Subgroups — #34053426

Order 19072.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₄₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₅₉₆	C₂	1	1	yes	maximal, characteristic
3	#23749702	C₂	1	2	yes	maximal, complemented
4	#23749695	C₂	1	4	yes	maximal, complemented
5	#23749690	C₂	1	2	yes	maximal, complemented
6	#23749588	C₂	1	2	yes	maximal, complemented
7	C₄×#14558720	C₂	1	1	yes	maximal, characteristic, complemented
8	#423×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₉₆	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
11	C₂×#14558720	C₂²	1	1	yes	characteristic
12	C₂×#14558720	C₂²	1	1	yes	characteristic
13	C₂×#14558719	C₂²	1	4	yes	complemented
14	M₁₆×D₁₄₉	—	2	8	no	—
15	C₂×C₈×D₁₄₉	C₂²	1	4	yes	complemented
16	C₄×D₅₉₆	C₄	1	1	yes	characteristic
17	C₄×D₅₉₆	C₄	1	2	yes	—
18	C₄×D₅₉₆	C₄	1	1	yes	characteristic
19	C₄×D₅₉₆	C₂²	1	1	yes	characteristic
20	C₄×D₅₉₆	C₂²	1	2	yes	—
21	C₄×D₅₉₆	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₄₉	C₄	1	1	yes	characteristic
23	C₂×#14558713	C₄	1	2	yes	—
24	#18011823	C₂²	1	4	yes	complemented
25	#18011823	C₂²	1	4	yes	complemented
26	#18011822	C₂²	1	2	yes	complemented
27	C₄×#11924746	C₂²	1	2	yes	complemented
28	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₁₄₉	C₂²	1	2	yes	complemented
31	C₄⋊C₈×C₁₄₉	C₂²	1	2	yes	complemented
32	C₂³×D₁₄₉	C₂×C₄	1	2	yes	—
33	C₂×D₅₉₆	—	2	4	no	—
34	C₂×D₅₉₆	C₂×C₄	1	2	yes	—
35	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
36	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₄₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₄₉	—	2	4	no	—
39	C₂×C₄×D₁₄₉	—	2	4	no	—
40	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₄₉	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
43	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
44	#14558720	—	2	2	no	—
45	#14558720	D₄	1	4	yes	complemented
46	#14558719	—	2	8	no	—
47	C₂×#11924746	C₂³	1	2	yes	—
48	C₂×#11924746	C₂³	1	2	yes	—
49	C₈×D₁₄₉	—	2	8	no	—
50	C₄⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
51	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
53	#14558713	C₂×C₄	1	2	yes	—
54	#14558713	C₂×C₄	1	2	yes	—
55	#14558713	C₂×C₄	1	2	yes	—
56	#14558713	C₂×C₄	1	2	yes	—
57	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
60	M₁₆×C₁₄₉	—	2	4	no	—
61	C₂×C₈×C₁₄₉	C₂³	1	4	yes	—
62	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₁₄₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
65	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
66	C₂²×D₁₄₉	—	2	4	no	—
67	C₂²×D₁₄₉	—	2	4	no	—
68	C₂²×D₁₄₉	—	4	2	no	—
69	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
70	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
71	D₅₉₆	—	2	4	no	—
72	D₅₉₆	—	2	2	no	—
73	D₅₉₆	M₁₆	1	4	yes	complemented
74	C₂×Dic₁₄₉	—	2	4	no	—
75	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
