Subgroups — #34053566

Order 19072.

484 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18011934	C₂	1	1	yes	maximal, characteristic
2	#23750107	C₂	1	2	yes	maximal, complemented
3	#23749728	C₂	1	2	yes	maximal, complemented
4	#23749723	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₁₁₉₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#23749707	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#23749625	C₂	1	2	yes	maximal, complemented
10	#23749579	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₁₆×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14558737	C₂²	1	2	yes	—
13	#18011934	C₄	1	8	yes	complemented
14	C₂×#14558723	C₂²	1	2	yes	—
15	C₂×D₁₁₉₂	C₂²	1	1	yes	characteristic
16	C₂×#14558719	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₄₉	C₂²	1	1	yes	characteristic
18	Q₈×Dic₁₄₉	C₂²	1	2	yes	—
19	C₄×D₅₉₆	—	2	2	no	—
20	C₄×D₅₉₆	C₂²	1	2	yes	—
21	#18011873	—	2	2	no	—
22	#18011873	C₂²	1	2	yes	—
23	#18011868	—	2	2	no	—
24	#18011868	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₁₄₉	—	2	2	no	—
26	C₄⋊C₄×D₁₄₉	C₂²	1	2	yes	—
27	#18011845	C₂²	1	1	yes	characteristic
28	#18011844	C₂²	1	2	yes	—
29	#18011835	C₂²	1	2	yes	—
30	#18011833	C₂²	1	2	yes	—
31	#18011831	C₂²	1	2	yes	—
32	#18011822	C₂²	1	1	yes	characteristic
33	#18011821	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₄₉	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₁₄₉	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₄₉	C₂²	1	2	yes	—
37	#107×C₁₄₉	C₂²	1	1	yes	characteristic
38	#103×C₁₄₉	C₂²	1	2	yes	—
39	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
40	#14558737	—	2	4	no	—
41	#14558737	C₂×C₄	1	8	yes	complemented
42	C₂×D₅₉₆	—	2	2	no	—
43	C₂×D₅₉₆	C₂³	1	2	yes	—
44	C₂×C₄×D₁₄₉	—	2	2	no	—
45	C₂×C₄×D₁₄₉	—	2	2	no	—
46	C₂×C₄×D₁₄₉	—	2	2	no	—
47	C₂×C₄×D₁₄₉	—	2	2	no	—
48	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
49	#14558723	C₂×C₄	1	8	yes	complemented
50	D₁₁₉₂	C₂×C₄	1	4	yes	complemented
51	#14558719	—	2	2	no	—
52	C₂×#11924746	C₂³	1	1	yes	characteristic
53	C₂×#11924746	C₂³	1	1	yes	characteristic
54	C₈×D₁₄₉	C₂×C₄	1	4	yes	complemented
55	C₄⋊Dic₁₄₉	—	2	2	no	—
56	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
57	#14558715	—	2	2	no	—
58	#14558715	D₄	1	1	yes	characteristic
59	#14558715	D₄	1	1	yes	characteristic
60	#14558713	—	2	2	no	—
61	#14558713	—	2	2	no	—
62	#14558713	D₄	1	1	yes	characteristic
63	#14558713	D₄	1	1	yes	characteristic
64	C₄×Dic₁₄₉	—	2	2	no	—
65	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₄₉	C₂³	1	2	yes	—
67	Q₁₆×C₁₄₉	C₂×C₄	1	4	yes	complemented
68	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₄₉	—	2	2	no	—
71	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
72	C₄²×C₁₄₉	—	2	2	no	—
73	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
74	C₂²×D₁₄₉	—	2	2	no	—
