Subgroups — #34058572

Order 19072.

992 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750353	C₂	1	4	yes	maximal, complemented
2	#23750347	C₂	1	4	yes	maximal, complemented
3	#23750308	C₂	1	8	yes	maximal, complemented
4	#23750233	C₂	1	4	yes	maximal, complemented
5	#23750201	C₂	1	1	yes	maximal, characteristic
6	#23750199	C₂	1	1	yes	maximal, characteristic
7	#23750179	C₂	1	4	yes	maximal, complemented
8	#23750178	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#18011909	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750102	C₂	1	1	yes	maximal, characteristic, complemented
11	#23750093	C₂	1	1	yes	maximal, characteristic
12	#519×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14558736	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
15	#18011911	C₂²	1	2	yes	—
16	#18011910	C₂²	1	2	yes	—
17	#18011909	C₂²	1	2	yes	—
18	#18011909	C₂²	1	2	yes	—
19	#18011909	C₂²	1	4	yes	—
20	#18011909	C₂²	1	2	yes	—
21	#18011906	C₂²	1	8	yes	—
22	#18011905	C₂²	1	8	yes	—
23	#18011903	C₂²	1	8	yes	—
24	#18011902	C₂²	1	2	yes	—
25	#18011901	C₂²	1	8	yes	—
26	#18011901	C₂²	1	8	yes	—
27	#18011900	C₂²	1	2	yes	—
28	#18011899	C₂²	1	2	yes	—
29	#18011895	C₂²	1	4	yes	—
30	#18011894	C₂²	1	4	yes	—
31	#18011891	C₂²	1	4	yes	—
32	#18011889	C₂²	1	8	yes	—
33	#18011888	C₂²	1	4	yes	—
34	#18011887	C₂²	1	4	yes	—
35	C₄×Dic₂₉₈	C₂²	1	4	yes	—
36	C₄×Dic₂₉₈	C₂²	1	4	yes	—
37	C₄×Dic₂₉₈	C₂²	1	8	yes	—
38	#18011876	C₂²	1	4	yes	—
39	#18011876	C₂²	1	2	yes	—
40	#18011873	C₂²	1	2	yes	—
41	#18011871	C₂²	1	4	yes	—
42	#18011870	C₂²	1	1	yes	characteristic
43	#18011868	C₂²	1	8	yes	—
44	#18011868	C₂²	1	2	yes	—
45	#18011867	C₂²	1	4	yes	—
46	C₂×#14558715	C₂²	1	4	yes	—
47	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
48	C₄²×D₁₄₉	C₂²	1	2	yes	—
49	C₂²×Q₈×C₁₄₉	C₂²	1	1	yes	characteristic
50	C₄×Q₈×C₁₄₉	C₂²	1	4	yes	—
51	C₄×Q₈×C₁₄₉	C₂²	1	4	yes	—
52	#117×C₁₄₉	C₂²	1	1	yes	characteristic
53	#117×C₁₄₉	C₂²	1	4	yes	—
54	#117×C₁₄₉	C₂²	1	1	yes	characteristic
55	#14558736	—	2	4	no	—
56	C₂×Dic₂₉₈	—	2	4	no	—
57	C₂×Dic₂₉₈	C₂³	1	1	yes	characteristic
58	C₂×Dic₂₉₈	C₂³	1	2	yes	—
59	C₂×Dic₂₉₈	C₂³	1	2	yes	—
60	C₂×Dic₂₉₈	C₂³	1	4	yes	—
61	C₂×D₅₉₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₄₉	C₂³	1	2	yes	—
63	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
64	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
65	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
66	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
67	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
68	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
69	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
70	#14558715	C₂³	1	4	yes	—
71	#14558715	C₂³	1	8	yes	—
72	#14558715	C₂³	1	8	yes	—
73	#14558715	C₂³	1	8	yes	—
74	#14558715	C₂³	1	8	yes	—
75	#14558715	C₂³	1	4	yes	—
76	#14558715	C₂³	1	4	yes	—
77	C₂²⋊Dic₁₄₉	C₂³	1	2	yes	—
78	#14558713	C₂³	1	4	yes	—
79	#14558713	C₂³	1	8	yes	—
80	#14558713	C₂³	1	8	yes	—
81	C₄×Dic₁₄₉	C₂³	1	8	yes	—
82	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
83	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
84	C₄×Dic₁₄₉	C₂³	1	2	yes	—
85	C₄×Dic₁₄₉	C₂³	1	2	yes	—
86	C₄×Dic₁₄₉	C₂³	1	4	yes	—
87	C₂×Q₈×C₁₄₉	—	2	4	no	—
88	C₂×Q₈×C₁₄₉	C₂³	1	2	yes	—
89	C₂×Q₈×C₁₄₉	C₂³	1	2	yes	—
90	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
