Subgroups — #34047585

Order 19072.

504 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×#18011868	C₂	1	1	yes	maximal, characteristic
2	#23749520	C₂	1	2	yes	maximal, complemented
3	#23749516	C₂	1	2	yes	maximal
4	#23749515	C₂	1	2	yes	maximal
5	#23749493	C₂	1	1	yes	maximal, characteristic, complemented
6	#23749492	C₂	1	1	yes	maximal, characteristic, complemented
7	#23749484	C₂	1	1	yes	maximal, characteristic
8	C₄×#14558715	C₂	1	2	yes	maximal, complemented
9	#23749476	C₂	1	1	yes	maximal, characteristic, complemented
10	#23749472	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
13	#18011868	C₄	1	8	yes	complemented
14	C₂×#14558715	C₂²	1	2	yes	—
15	C₂×#14558715	C₂²	1	2	yes	—
16	C₂×#14558715	C₂²	1	1	yes	characteristic
17	C₂×#14558715	C₂²	1	1	yes	characteristic
18	C₂×#14558713	C₂²	1	2	yes	—
19	C₂×#14558713	C₂²	1	2	yes	—
20	C₂×#14558713	C₂²	1	1	yes	characteristic
21	C₂×#14558713	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
25	#18011818	C₂²	1	1	yes	characteristic
26	#18011818	C₂²	1	1	yes	characteristic
27	#18011818	C₂²	1	2	yes	—
28	#18011818	C₂²	1	2	yes	—
29	#18011818	C₂²	1	2	yes	—
30	#18011818	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₄₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
34	#95×C₁₄₉	C₂²	1	1	yes	characteristic
35	#95×C₁₄₉	C₂²	1	1	yes	characteristic
36	C₂³×D₁₄₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
38	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
39	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₁₄₉	—	2	4	no	—
43	C₂×C₄×D₁₄₉	C₂×C₄	1	4	yes	—
44	#14558715	—	2	4	no	—
45	#14558715	—	2	4	no	—
46	#14558715	C₂×C₄	1	4	yes	—
47	#14558715	C₂×C₄	1	4	yes	—
48	#14558713	—	2	4	no	—
49	#14558713	—	4	2	no	—
50	#14558713	C₂×C₄	1	4	yes	—
51	#14558713	C₂×C₄	1	4	yes	—
52	C₄×Dic₁₄₉	—	2	4	no	—
53	C₄×Dic₁₄₉	—	4	2	no	—
54	C₄×Dic₁₄₉	C₂×C₄	1	4	yes	—
55	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
60	C₄⋊C₄×C₁₄₉	—	2	4	no	—
61	C₄²×C₁₄₉	—	4	2	no	—
62	C₄²×C₁₄₉	C₂×C₄	1	4	yes	—
63	C₂²×D₁₄₉	—	2	2	no	—
64	C₂²×D₁₄₉	—	2	2	no	—
65	C₂²×D₁₄₉	—	2	1	no	—
66	C₂²×D₁₄₉	—	2	1	no	—
67	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
68	C₂×Dic₁₄₉	—	2	2	no	—
69	C₂×Dic₁₄₉	—	2	1	no	—
70	C₂×Dic₁₄₉	—	2	1	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	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₄₉	#41	1	4	yes	—
81	C₄×D₁₄₉	—	2	8	no	—
82	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
83	C₂×C₄×C₁₄₉	—	2	2	no	—
84	C₂×C₄×C₁₄₉	—	2	2	no	—
85	C₂×C₄×C₁₄₉	—	2	2	no	—
86	C₂×C₄×C₁₄₉	—	2	1	no	—
87	C₂×C₄×C₁₄₉	—	2	2	no	—
88	C₂×C₄×C₁₄₉	—	2	1	no	—
89	C₂×C₄×C₁₄₉	—	2	2	no	—
90	C₂×C₄×C₁₄₉	—	2	2	no	—
91	C₂×C₄×C₁₄₉	—	2	2	no	—
92	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₄₉	#41	1	4	yes	—
96	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
97	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
98	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
99	C₂²×C₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×C₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
101	C₂×D₁₄₉	—	2	2	no	—
102	C₂×D₁₄₉	—	2	2	no	—
103	C₂×D₁₄₉	—	2	2	no	—
104	C₂×D₁₄₉	—	4	2	no	—
105	C₂×D₁₄₉	—	4	2	no	—
106	C₄×C₁₄₉	—	2	2	no	—
107	C₄×C₁₄₉	—	2	2	no	—
108	C₄×C₁₄₉	—	2	2	no	—
109	C₄×C₁₄₉	—	2	4	no	—
110	C₄×C₁₄₉	—	4	2	no	—
111	Dic₁₄₉	—	2	2	no	—
112	Dic₁₄₉	—	2	2	no	—
113	Dic₁₄₉	—	4	2	no	—
114	Dic₁₄₉	—	4	2	no	—
115	Dic₁₄₉	#117	1	4	yes	—
116	C₂×C₁₄₉	C₂×#117	1	1	yes	characteristic
117	C₂×C₁₄₉	C₄×#41	1	1	yes	characteristic
118	C₂×C₁₄₉	#520	1	1	yes	characteristic
119	C₂×C₁₄₉	#553	1	2	yes	—
120	C₂×C₁₄₉	#556	1	2	yes	—
121	D₁₄₉	—	4	2	no	—
122	C₁₄₉	#2347	1	1	yes	characteristic, complemented, Sylow
123	#2347	—	149	1	no	maximal, Sylow
