Subgroups — #34053399

Order 19072.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18011919	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#18011918	C₂	1	1	yes	maximal, characteristic, complemented
3	#23750171	C₂	1	1	yes	maximal, characteristic
4	#23749700	C₂	1	4	yes	maximal, complemented
5	#23749688	C₂	1	4	yes	maximal, complemented
6	#23749648	C₂	1	1	yes	maximal, characteristic, complemented
7	#23749591	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#408×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14558736	C₂²	1	1	yes	characteristic
11	#18011919	—	2	4	no	—
12	#18011918	—	2	4	no	—
13	C₂×#14558720	C₂²	1	1	yes	characteristic
14	C₂×#14558720	C₂²	1	2	yes	—
15	C₂×#14558719	C₂²	1	2	yes	—
16	C₂²×#11924746	C₂²	1	2	yes	—
17	C₂²×#11924746	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₄₉	C₂²	1	2	yes	—
19	#18011909	C₄	1	1	yes	characteristic
20	#18011907	C₄	1	1	yes	characteristic
21	#18011900	C₄	1	1	yes	characteristic
22	#18011898	C₄	1	1	yes	characteristic
23	#18011883	C₄	1	4	yes	—
24	C₄×C₂²⋊D₁₄₉	C₂²	1	4	yes	—
25	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
26	#18011840	C₂²	1	4	yes	—
27	#18011823	C₂²	1	4	yes	—
28	C₈×Dic₁₄₉	C₂²	1	2	yes	—
29	C₈×Dic₁₄₉	C₂²	1	2	yes	—
30	C₂×#41×C₁₄₉	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₁₄₉	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
33	C₂²⋊C₈×C₁₄₉	C₂²	1	4	yes	—
34	#14558736	—	2	2	no	—
35	#14558736	—	2	2	no	—
36	C₂×Dic₂₉₈	C₂×C₄	1	1	yes	characteristic
37	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
42	#14558720	—	2	4	no	—
43	#14558720	—	2	2	no	—
44	#14558719	—	2	4	no	—
45	C₂×#11924746	—	2	4	no	—
46	C₂×#11924746	D₄	1	4	yes	complemented
47	C₂×#11924746	C₂³	1	2	yes	—
48	C₂×#11924746	C₂³	1	1	yes	characteristic
49	C₂×#11924746	C₂³	1	1	yes	characteristic
50	C₈×D₁₄₉	—	2	4	no	—
51	#14558715	C₂×C₄	1	4	yes	—
52	C₂²⋊Dic₁₄₉	C₂×C₄	1	4	yes	—
53	#14558713	C₂×C₄	1	4	yes	—
54	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₄₉	C₂³	1	2	yes	—
57	#41×C₁₄₉	—	2	2	no	—
58	#41×C₁₄₉	—	2	2	no	—
59	C₂×Q₈×C₁₄₉	C₂×C₄	1	1	yes	characteristic
60	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
61	C₂×D₄×C₁₄₉	C₂×C₄	1	2	yes	—
62	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
63	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
64	M₁₆×C₁₄₉	—	2	2	no	—
65	C₂×C₈×C₁₄₉	—	2	2	no	—
66	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
67	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
68	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
69	Dic₂₉₈	—	2	2	no	—
70	D₅₉₆	—	2	2	no	—
71	C₂²⋊D₁₄₉	—	2	4	no	—
72	C₂²⋊D₁₄₉	—	2	8	no	—
73	C₂×Dic₁₄₉	—	2	4	no	—
74	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
77	C₄×D₁₄₉	—	2	2	no	—
78	C₄×D₁₄₉	—	2	2	no	—
79	#11924746	—	2	2	no	—
80	#11924746	C₂×D₄	1	2	yes	—
81	#11924746	C₂×D₄	1	2	yes	—
82	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
83	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
84	Q₈×C₁₄₉	—	2	2	no	—
85	D₄×C₁₄₉	—	2	4	no	—
86	D₄×C₁₄₉	—	2	2	no	—
87	C₂×C₄×C₁₄₉	—	2	2	no	—
88	C₂×C₄×C₁₄₉	—	2	2	no	—
89	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₄₉	#41	1	2	yes	—
93	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₄₉	—	2	2	no	—
95	C₈×C₁₄₉	—	2	2	no	—
96	C₂²×C₁₄₉	—	2	2	no	—
97	C₂²×C₁₄₉	—	2	2	no	—
98	C₂²×C₁₄₉	—	2	4	no	—
99	C₂²×C₁₄₉	C₄×D₄	1	2	yes	—
100	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₄₉	—	2	2	no	—
102	C₂×D₁₄₉	—	2	4	no	—
103	C₄×C₁₄₉	—	2	2	no	—
104	C₄×C₁₄₉	C₄×D₄	1	2	yes	—
105	C₄×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₄₉	C₂×#41	1	1	yes	characteristic
107	Dic₁₄₉	—	2	2	no	—
108	Dic₁₄₉	—	2	2	no	—
109	Dic₁₄₉	#131	1	4	yes	complemented
110	C₂×C₁₄₉	—	2	2	no	—
111	C₂×C₁₄₉	—	4	2	no	—
112	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₁₄₉	C₂×#131	1	2	yes	—