91	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₁₄₉	C₂³	1	8	yes	—
93	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
94	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
95	C₂²⋊C₄×C₁₄₉	C₂³	1	2	yes	—
96	C₂²⋊C₄×C₁₄₉	C₂³	1	2	yes	—
97	C₄²×C₁₄₉	C₂³	1	8	yes	—
98	C₄²×C₁₄₉	C₂³	1	2	yes	—
99	C₄²×C₁₄₉	C₂³	1	2	yes	—
100	C₂²×D₁₄₉	C₂⁴	1	2	yes	—
101	Dic₂₉₈	—	2	4	no	—
102	Dic₂₉₈	—	2	4	no	—
103	Dic₂₉₈	—	2	2	no	—
104	Dic₂₉₈	—	2	8	no	—
105	D₅₉₆	—	2	2	no	—
106	C₂²⋊D₁₄₉	—	2	4	no	—
107	C₂×Dic₁₄₉	—	2	4	no	—
108	C₂×Dic₁₄₉	—	2	4	no	—
109	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
110	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
111	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
112	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
113	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
114	C₄×D₁₄₉	—	2	4	no	—
115	C₄×D₁₄₉	—	2	8	no	—
116	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
117	Q₈×C₁₄₉	—	2	4	no	—
118	Q₈×C₁₄₉	—	2	4	no	—
119	C₂×C₄×C₁₄₉	—	2	4	no	—
120	C₂×C₄×C₁₄₉	#41	1	4	yes	—
121	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
122	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
123	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
124	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
126	C₂²×C₁₄₉	—	2	2	no	—
127	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
128	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
129	C₂×D₁₄₉	—	2	2	no	—
130	C₂×D₁₄₉	—	2	4	no	—
131	C₄×C₁₄₉	—	2	4	no	—
132	C₄×C₁₄₉	—	2	4	no	—
133	C₄×C₁₄₉	—	2	4	no	—
134	C₄×C₁₄₉	C₂×#41	1	2	yes	—
135	C₄×C₁₄₉	C₂×#41	1	2	yes	—
136	Dic₁₄₉	—	2	2	no	—
137	Dic₁₄₉	—	2	4	no	—
138	Dic₁₄₉	—	2	2	no	—
139	Dic₁₄₉	—	2	4	no	—
140	Dic₁₄₉	2₋¹⁺⁴	1	4	yes	—
141	C₂×C₁₄₉	—	2	2	no	—
142	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
143	C₂×C₁₄₉	C₂×2₋¹⁺⁴	1	2	yes	—
144	D₁₄₉	—	4	2	no	—
145	C₁₄₉	#3509	1	1	yes	characteristic, complemented, Sylow
146	#3509	—	149	1	no	maximal, Sylow
147	#557	—	149	4	no	—
148	#556	—	149	4	no	—
149	#552	—	149	8	no	—
150	#541	—	149	4	no	—
151	#536	—	149	1	no	—
152	#536	—	149	1	no	—
153	#533	—	149	4	no	—
154	#533	—	149	1	no	—
155	C₂×C₄⋊Q₈	—	149	1	no	—
156	#519	—	149	1	no	—
157	#519	D₁₄₉	1	1	yes	characteristic, complemented
158	C₄×#41	—	149	1	no	—
159	C₂×#41	—	149	1	no	—
160	C₂²×Q₈	—	149	1	no	—
161	C₂²×Q₈	C₂×D₁₄₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	149	2	no	—
163	C₄⋊Q₈	—	149	2	no	—
164	C₄⋊Q₈	—	149	4	no	—
165	C₄⋊Q₈	—	149	2	no	—
166	C₄⋊Q₈	—	149	2	no	—
167	C₄⋊Q₈	—	149	2	no	—
168	#126	—	149	8	no	—
169	#126	—	149	8	no	—
170	#125	—	149	2	no	—
171	#125	—	149	8	no	—
172	#125	—	149	8	no	—
173	#125	—	149	8	no	—
174	#124	—	149	2	no	—
175	#124	—	149	2	no	—
176	#123	—	149	4	no	—
177	#123	—	149	4	no	—
178	C₂²⋊Q₈	—	149	4	no	—
179	C₂²⋊Q₈	—	149	4	no	—
180	C₂²⋊Q₈	—	149	8	no	—
181	C₂²⋊Q₈	—	149	4	no	—
182	C₄×Q₈	—	149	2	no	—
183	C₄×Q₈	—	149	4	no	—
184	C₄×Q₈	—	149	4	no	—
185	C₄×Q₈	—	149	8	no	—
186	C₄×Q₈	—	149	4	no	—
187	C₄×Q₈	C₂×D₁₄₉	1	4	yes	—
188	C₄×Q₈	C₂×D₁₄₉	1	4	yes	—
189	C₄×D₄	—	149	2	no	—
190	C₄×D₄	—	149	4	no	—
191	#117	—	149	1	no	—
192	#117	—	149	4	no	—
193	#117	—	149	2	no	—
194	#117	—	149	8	no	—
195	#117	C₂×D₁₄₉	1	1	yes	characteristic
196	#117	C₂×D₁₄₉	1	1	yes	characteristic
197	#117	C₂×D₁₄₉	1	4	yes	—
198	C₂×C₄⋊C₄	—	149	4	no	—
199	C₂×C₄²	—	149	1	no	—
200	C₂×C₄²	—	149	2	no	—
