Subgroups — #34053756

Order 19072.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750172	C₂	1	1	yes	maximal, characteristic
2	#23749702	C₂	1	1	yes	maximal, characteristic, complemented
3	#23749701	C₂	1	2	yes	maximal, complemented
4	C₈×D₅₉₆	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#11924746	C₂	1	1	yes	maximal, characteristic, complemented
6	#23749688	C₂	1	2	yes	maximal, complemented
7	#23749677	C₂	1	1	yes	maximal, characteristic, complemented
8	#23749676	C₂	1	1	yes	maximal, characteristic, complemented
9	#23749593	C₂	1	2	yes	maximal, complemented
10	#23749590	C₂	1	2	yes	maximal, complemented
11	#436×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14558736	C₂²	1	2	yes	—
13	C₂×#14558719	C₂²	1	2	yes	—
14	C₂²×#11924746	C₂²	1	2	yes	—
15	C₂×C₈×D₁₄₉	C₂²	1	2	yes	—
16	#18011910	C₄	1	1	yes	characteristic
17	#18011907	C₄	1	1	yes	characteristic
18	#18011897	C₄	1	2	yes	—
19	#18011886	C₄	1	1	yes	characteristic
20	#18011884	C₄	1	2	yes	—
21	#18011882	C₄	1	1	yes	characteristic
22	C₄×D₅₉₆	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₄₉	C₂²	1	2	yes	—
24	C₄²×D₁₄₉	C₂²	1	1	yes	characteristic
25	C₄⋊#11924746	C₂²	1	1	yes	characteristic
26	#18011840	C₂²	1	1	yes	characteristic
27	#18011840	C₂²	1	1	yes	characteristic
28	C₂²⋊#11924746	C₂²	1	2	yes	—
29	#18011823	C₂²	1	1	yes	characteristic
30	#18011823	C₂²	1	1	yes	characteristic
31	#18011823	C₂²	1	2	yes	—
32	#18011823	C₂²	1	2	yes	—
33	#18011823	C₂²	1	2	yes	—
34	C₄×#11924746	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₄₉	C₂²	1	2	yes	—
36	C₂×M₁₆×C₁₄₉	C₂²	1	2	yes	—
37	C₄×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₄₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₁₄₉	C₂²	1	2	yes	—
40	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
41	#14558736	—	4	2	no	—
42	#14558736	—	4	2	no	—
43	C₂×Dic₂₉₈	C₂×C₄	1	2	yes	—
44	C₂×D₅₉₆	C₂×C₄	1	1	yes	characteristic
45	C₂×D₅₉₆	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₄₉	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₄₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
52	C₂×C₄×D₁₄₉	C₂³	1	1	yes	characteristic
53	#14558719	—	4	2	no	—
54	C₂×#11924746	—	4	2	no	—
55	C₂×#11924746	C₂³	1	2	yes	—
56	C₂×#11924746	C₂³	1	1	yes	characteristic
57	C₂×#11924746	C₂³	1	1	yes	characteristic
58	C₈×D₁₄₉	—	4	2	no	—
59	C₄⋊Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
60	#14558715	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₄₉	C₂×C₄	1	2	yes	—
62	#14558713	C₂×C₄	1	2	yes	—
63	#14558713	C₂×C₄	1	2	yes	—
64	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₄₉	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
69	M₁₆×C₁₄₉	—	4	2	no	—
70	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₄₉	C₂³	1	2	yes	—
72	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₄₉	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₄₉	C₂×C₄	1	2	yes	—
75	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