114	D₁₄₉	—	4	2	no	—
115	C₁₄₉	#2931	1	1	yes	characteristic, complemented, Sylow
116	#2931	—	149	1	no	maximal, Sylow
117	C₂×#131	—	149	1	no	—
118	C₂×#131	—	149	1	no	—
119	#532	—	149	1	no	—
120	#436	—	149	4	no	—
121	C₈×D₄	—	149	4	no	—
122	#423	—	149	1	no	—
123	#408	—	149	1	no	—
124	#408	D₁₄₉	1	1	yes	characteristic, complemented
125	C₂×C₄×C₈	—	149	1	no	—
126	C₂×#41	—	149	1	no	—
127	C₂×#41	C₂×D₁₄₉	1	1	yes	characteristic
128	#131	—	298	4	no	—
129	#131	—	298	4	no	—
130	C₂×M₁₆	—	149	1	no	—
131	C₂×M₁₆	—	149	2	no	—
132	C₂×M₁₆	—	149	2	no	—
133	C₂×M₁₆	C₂×D₁₄₉	1	1	yes	characteristic
134	C₂²×C₈	—	149	2	no	—
135	C₂²×C₈	—	149	2	no	—
136	C₂²×C₈	—	149	1	no	—
137	C₂²×C₈	C₂×D₁₄₉	1	1	yes	characteristic
138	C₄⋊Q₈	—	149	1	no	—
139	#127	—	149	1	no	—
140	#124	—	149	1	no	—
141	#124	—	149	1	no	—
142	#121	—	149	4	no	—
143	C₄×D₄	—	149	4	no	—
144	C₂×C₄²	—	149	1	no	—
145	C₄⋊C₈	—	149	4	no	—
146	C₂²⋊C₈	—	149	4	no	—
147	C₂²⋊C₈	C₂×D₁₄₉	1	4	yes	—
148	C₄×C₈	—	149	2	no	—
149	C₄×C₈	—	149	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	#41	—	298	2	no	—
153	#41	—	298	2	no	—
154	C₂×Q₈	—	149	1	no	—
155	C₂×Q₈	C₄×D₁₄₉	1	1	yes	characteristic
156	C₂×D₄	—	149	4	no	—
157	C₂×D₄	—	149	2	no	—
158	C₂×D₄	—	149	1	no	—
159	C₂×D₄	C₄×D₁₄₉	1	2	yes	—
160	C₂×D₄	C₄×D₁₄₉	1	1	yes	characteristic
161	C₂²×C₄	—	149	2	no	—
162	C₂²×C₄	—	149	2	no	—
163	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	298	4	no	—
167	M₁₆	—	298	2	no	—
168	M₁₆	—	298	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	298	4	no	—
171	C₂×C₈	—	298	4	no	—
172	C₂×C₈	—	149	2	no	—
173	C₂×C₈	—	149	4	no	—
174	C₂×C₈	—	149	1	no	—
175	C₂×C₈	—	149	1	no	—
176	C₂×C₈	C₂²×D₁₄₉	1	2	yes	—
177	C₂×C₈	C₂²×D₁₄₉	1	2	yes	—
178	C₄⋊C₄	—	149	4	no	—
179	C₂²⋊C₄	—	149	4	no	—
180	C₂²⋊C₄	—	149	4	no	—
181	C₄²	—	149	1	no	—
182	C₄²	—	149	1	no	—
183	C₄²	—	149	2	no	—
184	C₂³	—	149	2	no	—
185	C₂³	C₂×C₄×D₁₄₉	1	2	yes	—
186	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	298	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	298	4	no	—
192	D₄	—	298	8	no	—
193	D₄	—	298	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	298	4	no	—
197	C₂×C₄	—	298	2	no	—
198	C₂×C₄	—	298	2	no	—
199	C₂×C₄	—	149	2	no	—
200	C₂×C₄	—	149	2	no	—
201	C₂×C₄	—	149	2	no	—
202	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
204	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
205	C₂×C₄	#14558735	1	2	yes	—
206	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	298	2	no	—
210	C₈	—	149	2	no	—
211	C₈	—	149	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	298	2	no	—
216	C₂²	—	298	4	no	—
217	C₂²	#18011871	1	2	yes	—
218	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	298	2	no	—
221	C₄	—	298	2	no	—
222	C₄	—	149	4	no	—
223	C₄	#18011871	1	2	yes	—
224	C₄	C₂×D₄×D₁₄₉	1	1	yes	characteristic
225	C₄	C₂×#14558735	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	596	2	no	—
229	C₂	C₂×#18011871	1	1	yes	characteristic
230	C₂	#131×D₁₄₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₄₉
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#408×C₁₄₉	C₂
prime radical	#23750171	C₂
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	#408	D₁₄₉
hypocenter	C₁₄₉	#2931
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#18011919	C₂	semidirect
2. C₂×#18011918	C₂	semidirect
4. #23749700	C₂	semidirect
5. #23749688	C₂	semidirect
6. #23749648	C₂	semidirect
7. #23749591	C₂	semidirect
8. C₂×C₈×Dic₁₄₉	C₂	semidirect
9. #408×C₁₄₉	C₂	semidirect
46. C₂×#11924746	D₄	semidirect
109. Dic₁₄₉	#131	semidirect
115. C₁₄₉	#2931	semidirect
124. #408	D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18011919	2
C₂×#18011918	2
#23750171	2
#23749700	2
#23749688	2
#23749648	2
#23749591	2
C₂×C₈×Dic₁₄₉	2
#408×C₁₄₉	2
#2931	149