201	#41	—	298	4	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	298	4	no	—
204	C₂×Q₈	—	149	4	no	—
205	C₂×Q₈	—	149	2	no	—
206	C₂×Q₈	—	149	2	no	—
207	C₂×Q₈	—	149	1	no	—
208	C₂×Q₈	C₂²×D₁₄₉	1	2	yes	—
209	C₂×Q₈	C₂²×D₁₄₉	1	2	yes	—
210	C₂×D₄	—	149	2	no	—
211	C₂×D₄	—	149	1	no	—
212	C₂²×C₄	—	149	2	no	—
213	C₂²×C₄	—	149	2	no	—
214	C₂²×C₄	—	149	4	no	—
215	C₂²×C₄	—	149	2	no	—
216	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
217	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
218	C₄⋊C₄	—	149	8	no	—
219	C₄⋊C₄	—	149	8	no	—
220	C₄⋊C₄	—	149	4	no	—
221	C₄⋊C₄	—	149	4	no	—
222	C₄⋊C₄	—	149	2	no	—
223	C₄⋊C₄	—	149	8	no	—
224	C₄⋊C₄	—	149	8	no	—
225	C₄⋊C₄	—	149	8	no	—
226	C₄⋊C₄	—	149	4	no	—
227	C₄⋊C₄	—	149	4	no	—
228	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
229	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
230	C₄⋊C₄	C₂²×D₁₄₉	1	8	yes	—
231	C₂²⋊C₄	—	149	2	no	—
232	C₂²⋊C₄	—	149	4	no	—
233	C₂²⋊C₄	—	149	8	no	—
234	C₂²⋊C₄	—	149	8	no	—
235	C₂²⋊C₄	C₂²×D₁₄₉	1	2	yes	—
236	C₂²⋊C₄	C₂²×D₁₄₉	1	2	yes	—
237	C₄²	—	149	1	no	—
238	C₄²	—	149	1	no	—
239	C₄²	—	149	4	no	—
240	C₄²	—	149	2	no	—
241	C₄²	—	149	8	no	—
242	C₄²	—	149	2	no	—
243	C₄²	C₂²×D₁₄₉	1	2	yes	—
244	C₄²	C₂²×D₁₄₉	1	8	yes	—
245	C₄²	C₂²×D₁₄₉	1	2	yes	—
246	C₂³	—	149	2	no	—
247	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
248	Q₈	—	2	4	no	—
249	Q₈	—	2	4	no	—
250	Q₈	—	298	8	no	—
251	Q₈	—	298	4	no	—
252	Q₈	—	298	4	no	—
253	Q₈	—	298	2	no	—
254	D₄	—	298	4	no	—
255	D₄	—	298	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	298	8	no	—
258	C₂×C₄	—	298	4	no	—
259	C₂×C₄	—	298	4	no	—
260	C₂×C₄	—	298	4	no	—
261	C₂×C₄	—	149	4	no	—
262	C₂×C₄	—	149	2	no	—
263	C₂×C₄	—	149	4	no	—
264	C₂×C₄	—	149	2	no	—
265	C₂×C₄	—	149	2	no	—
266	C₂×C₄	#14558736	1	4	yes	—
267	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
268	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
269	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
270	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
271	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
272	C₂²	—	2	2	no	—
273	C₂²	—	298	2	no	—
274	C₂²	—	298	4	no	—
275	C₂²	C₂×#14558736	1	2	yes	—
276	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	4	no	—
280	C₄	—	298	2	no	—
281	C₄	—	298	4	no	—
282	C₄	—	298	2	no	—
283	C₄	—	298	4	no	—
284	C₄	—	149	4	no	—
285	C₄	C₂×#14558736	1	2	yes	—
286	C₄	C₂×#14558736	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	596	2	no	—
289	C₂	C₂²×#14558736	1	1	yes	characteristic
290	C₂	2₋¹⁺⁴×D₁₄₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	#519×C₁₄₉	C₂
prime radical	C₂×#14558736	C₂²
hypercenter	#519	D₁₄₉
hypocenter	C₁₄₉	#3509
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #519

Complements

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

N	H ≅ G/N	type
1. #23750353	C₂	semidirect
2. #23750347	C₂	semidirect
3. #23750308	C₂	semidirect
4. #23750233	C₂	semidirect
7. #23750179	C₂	semidirect
8. #23750178	C₂	semidirect
9. C₂×#18011909	C₂	semidirect
10. #23750102	C₂	semidirect
12. #519×C₁₄₉	C₂	semidirect
145. C₁₄₉	#3509	semidirect
157. #519	D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23750353	2
#23750347	2
#23750308	2
#23750233	2
#23750201	2
#23750199	2
#23750179	2
#23750178	2
C₂×#18011909	2
#23750102	2
#23750093	2
#519×C₁₄₉	2
#3509	149