124	C₂×#117	—	149	1	no	—
125	#389	—	149	2	no	—
126	#388	—	149	2	no	—
127	#388	—	149	2	no	—
128	#383	—	149	1	no	—
129	#383	—	149	1	no	—
130	#380	—	149	1	no	—
131	C₄×C₄⋊C₄	—	149	2	no	—
132	C₄×C₄⋊C₄	—	149	1	no	—
133	C₄×C₄⋊C₄	D₁₄₉	1	1	yes	characteristic, complemented
134	C₄×C₂²⋊C₄	—	149	1	no	—
135	C₂³×C₄	—	149	1	no	—
136	#117	—	149	8	no	—
137	C₂×C₄⋊C₄	—	149	2	no	—
138	C₂×C₄⋊C₄	—	149	2	no	—
139	C₂×C₄⋊C₄	—	149	1	no	—
140	C₂×C₄⋊C₄	—	149	1	no	—
141	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
142	C₂×C₂²⋊C₄	—	149	2	no	—
143	C₂×C₂²⋊C₄	—	149	2	no	—
144	C₂×C₂²⋊C₄	—	149	1	no	—
145	C₂×C₂²⋊C₄	—	149	1	no	—
146	C₂×C₄²	—	149	2	no	—
147	C₂×C₄²	—	149	2	no	—
148	C₂×C₄²	—	149	1	no	—
149	C₂×C₄²	C₂×D₁₄₉	1	2	yes	—
150	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
151	#95	—	149	1	no	—
152	#95	—	149	1	no	—
153	#95	—	149	2	no	—
154	#95	—	149	2	no	—
155	#95	—	149	2	no	—
156	#95	—	149	2	no	—
157	#95	C₂×D₁₄₉	1	1	yes	characteristic
158	#95	C₂×D₁₄₉	1	1	yes	characteristic
159	C₂⁴	—	149	1	no	—
160	C₂²×C₄	—	298	4	no	—
161	C₂²×C₄	—	149	2	no	—
162	C₂²×C₄	—	149	2	no	—
163	C₂²×C₄	—	149	1	no	—
164	C₂²×C₄	—	149	1	no	—
165	C₂²×C₄	—	149	4	no	—
166	C₂²×C₄	—	149	1	no	—
167	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
168	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	298	4	no	—
174	C₄⋊C₄	—	298	4	no	—
175	C₄⋊C₄	—	149	4	no	—
176	C₄⋊C₄	—	149	4	no	—
177	C₂²⋊C₄	—	298	4	no	—
178	C₂²⋊C₄	—	596	2	no	—
179	C₂²⋊C₄	—	149	4	no	—
180	C₂²⋊C₄	—	149	4	no	—
181	C₄²	—	4	2	no	—
182	C₄²	—	298	4	no	—
183	C₄²	—	596	2	no	—
184	C₄²	—	149	4	no	—
185	C₄²	C₄×D₁₄₉	1	4	yes	—
186	C₂³	—	298	2	no	—
187	C₂³	—	298	2	no	—
188	C₂³	—	298	1	no	—
189	C₂³	—	298	1	no	—
190	C₂³	—	149	2	no	—
191	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	298	8	no	—
202	C₂×C₄	—	298	2	no	—
203	C₂×C₄	—	298	1	no	—
204	C₂×C₄	—	298	1	no	—
205	C₂×C₄	—	298	2	no	—
206	C₂×C₄	—	298	2	no	—
207	C₂×C₄	—	298	2	no	—
208	C₂×C₄	—	298	2	no	—
209	C₂×C₄	—	298	2	no	—
210	C₂×C₄	—	298	2	no	—
211	C₂×C₄	—	149	2	no	—
212	C₂×C₄	—	149	2	no	—
213	C₂×C₄	—	149	2	no	—
214	C₂×C₄	—	149	4	no	—
215	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
216	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
217	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
218	C₂×C₄	#14558736	1	4	yes	—
219	C₂²	—	298	2	no	—
220	C₂²	—	298	2	no	—
221	C₂²	—	298	2	no	—
222	C₂²	—	596	2	no	—
223	C₂²	—	596	2	no	—
224	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
225	C₂²	#41×D₁₄₉	1	2	yes	—
226	C₂²	C₂×#14558736	1	2	yes	—
227	C₂²	#18011956	1	1	yes	characteristic
228	C₂²	#18011959	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	298	2	no	—
235	C₄	—	298	2	no	—
236	C₄	—	596	2	no	—
237	C₄	—	596	2	no	—
238	C₄	—	149	4	no	—
239	C₂	—	596	2	no	—
240	C₂	#117×D₁₄₉	1	1	yes	characteristic
241	C₂	C₄×#14558736	1	1	yes	characteristic
242	C₂	#23750106	1	1	yes	characteristic
243	C₂	#23750324	1	2	yes	—
244	C₂	#23750347	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₄₉
commutator	C₂²×C₁₄₉	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₁₄₉	C₂
socle, cosocle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂³×D₁₄₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₄₉
hypocenter	C₁₄₉	#2347
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #23749520	C₂	semidirect
5. #23749493	C₂	semidirect
6. #23749492	C₂	semidirect
8. C₄×#14558715	C₂	semidirect
9. #23749476	C₂	semidirect
11. C₄×C₄⋊C₄×C₁₄₉	C₂	semidirect
13. #18011868	C₄	semidirect
122. C₁₄₉	#2347	semidirect
133. C₄×C₄⋊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×#18011868	2
#23749520	2
#23749516	2
#23749515	2
#23749493	2
#23749492	2
#23749484	2
C₄×#14558715	2
#23749476	2
#23749472	2
C₄×C₄⋊C₄×C₁₄₉	2
#2347	149