76	C₂²×D₁₄₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₄₉	C₂²×C₄	1	2	yes	—
78	Dic₂₉₈	—	4	2	no	—
79	D₅₉₆	—	2	2	no	—
80	D₅₉₆	—	4	2	no	—
81	C₂²⋊D₁₄₉	—	2	4	no	—
82	C₂²⋊D₁₄₉	—	4	2	no	—
83	C₂²⋊D₁₄₉	—	4	2	no	—
84	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₁₄₉	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₄₉	C₂²×C₄	1	1	yes	characteristic
88	C₄×D₁₄₉	—	2	2	no	—
89	C₄×D₁₄₉	—	2	1	no	—
90	C₄×D₁₄₉	—	2	1	no	—
91	C₄×D₁₄₉	—	2	2	no	—
92	C₄×D₁₄₉	—	2	2	no	—
93	C₄×D₁₄₉	—	2	2	no	—
94	#11924746	—	2	2	no	—
95	#11924746	—	2	1	no	—
96	#11924746	—	2	1	no	—
97	C₂³×C₁₄₉	C₂²×C₄	1	2	yes	—
98	D₄×C₁₄₉	—	2	2	no	—
99	C₂×C₄×C₁₄₉	—	2	2	no	—
100	C₂×C₄×C₁₄₉	—	2	2	no	—
101	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₄₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₁₄₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
105	C₈×C₁₄₉	—	2	1	no	—
106	C₈×C₁₄₉	—	2	2	no	—
107	C₈×C₁₄₉	—	2	1	no	—
108	C₂²×C₁₄₉	—	2	2	no	—
109	C₂²×C₁₄₉	—	2	2	no	—
110	C₂²×C₁₄₉	—	2	2	no	—
111	C₂²×C₁₄₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₁₄₉	—	2	1	no	—
113	C₂×D₁₄₉	—	2	2	no	—
114	C₂×D₁₄₉	—	2	1	no	—
115	C₂×D₁₄₉	—	2	1	no	—
116	C₂×D₁₄₉	—	2	2	no	—
117	C₂×D₁₄₉	—	2	2	no	—
118	C₄×C₁₄₉	—	2	2	no	—
119	C₄×C₁₄₉	—	2	1	no	—
120	C₄×C₁₄₉	#131	1	2	yes	—
121	C₄×C₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₁₄₉	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₁₄₉	—	2	1	no	—
124	Dic₁₄₉	—	2	2	no	—
125	Dic₁₄₉	—	2	1	no	—
126	Dic₁₄₉	#131	1	2	yes	—
127	C₂×C₁₄₉	—	4	2	no	—
128	C₂×C₁₄₉	#518	1	1	yes	characteristic
129	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₄₉	C₂×#131	1	1	yes	characteristic
131	D₁₄₉	—	4	1	no	—
132	D₁₄₉	—	4	2	no	—
133	C₁₄₉	#2979	1	1	yes	characteristic, complemented, Sylow
134	#2979	—	149	1	no	maximal, Sylow
135	#532	—	149	1	no	—
136	#436	—	149	2	no	—
137	#436	—	149	1	no	—
138	#436	D₁₄₉	1	1	yes	characteristic, complemented
139	C₈×D₄	—	149	1	no	—
140	C₈×D₄	—	149	2	no	—
141	C₈×D₄	—	149	1	no	—
142	#431	—	149	1	no	—
143	#431	—	149	1	no	—
144	#408	—	149	2	no	—
145	#408	—	149	2	no	—
146	C₂×#41	—	149	2	no	—
147	C₂×M₁₆	—	149	2	no	—
148	C₂×M₁₆	C₂×D₁₄₉	1	2	yes	—
149	C₂²×C₈	—	149	2	no	—
150	C₂²×C₈	—	149	2	no	—
151	C₄⋊Q₈	—	149	1	no	—
152	#127	—	149	1	no	—
153	#124	—	149	2	no	—
154	#121	—	149	1	no	—
155	#121	—	149	1	no	—
156	#121	—	149	2	no	—
157	C₄×D₄	—	149	2	no	—
158	C₄×D₄	—	149	1	no	—
159	C₄×D₄	C₂×D₁₄₉	1	1	yes	characteristic
160	C₂×C₄²	—	149	1	no	—
161	C₄⋊C₈	—	149	1	no	—
162	C₄⋊C₈	—	149	1	no	—
163	C₄⋊C₈	—	149	1	no	—
164	C₄⋊C₈	C₂×D₁₄₉	1	1	yes	characteristic
165	C₂²⋊C₈	—	149	2	no	—
166	C₂²⋊C₈	—	149	2	no	—
167	C₂²⋊C₈	—	149	1	no	—
168	C₂²⋊C₈	—	149	1	no	—
169	C₂²⋊C₈	—	149	2	no	—
170	C₂²⋊C₈	—	149	2	no	—
171	C₂²⋊C₈	C₂×D₁₄₉	1	2	yes	—
172	C₄×C₈	—	149	2	no	—
173	C₄×C₈	—	149	1	no	—
174	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
175	#41	—	596	2	no	—
176	#41	—	596	2	no	—
177	C₂×Q₈	—	149	2	no	—
178	C₂×D₄	—	149	2	no	—
179	C₂×D₄	—	149	2	no	—