77	C₄×D₁₄₉	—	2	4	no	—
78	C₄×D₁₄₉	—	2	4	no	—
79	C₄×D₁₄₉	—	2	2	no	—
80	C₄×D₁₄₉	—	2	2	no	—
81	C₄×D₁₄₉	—	2	2	no	—
82	C₄×D₁₄₉	—	2	2	no	—
83	#11924746	—	2	2	no	—
84	#11924746	C₂×D₄	1	4	yes	complemented
85	C₂³×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₄₉	—	2	2	no	—
87	C₂×C₄×C₁₄₉	—	4	1	no	—
88	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₄₉	#41	1	2	yes	—
95	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
96	C₈×C₁₄₉	—	2	4	no	—
97	C₂²×C₁₄₉	—	2	1	no	—
98	C₂²×C₁₄₉	—	2	1	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	2	no	—
103	C₂×D₁₄₉	—	2	2	no	—
104	C₂×D₁₄₉	—	2	2	no	—
105	C₂×D₁₄₉	—	2	2	no	—
106	C₂×D₁₄₉	—	4	8	no	—
107	C₂×D₁₄₉	C₂×M₁₆	1	4	yes	complemented
108	C₄×C₁₄₉	—	2	1	no	—
109	C₄×C₁₄₉	—	2	1	no	—
110	C₄×C₁₄₉	—	2	1	no	—
111	C₄×C₁₄₉	C₄×D₄	1	2	yes	—
112	C₄×C₁₄₉	C₂×M₁₆	1	2	yes	complemented
113	C₄×C₁₄₉	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₁₄₉	C₂×#41	1	1	yes	characteristic
115	Dic₁₄₉	—	2	2	no	—
116	Dic₁₄₉	—	2	2	no	—
117	C₂×C₁₄₉	—	2	2	no	—
118	C₂×C₁₄₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂×C₁₄₉	C₂²×M₁₆	1	1	yes	characteristic
120	C₂×C₁₄₉	#568	1	1	yes	characteristic
121	D₁₄₉	—	2	4	no	—
122	D₁₄₉	—	4	2	no	—
123	C₁₄₉	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
124	D₄×M₁₆	—	149	1	no	maximal, Sylow
125	C₂²×M₁₆	—	149	2	no	—
126	C₂×C₄×D₄	—	149	1	no	—
127	#436	—	149	2	no	—
128	#435	—	149	4	no	—
129	C₈×D₄	—	149	2	no	—
130	#423	D₁₄₉	1	1	yes	characteristic, complemented
131	#407	—	149	2	no	—
132	C₄×M₁₆	—	149	1	no	—
133	C₂²×D₄	—	149	1	no	—
134	C₂³×C₄	—	149	2	no	—
135	C₂×M₁₆	—	298	8	no	—
136	C₂×M₁₆	—	149	1	no	—
137	C₂×M₁₆	—	149	1	no	—
138	C₂×M₁₆	—	149	4	no	—
139	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
140	C₂²×C₈	—	149	4	no	—
141	C₄×D₄	—	149	2	no	—
142	C₄×D₄	—	149	1	no	—
143	C₄×D₄	—	149	1	no	—
144	C₄×D₄	—	149	1	no	—
145	C₄×D₄	—	149	2	no	—
146	C₄×D₄	—	149	1	no	—
147	C₂×C₄⋊C₄	—	149	1	no	—
148	C₂×C₂²⋊C₄	—	149	2	no	—
149	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
150	C₄⋊C₈	C₂×D₁₄₉	1	2	yes	complemented
151	C₄⋊C₈	C₂×D₁₄₉	1	2	yes	complemented
152	C₂²⋊C₈	—	149	4	no	—
153	C₂²⋊C₈	—	149	4	no	—
154	#97	—	149	2	no	—
155	C₄×C₈	—	149	2	no	—
156	C₂⁴	—	149	2	no	—
157	C₂×D₄	—	298	4	no	—
158	C₂×D₄	—	149	2	no	—
159	C₂×D₄	—	149	1	no	—
160	C₂×D₄	—	149	1	no	—
161	C₂²×C₄	—	298	4	no	—
162	C₂²×C₄	—	298	4	no	—
163	C₂²×C₄	—	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	C₄×D₁₄₉	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₁₄₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
171	M₁₆	—	2	4	no	—
172	M₁₆	—	298	8	no	—
173	M₁₆	—	298	2	no	—
174	M₁₆	—	149	4	no	—
175	C₂×C₈	—	298	8	no	—
176	C₂×C₈	—	149	2	no	—
177	C₂×C₈	—	149	2	no	—
178	C₂×C₈	C₂²×D₁₄₉	1	4	yes	—
179	C₄⋊C₄	—	149	2	no	—
180	C₄⋊C₄	—	149	1	no	—