75	C₂²×D₁₄₉	C₂×D₄	1	1	yes	characteristic
76	D₅₉₆	—	2	8	no	—
77	D₅₉₆	—	2	2	no	—
78	D₅₉₆	C₂²×C₄	1	4	yes	—
79	C₂×Dic₁₄₉	—	2	2	no	—
80	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₁₄₉	C₂×D₄	1	1	yes	characteristic
83	C₄×D₁₄₉	—	2	8	no	—
84	C₄×D₁₄₉	—	8	2	no	—
85	C₄×D₁₄₉	—	8	2	no	—
86	C₄×D₁₄₉	—	8	2	no	—
87	C₄×D₁₄₉	C₂²×C₄	1	2	yes	—
88	C₄×D₁₄₉	#41	1	2	yes	—
89	#11924746	—	2	1	no	—
90	#11924746	C₂²×C₄	1	2	yes	—
91	Q₈×C₁₄₉	—	2	2	no	—
92	Q₈×C₁₄₉	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₁₄₉	—	2	2	no	—
94	C₂×C₄×C₁₄₉	—	2	2	no	—
95	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
98	C₈×C₁₄₉	—	2	1	no	—
99	C₈×C₁₄₉	C₂²×C₄	1	2	yes	—
100	C₂²×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×D₁₄₉	—	2	1	no	—
102	C₂×D₁₄₉	—	2	1	no	—
103	C₂×D₁₄₉	—	2	4	no	—
104	C₂×D₁₄₉	—	4	2	no	—
105	C₂×D₁₄₉	—	4	2	no	—
106	C₂×D₁₄₉	C₄×D₄	1	2	yes	—
107	C₄×C₁₄₉	—	2	4	no	—
108	C₄×C₁₄₉	—	2	1	no	—
109	C₄×C₁₄₉	—	2	1	no	—
110	C₄×C₁₄₉	—	4	2	no	—
111	C₄×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₁₄₉	C₂×#41	1	1	yes	characteristic
113	Dic₁₄₉	—	2	1	no	—
114	Dic₁₄₉	—	2	1	no	—
115	Dic₁₄₉	—	4	2	no	—
116	Dic₁₄₉	C₄×D₄	1	2	yes	—
117	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂×C₁₄₉	#576	1	1	yes	characteristic
119	C₂×C₁₄₉	#578	1	1	yes	characteristic
120	D₁₄₉	—	2	2	no	—
121	D₁₄₉	—	4	4	no	—
122	C₁₄₉	#2951	1	1	yes	characteristic, complemented, Sylow
123	#2951	—	149	1	no	maximal, Sylow
124	C₂×#135	—	149	1	no	—
125	#520	—	149	2	no	—
126	#440	—	149	2	no	—
127	#440	—	149	2	no	—
128	C₄×Q₁₆	—	149	1	no	—
129	C₄×Q₁₆	D₁₄₉	1	1	yes	characteristic, complemented
130	C₄×D₈	—	149	1	no	—
131	C₄×D₈	—	149	1	no	—
132	C₂×#107	—	149	1	no	—
133	#417	—	149	2	no	—
134	#405	—	149	1	no	—
135	C₂×#41	—	149	2	no	—
136	#135	—	149	8	no	—
137	C₂×Q₁₆	C₂×D₁₄₉	1	1	yes	characteristic
138	C₂×SD₁₆	—	149	2	no	—
139	C₂×D₈	—	149	1	no	—
140	C₂×M₁₆	—	149	1	no	—
141	C₂²×C₈	—	149	1	no	—
142	C₄×Q₈	—	149	2	no	—
143	C₄×Q₈	C₂×D₁₄₉	1	2	yes	—
144	C₄×D₄	—	298	2	no	—
145	C₄×D₄	—	298	2	no	—
146	C₄×D₄	—	149	2	no	—
147	C₄×D₄	—	149	2	no	—
148	#117	—	298	2	no	—
149	#117	—	149	1	no	—
150	C₂×C₄⋊C₄	—	298	2	no	—
151	C₂×C₄⋊C₄	—	149	2	no	—
152	#107	—	149	1	no	—
153	#107	—	149	2	no	—
154	#107	C₂×D₁₄₉	1	1	yes	characteristic
155	#103	—	149	2	no	—
156	#103	C₂×D₁₄₉	1	2	yes	—
157	#102	—	149	2	no	—
158	#102	—	149	2	no	—
159	#97	—	149	1	no	—
160	#97	—	149	1	no	—
161	C₄×C₈	—	149	1	no	—
162	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
163	#41	—	298	4	no	—
164	#41	—	149	8	no	—
165	C₂×Q₈	C₂²×D₁₄₉	1	2	yes	—
166	C₂×D₄	—	298	2	no	—
167	C₂×D₄	—	149	2	no	—
168	C₂²×C₄	—	298	2	no	—
169	C₂²×C₄	—	298	2	no	—
170	C₂²×C₄	—	298	2	no	—
171	C₂²×C₄	—	298	2	no	—
172	C₂²×C₄	—	149	1	no	—
173	Q₁₆	C₄×D₁₄₉	1	4	yes	complemented
174	SD₁₆	—	149	8	no	—
175	D₈	—	149	4	no	—
176	M₁₆	—	298	2	no	—
177	C₂×C₈	—	149	1	no	—
178	C₂×C₈	—	149	1	no	—
