Subgroups — #34061191

Order 19072.

428 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750576	C₂	1	2	yes	maximal
2	#23750574	C₂	1	2	yes	maximal
3	#23750574	C₂	1	2	yes	maximal
4	#23750574	C₂	1	2	yes	maximal
5	#23750574	C₂	1	2	yes	maximal
6	#23750570	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×#10597010	C₂	1	2	yes	maximal
8	C₄×C₄⋊#10597010	C₂	1	1	yes	maximal, characteristic
9	#125×D₁₄₉	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊#10597010	C₂²	1	2	yes	—
11	C₂×C₄⋊#10597010	C₂²	1	2	yes	—
12	C₂×C₄⋊#10597010	C₂²	1	2	yes	—
13	C₂×C₄×#10597010	C₂²	1	2	yes	—
14	C₂×C₄×#10597010	C₂²	1	1	yes	characteristic
15	C₂×C₄×#10597010	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10597010	C₂²	1	2	yes	—
17	(C₂×#10597010)⋊C₄	C₂²	1	2	yes	—
18	(C₂×#10597010)⋊C₄	C₂²	1	2	yes	—
19	(C₂×#10597010)⋊C₄	C₂²	1	2	yes	—
20	(C₂×#10597010)⋊C₄	C₂²	1	2	yes	—
21	(C₂×#10597010)⋊C₄	C₂²	1	1	yes	characteristic
22	(C₂×#10597010)⋊C₄	C₂²	1	1	yes	characteristic
23	(C₂×#10597010)⋊C₄	C₂²	1	2	yes	—
24	(C₂×#10597010)⋊C₄	C₂²	1	2	yes	—
25	(C₂×#10597010)⋊C₄	C₂²	1	2	yes	—
26	#18011903	C₄	1	1	yes	characteristic, complemented
27	#18011902	C₄	1	1	yes	characteristic, complemented
28	#18011902	C₄	1	1	yes	characteristic, complemented
29	#18011901	C₄	1	2	yes	complemented
30	#18011901	C₄	1	2	yes	complemented
31	C₄⋊C₄×D₁₄₉	C₂²	1	2	yes	—
32	C₄⋊C₄×D₁₄₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₄₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₄₉	C₂²	1	2	yes	—
35	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
36	#125×C₁₄₉	C₄	1	1	yes	characteristic, complemented
37	C₂²×#10597010	C₂³	1	2	yes	—
38	C₂²×#10597010	C₂³	1	2	yes	—
39	C₂²×#10597010	C₂³	1	2	yes	—
40	C₂²×#10597010	C₂³	1	2	yes	—
41	C₄⋊#10597010	—	2	4	no	—
42	C₄⋊#10597010	—	2	4	no	—
43	C₄⋊#10597010	—	4	2	no	—
44	C₄×#10597010	—	2	4	no	—
45	C₄×#10597010	—	4	1	no	—
46	C₄×#10597010	—	4	1	no	—
47	C₄×#10597010	—	4	2	no	—
48	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
51	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
52	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
53	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
54	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
55	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
57	#14558715	C₂×C₄	1	2	yes	—
58	#14558715	C₂×C₄	1	2	yes	—
59	#14558715	C₂×C₄	1	2	yes	—
60	#14558715	C₂×C₄	1	2	yes	—
61	#14558715	C₂×C₄	1	2	yes	—
62	#14558715	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
68	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
71	C₂×#10597010	—	2	2	no	—