180	C₂×D₄	—	149	2	no	—
181	C₂×D₄	—	149	1	no	—
182	C₂×D₄	—	149	2	no	—
183	C₂×D₄	C₄×D₁₄₉	1	1	yes	characteristic
184	C₂²×C₄	—	149	1	no	—
185	C₂²×C₄	—	149	1	no	—
186	C₂²×C₄	—	149	1	no	—
187	C₂²×C₄	—	149	2	no	—
188	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
189	M₁₆	—	4	2	no	—
190	M₁₆	—	596	2	no	—
191	C₂×C₈	—	596	2	no	—
192	C₂×C₈	—	596	2	no	—
193	C₂×C₈	—	149	2	no	—
194	C₂×C₈	—	149	1	no	—
195	C₂×C₈	—	149	1	no	—
196	C₂×C₈	C₂²×D₁₄₉	1	2	yes	—
197	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
199	C₄⋊C₄	—	149	2	no	—
200	C₄⋊C₄	—	149	1	no	—
201	C₄⋊C₄	C₄×D₁₄₉	1	1	yes	characteristic
202	C₂²⋊C₄	—	149	2	no	—
203	C₂²⋊C₄	—	149	2	no	—
204	C₂²⋊C₄	—	149	2	no	—
205	C₂²⋊C₄	C₄×D₁₄₉	1	2	yes	—
206	C₄²	—	149	1	no	—
207	C₄²	—	149	1	no	—
208	C₄²	—	149	1	no	—
209	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
210	C₂³	—	149	2	no	—
211	C₂³	—	149	1	no	—
212	C₂³	C₂×C₄×D₁₄₉	1	2	yes	—
213	Q₈	—	596	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	298	4	no	—
216	D₄	—	298	2	no	—
217	D₄	—	596	2	no	—
218	D₄	—	596	2	no	—
219	D₄	—	596	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	298	2	no	—
223	C₂×C₄	—	298	2	no	—
224	C₂×C₄	—	298	2	no	—
225	C₂×C₄	—	298	1	no	—
226	C₂×C₄	—	298	2	no	—
227	C₂×C₄	—	298	1	no	—
228	C₂×C₄	—	149	1	no	—
229	C₂×C₄	—	149	1	no	—
230	C₂×C₄	—	149	2	no	—
231	C₂×C₄	—	149	1	no	—
232	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₄₉	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₁₄₉	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	298	1	no	—
240	C₈	—	298	2	no	—
241	C₈	—	298	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	298	1	no	—
246	C₂²	—	298	2	no	—
247	C₂²	—	298	1	no	—
248	C₂²	—	298	2	no	—
249	C₂²	—	298	1	no	—
250	C₂²	—	298	2	no	—
251	C₂²	C₂²×C₄×D₁₄₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	298	1	no	—
255	C₄	—	298	2	no	—
256	C₄	—	298	1	no	—
257	C₄	—	149	2	no	—
258	C₄	#18011919	1	2	yes	—
259	C₄	#18011956	1	1	yes	characteristic
260	C₄	#18011957	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	596	1	no	—
263	C₂	—	596	2	no	—
264	C₂	#23750097	1	1	yes	characteristic
265	C₂	#131×D₁₄₉	1	1	yes	characteristic
266	C₂	C₂×#18011919	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₄₉
commutator, socle	C₂²×C₁₄₉	C₂³×C₄
Fitting	#436×C₁₄₉	C₂
prime radical	#23750172	C₂
cosocle	C₂×C₄×C₁₄₉	C₂⁴
hypercenter	#436	D₁₄₉
hypocenter	C₁₄₉	#2979
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. #23749702	C₂	semidirect
3. #23749701	C₂	semidirect
4. C₈×D₅₉₆	C₂	semidirect
5. D₄×#11924746	C₂	semidirect
6. #23749688	C₂	semidirect
7. #23749677	C₂	semidirect
8. #23749676	C₂	semidirect
9. #23749593	C₂	semidirect
10. #23749590	C₂	semidirect
11. #436×C₁₄₉	C₂	semidirect
133. C₁₄₉	#2979	semidirect
138. #436	D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23750172	2
#23749702	2
#23749701	2
C₈×D₅₉₆	2
D₄×#11924746	2
#23749688	2
#23749677	2
#23749676	2
#23749593	2
#23749590	2
#436×C₁₄₉	2
#2979	149