181	C₄⋊C₄	—	149	1	no	—
182	C₂²⋊C₄	—	149	2	no	—
183	C₂²⋊C₄	—	149	2	no	—
184	C₂²⋊C₄	—	149	2	no	—
185	C₂²⋊C₄	—	149	2	no	—
186	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
187	C₄²	C₄×D₁₄₉	1	1	yes	characteristic
188	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
189	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
190	C₂³	—	298	4	no	—
191	C₂³	—	298	4	no	—
192	C₂³	—	596	2	no	—
193	C₂³	—	149	2	no	—
194	C₂³	—	149	2	no	—
195	C₂³	C₂×C₄×D₁₄₉	1	1	yes	characteristic
196	D₄	—	298	4	no	—
197	D₄	—	298	2	no	—
198	D₄	—	149	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	298	4	no	—
202	C₂×C₄	—	298	2	no	—
203	C₂×C₄	—	298	2	no	—
204	C₂×C₄	—	298	2	no	—
205	C₂×C₄	—	298	2	no	—
206	C₂×C₄	—	298	4	no	—
207	C₂×C₄	—	298	4	no	—
208	C₂×C₄	—	149	2	no	—
209	C₂×C₄	—	149	2	no	—
210	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
215	C₂×C₄	D₄×D₁₄₉	1	2	yes	—
216	C₂×C₄	#14558737	1	2	yes	—
217	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	298	2	no	—
220	C₈	—	149	4	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	298	2	no	—
224	C₂²	—	298	2	no	—
225	C₂²	—	298	2	no	—
226	C₂²	—	298	2	no	—
227	C₂²	—	298	2	no	—
228	C₂²	—	596	8	no	—
229	C₂²	—	149	4	no	—
230	C₂²	#18011873	1	2	yes	—
231	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	298	2	no	—
236	C₄	—	298	2	no	—
237	C₄	#18011873	1	2	yes	—
238	C₄	M₁₆×D₁₄₉	1	2	yes	complemented
239	C₄	C₂×D₄×D₁₄₉	1	1	yes	characteristic
240	C₄	C₂×#14558737	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	298	4	no	—
243	C₂	—	596	2	no	—
244	C₂	C₂×#18011873	1	1	yes	characteristic
245	C₂	C₂×M₁₆×D₁₄₉	1	1	yes	characteristic
246	C₂	#23750391	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₄₉
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#423×C₁₄₉	C₂
prime radical	C₂²×D₅₉₆	C₄
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	#423	D₁₄₉
hypocenter	C₁₄₉	D₄×M₁₆
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₁₄₉	C₂	semidirect
3. #23749702	C₂	semidirect
4. #23749695	C₂	semidirect
5. #23749690	C₂	semidirect
6. #23749588	C₂	semidirect
7. C₄×#14558720	C₂	semidirect
8. #423×C₁₄₉	C₂	semidirect
13. C₂×#14558719	C₂²	semidirect
15. C₂×C₈×D₁₄₉	C₂²	semidirect
24. #18011823	C₂²	semidirect
25. #18011823	C₂²	semidirect
26. #18011822	C₂²	semidirect
27. C₄×#11924746	C₂²	semidirect
30. C₄⋊C₈×C₁₄₉	C₂²	semidirect
31. C₄⋊C₈×C₁₄₉	C₂²	semidirect
45. #14558720	D₄	semidirect
73. D₅₉₆	M₁₆	semidirect
84. #11924746	C₂×D₄	semidirect
107. C₂×D₁₄₉	C₂×M₁₆	semidirect
112. C₄×C₁₄₉	C₂×M₁₆	semidirect
123. C₁₄₉	D₄×M₁₆	semidirect
130. #423	D₁₄₉	semidirect
150. C₄⋊C₈	C₂×D₁₄₉	semidirect
151. C₄⋊C₈	C₂×D₁₄₉	semidirect
238. C₄	M₁₆×D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₄₉	2
C₂×C₄×D₅₉₆	2
#23749702	2
#23749695	2
#23749690	2
#23749588	2
C₄×#14558720	2
#423×C₁₄₉	2
D₄×M₁₆	149