72	C₂×#10597010	—	2	2	no	—
73	C₂×#10597010	—	2	2	no	—
74	C₂×#10597010	—	2	2	no	—
75	C₂×#10597010	—	2	2	no	—
76	C₂×#10597010	—	2	2	no	—
77	C₂×#10597010	—	2	2	no	—
78	C₂×#10597010	—	2	2	no	—
79	C₂×#10597010	—	2	2	no	—
80	C₂×#10597010	—	2	2	no	—
81	C₂×#10597010	—	2	2	no	—
82	C₂×#10597010	#41	1	4	yes	—
83	C₂²×D₁₄₉	C₂⁴	1	1	yes	characteristic
84	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
89	C₄×D₁₄₉	—	2	2	no	—
90	C₄×D₁₄₉	—	2	2	no	—
91	C₄×D₁₄₉	—	2	2	no	—
92	C₄×D₁₄₉	—	2	1	no	—
93	C₄×D₁₄₉	—	2	2	no	—
94	C₄×D₁₄₉	—	2	2	no	—
95	C₄×D₁₄₉	—	2	1	no	—
96	C₄×D₁₄₉	—	2	2	no	—
97	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
102	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₄₉	C₂×#41	1	2	yes	—
104	C₂×D₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂×D₁₄₉	2₋¹⁺⁴	1	2	yes	—
106	C₂×D₁₄₉	2₋¹⁺⁴	1	1	yes	characteristic
107	#10597010	—	2	4	no	—
108	#10597010	—	4	2	no	—
109	#10597010	—	4	2	no	—
110	#10597010	—	4	2	no	—
111	C₄×C₁₄₉	—	2	1	no	—
112	C₄×C₁₄₉	—	2	2	no	—
113	C₄×C₁₄₉	—	2	1	no	—
114	C₄×C₁₄₉	—	2	1	no	—
115	C₄×C₁₄₉	—	2	2	no	—
116	Dic₁₄₉	—	2	2	no	—
117	Dic₁₄₉	—	2	1	no	—
118	Dic₁₄₉	—	2	1	no	—
119	Dic₁₄₉	—	2	1	no	—
120	Dic₁₄₉	—	2	2	no	—
121	C₂×C₁₄₉	C₄×#41	1	1	yes	characteristic
122	C₂×C₁₄₉	#519	1	1	yes	characteristic
123	C₂×C₁₄₉	#520	1	1	yes	characteristic
124	D₁₄₉	#539	1	2	yes	—
125	D₁₄₉	#541	1	2	yes	—
126	C₁₄₉	#2336	1	1	yes	characteristic, complemented, Sylow
127	#2336	—	149	1	no	maximal, Sylow
128	C₂×#125	—	149	1	no	—
129	#389	—	149	2	no	—
130	#387	—	149	2	no	—
131	#387	—	149	2	no	—
132	#387	—	149	2	no	—
133	#387	—	149	2	no	—
134	#383	—	149	1	no	—
135	C₄×C₄⋊C₄	—	149	1	no	—
136	C₄×C₄⋊C₄	—	149	2	no	—
137	#125	—	149	2	no	—
138	#125	—	149	2	no	—
139	#125	—	149	1	no	—
140	#125	—	149	1	no	—
141	#125	—	149	1	no	—
142	#125	#10597010	1	1	yes	characteristic, complemented
143	C₂×C₄⋊C₄	—	149	2	no	—
144	C₂×C₄⋊C₄	—	149	2	no	—
145	C₂×C₄⋊C₄	—	149	2	no	—
146	C₂×C₄⋊C₄	—	149	2	no	—
147	C₂×C₄⋊C₄	—	149	1	no	—
148	C₂×C₄⋊C₄	—	149	1	no	—
149	C₂×C₄⋊C₄	—	149	2	no	—
150	C₂×C₄²	—	149	2	no	—
151	C₂×C₄²	—	149	1	no	—
152	C₂×C₄²	—	149	1	no	—
153	C₂×C₄²	—	149	2	no	—
154	C₂×C₄²	—	149	1	no	—
155	#95	—	149	2	no	—
156	#95	—	149	2	no	—
157	#95	—	149	2	no	—
158	#95	—	149	1	no	—
159	#95	—	149	2	no	—
160	#95	—	149	1	no	—
161	#95	—	149	2	no	—
162	#95	—	149	2	no	—
163	#95	—	149	2	no	—
164	C₂²×C₄	—	149	2	no	—
165	C₂²×C₄	—	149	2	no	—
166	C₂²×C₄	—	149	2	no	—
167	C₂²×C₄	—	149	2	no	—
168	C₂²×C₄	—	149	1	no	—
169	C₂²×C₄	—	149	1	no	—
170	C₂²×C₄	—	149	2	no	—
171	C₂²×C₄	—	149	2	no	—
172	C₂²×C₄	—	149	1	no	—
173	C₄⋊C₄	—	298	4	no	—
174	C₄⋊C₄	—	298	4	no	—