179	C₂×C₈	—	149	4	no	—
180	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	298	2	no	—
184	C₄⋊C₄	—	298	2	no	—
185	C₄⋊C₄	—	149	2	no	—
186	C₄⋊C₄	—	149	1	no	—
187	C₄⋊C₄	—	149	1	no	—
188	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
189	C₂²⋊C₄	—	298	2	no	—
190	C₂²⋊C₄	—	298	2	no	—
191	C₂²⋊C₄	—	149	1	no	—
192	C₂²⋊C₄	—	149	1	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	298	2	no	—
195	C₄²	—	149	1	no	—
196	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
197	C₂³	—	298	2	no	—
198	C₂³	—	149	1	no	—
199	Q₈	—	2	2	no	—
200	Q₈	C₂×C₄×D₁₄₉	1	4	yes	—
201	D₄	—	298	8	no	—
202	D₄	—	298	2	no	—
203	D₄	—	149	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	298	8	no	—
207	C₂×C₄	—	298	2	no	—
208	C₂×C₄	—	149	1	no	—
209	C₂×C₄	—	149	1	no	—
210	C₂×C₄	—	149	1	no	—
211	C₂×C₄	—	149	2	no	—
212	C₂×C₄	—	149	2	no	—
213	C₂×C₄	—	1192	2	no	—
214	C₂×C₄	—	1192	2	no	—
215	C₂×C₄	—	1192	2	no	—
216	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
217	C₂×C₄	D₄×D₁₄₉	1	1	yes	characteristic
218	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	298	1	no	—
221	C₈	—	149	2	no	—
222	C₈	C₂×C₄×D₁₄₉	1	2	yes	—
223	C₂²	—	298	1	no	—
224	C₂²	—	298	1	no	—
225	C₂²	—	298	4	no	—
226	C₂²	—	596	2	no	—
227	C₂²	—	596	2	no	—
228	C₂²	—	149	2	no	—
229	C₂²	C₂×D₄×D₁₄₉	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	4	no	—
233	C₄	—	4	2	no	—
234	C₄	—	298	1	no	—
235	C₄	—	298	1	no	—
236	C₄	—	596	2	no	—
237	C₄	—	149	2	no	—
238	C₄	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
239	C₄	#41×D₁₄₉	1	1	yes	characteristic
240	C₂	—	298	2	no	—
241	C₂	—	596	4	no	—
242	C₂	C₄×D₄×D₁₄₉	1	1	yes	characteristic
243	C₂	#23750427	1	1	yes	characteristic
244	C₂	#23750438	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₄₉
commutator	C₄×C₁₄₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₄₉
Fitting	C₄×Q₁₆×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂²×D₄
prime radical	C₂×#18011934	C₂
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	C₄×Q₁₆	D₁₄₉
hypocenter	C₁₄₉	#2951
LCS	G ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #23750107	C₂	semidirect
3. #23749728	C₂	semidirect
4. #23749723	C₂	semidirect
5. Q₁₆×Dic₁₄₉	C₂	semidirect
6. C₄×D₁₁₉₂	C₂	semidirect
7. #23749707	C₂	semidirect
8. #107×D₁₄₉	C₂	semidirect
9. #23749625	C₂	semidirect
10. #23749579	C₂	semidirect
11. C₄×Q₁₆×C₁₄₉	C₂	semidirect
13. #18011934	C₄	semidirect
41. #14558737	C₂×C₄	semidirect
49. #14558723	C₂×C₄	semidirect
50. D₁₁₉₂	C₂×C₄	semidirect
54. C₈×D₁₄₉	C₂×C₄	semidirect
67. Q₁₆×C₁₄₉	C₂×C₄	semidirect
122. C₁₄₉	#2951	semidirect
129. C₄×Q₁₆	D₁₄₉	semidirect
173. Q₁₆	C₄×D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18011934	2
#23750107	2
#23749728	2
#23749723	2
Q₁₆×Dic₁₄₉	2
C₄×D₁₁₉₂	2
#23749707	2
#107×D₁₄₉	2
#23749625	2
#23749579	2
C₄×Q₁₆×C₁₄₉	2
#2951	149