yes	—
188	C₂²×C₄	C₂×Dic₁₄₉	1	4	yes	complemented
189	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
194	C₄⋊C₄	—	596	2	no	—
195	C₄⋊C₄	C₂×Dic₁₄₉	1	4	yes	complemented
196	C₂²⋊C₄	—	298	4	no	—
197	C₂²⋊C₄	—	298	4	no	—
198	C₂²⋊C₄	—	596	2	no	—
199	C₂²⋊C₄	C₂×Dic₁₄₉	1	4	yes	—
200	C₂²⋊C₄	C₂×Dic₁₄₉	1	4	yes	complemented
201	C₄²	—	298	4	no	—
202	C₄²	—	596	2	no	—
203	C₄²	C₂×Dic₁₄₉	1	4	yes	complemented
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂²×Dic₁₄₉	1	2	yes	—
211	C₂³	C₂²×Dic₁₄₉	1	2	yes	—
212	C₂³	#14558736	1	2	yes	—
213	C₂³	#14558736	1	2	yes	—
214	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
215	D₄	—	2	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	298	2	no	—
225	C₂×C₄	—	298	2	no	—
226	C₂×C₄	—	298	2	no	—
227	C₂×C₄	—	298	2	no	—
228	C₂×C₄	—	298	2	no	—
229	C₂×C₄	—	298	2	no	—
230	C₂×C₄	—	298	2	no	—
231	C₂×C₄	—	298	2	no	—
232	C₂×C₄	—	298	2	no	—
233	C₂×C₄	—	298	2	no	—
234	C₂×C₄	—	298	2	no	—
235	C₂×C₄	—	596	8	no	—
236	C₂×C₄	—	149	4	no	—
237	C₂×C₄	C₂²×Dic₁₄₉	1	2	yes	—
238	C₂×C₄	C₂²×Dic₁₄₉	1	2	yes	—
239	C₂×C₄	C₂²×Dic₁₄₉	1	2	yes	—
240	C₂×C₄	C₂²×Dic₁₄₉	1	2	yes	—
241	C₂×C₄	C₂²×Dic₁₄₉	1	2	yes	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	#18011870	1	4	yes	—
252	C₂²	C₂³×Dic₁₄₉	1	1	yes	characteristic
253	C₂²	#41×D₁₄₉	1	2	yes	—
254	C₂²	C₂×#14558736	1	1	yes	characteristic
255	C₂²	C₂×#14558736	1	1	yes	characteristic
256	C₂²	#18011956	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	298	4	no	—
263	C₄	—	596	2	no	—
264	C₄	—	596	2	no	—
265	C₄	—	596	2	no	—
266	C₂	—	2	4	no	—
267	C₂	—	4	2	no	—
268	C₂	C₂×#18011870	1	1	yes	characteristic
269	C₂	#41×Dic₁₄₉	1	1	yes	characteristic
270	C₂	#23750100	1	1	yes	characteristic
271	C₂	#23750304	1	2	yes	—
272	C₂	#23750324	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₄₉
commutator	C₂²×C₁₄₉	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₄₉	C₂
socle, cosocle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂²×D₄×C₁₄₉	C₂²
hypercenter	C₂×C₄×D₄	D₁₄₉
hypocenter	C₁₄₉	#2353
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #23749518	C₂	semidirect
2. #23749511	C₂	semidirect
4. #23749493	C₂	semidirect
5. #23749488	C₂	semidirect
6. #23749485	C₂	semidirect
7. C₂²⋊C₄×Dic₁₄₉	C₂	semidirect
8. C₄×C₂²⋊Dic₁₄₉	C₂	semidirect
9. #23749471	C₂	semidirect
10. C₂×#18011818	C₂	semidirect
31. C₄×D₄×C₁₄₉	C₄	semidirect
57. C₂²×C₄×C₁₄₉	C₂×C₄	semidirect
63. C₄⋊C₄×C₁₄₉	C₂×C₄	semidirect
65. C₂²⋊C₄×C₁₄₉	C₂×C₄	semidirect
66. C₄²×C₁₄₉	C₂×C₄	semidirect
136. C₁₄₉	#2353	semidirect
153. C₄×D₄	Dic₁₄₉	semidirect
188. C₂²×C₄	C₂×Dic₁₄₉	semidirect
195. C₄⋊C₄	C₂×Dic₁₄₉	semidirect
200. C₂²⋊C₄	C₂×Dic₁₄₉	semidirect
203. C₄²	C₂×Dic₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23749518	2
#23749511	2
#23749505	2
#23749493	2
#23749488	2
#23749485	2
C₂²⋊C₄×Dic₁₄₉	2
C₄×C₂²⋊Dic₁₄₉	2
#23749471	2
C₂×#18011818	2
C₂×C₄×D₄×C₁₄₉	2
#2353	149