175	C₄⋊C₄	—	596	2	no	—
176	C₄⋊C₄	—	149	2	no	—
177	C₄⋊C₄	—	149	2	no	—
178	C₄⋊C₄	—	149	2	no	—
179	C₄⋊C₄	—	149	2	no	—
180	C₄⋊C₄	—	149	2	no	—
181	C₄⋊C₄	—	149	2	no	—
182	C₄⋊C₄	—	149	2	no	—
183	C₄⋊C₄	—	149	2	no	—
184	C₄⋊C₄	—	149	1	no	—
185	C₄⋊C₄	—	149	1	no	—
186	C₄⋊C₄	C₂×#10597010	1	2	yes	—
187	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
188	C₄⋊C₄	C₂×#10597010	1	1	yes	characteristic
189	C₄⋊C₄	C₂×#10597010	1	2	yes	—
190	C₄²	—	298	4	no	—
191	C₄²	—	596	1	no	—
192	C₄²	—	596	1	no	—
193	C₄²	—	596	2	no	—
194	C₄²	—	149	1	no	—
195	C₄²	—	149	1	no	—
196	C₄²	—	149	1	no	—
197	C₄²	C₂×#10597010	1	1	yes	characteristic
198	C₂³	—	149	1	no	—
199	C₂×C₄	—	298	2	no	—
200	C₂×C₄	—	298	1	no	—
201	C₂×C₄	—	298	2	no	—
202	C₂×C₄	—	298	2	no	—
203	C₂×C₄	—	298	2	no	—
204	C₂×C₄	—	298	2	no	—
205	C₂×C₄	—	298	1	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	—	298	2	no	—
212	C₂×C₄	—	298	2	no	—
213	C₂×C₄	—	298	2	no	—
214	C₂×C₄	—	298	2	no	—
215	C₂×C₄	—	298	2	no	—
216	C₂×C₄	—	298	2	no	—
217	C₂×C₄	—	298	2	no	—
218	C₂×C₄	—	149	4	no	—
219	C₂×C₄	—	149	1	no	—
220	C₂×C₄	—	149	1	no	—
221	C₂×C₄	—	149	1	no	—
222	C₂×C₄	—	149	2	no	—
223	C₂×C₄	—	149	2	no	—
224	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
225	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
226	C₂×C₄	C₂²×#10597010	1	1	yes	characteristic
227	C₂×C₄	C₂²×#10597010	1	2	yes	—
228	C₂×C₄	C₂²×#10597010	1	2	yes	—
229	C₂²	—	149	2	no	—
230	C₂²	—	149	2	no	—
231	C₂²	—	149	1	no	—
232	C₂²	—	149	1	no	—
233	C₂²	C₂³×#10597010	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	2	no	—
239	C₄	—	298	4	no	—
240	C₄	—	298	2	no	—
241	C₄	—	298	1	no	—
242	C₄	—	298	1	no	—
243	C₄	—	298	1	no	—
244	C₄	—	298	2	no	—
245	C₄	—	596	2	no	—
246	C₄	—	596	2	no	—
247	C₄	—	596	2	no	—
248	C₂	—	149	2	no	—
249	C₂	—	149	2	no	—
250	C₂	#41×#10597010	1	1	yes	characteristic
251	C₂	#23750693	1	1	yes	characteristic
252	C₂	#23750694	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10597010
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#125×C₁₄₉	C₄
prime radical, cosocle	C₂²×D₁₄₉	C₂⁴
hypercenter	#125	#10597010
hypocenter	C₁₄₉	#2336
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
26. #18011903	C₄	semidirect
27. #18011902	C₄	semidirect
28. #18011902	C₄	semidirect
29. #18011901	C₄	semidirect
30. #18011901	C₄	semidirect
36. #125×C₁₄₉	C₄	semidirect
126. C₁₄₉	#2336	semidirect
142. #125	#10597010	semidirect

Maximal subgroups

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

subgroup	index
#23750576	2
#23750574	2
#23750570	2
C₄⋊C₄×#10597010	2
C₄×C₄⋊#10597010	2
#125×D₁₄₉	2
#2336	149