Subgroups — #34077942

Order 19328.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18012765	C₂	1	1	yes	maximal, characteristic
2	#23752692	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752322	C₂	1	2	yes	maximal, complemented
4	#23752312	C₂	1	2	yes	maximal, complemented
5	C₄×#14558992	C₂	1	2	yes	maximal, complemented
6	C₄×#14558989	C₂	1	2	yes	maximal, complemented
7	#23752255	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752206	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#18012660	C₂	1	1	yes	maximal, characteristic, complemented
10	#23752168	C₂	1	1	yes	maximal, characteristic, complemented
11	#518×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14559004	C₂²	1	1	yes	characteristic
13	#18012765	C₄	1	8	yes	complemented
14	C₂×#14558992	C₂²	1	2	yes	—
15	C₂×#14558989	C₂²	1	2	yes	—
16	C₂×#14558988	C₂²	1	1	yes	characteristic
17	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₃₀₂	C₂²	1	2	yes	complemented
19	C₄×D₆₀₄	C₂²	1	2	yes	complemented
20	(C₂²⋊D₁₅₁)⋊C₄	—	2	2	no	—
21	#18012697	—	2	1	no	—
22	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₁₅₁	—	2	1	no	—
24	#18012672	C₂²	1	2	yes	complemented
25	#18012670	C₂²	1	2	yes	complemented
26	#18012662	C₂²	1	1	yes	characteristic, complemented
27	#18012662	C₂²	1	1	yes	characteristic, complemented
28	#18012660	C₂²	1	2	yes	—
29	#18012660	C₂²	1	2	yes	—
30	#18012659	C₂²	1	1	yes	characteristic, complemented
31	#18012659	C₂²	1	1	yes	characteristic, complemented
32	#18012650	C₂²	1	2	yes	complemented
33	C₄×C₁₅₁⋊C₈	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₅₁	—	2	1	no	—
36	C₄×D₄×C₁₅₁	—	2	1	no	—
37	C₄×D₄×C₁₅₁	C₂²	1	2	yes	—
38	C₄×D₄×C₁₅₁	C₂²	1	2	yes	—
39	#117×C₁₅₁	C₂²	1	1	yes	characteristic
40	#117×C₁₅₁	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₅₁	—	2	2	no	—
42	#14559004	—	2	2	no	—
43	#14559004	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
45	C₂×D₆₀₄	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₅₁	—	2	1	no	—
47	C₂²×Dic₁₅₁	—	2	1	no	—
48	C₂×C₄×D₁₅₁	—	2	1	no	—
49	C₂×C₄×D₁₅₁	—	2	2	no	—
50	#14558992	C₂×C₄	1	8	yes	complemented
51	#14558989	C₂×C₄	1	8	yes	complemented
52	#14558988	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₅₁⋊C₈	—	2	2	no	—
54	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
55	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
57	C₄⋊Dic₁₅₁	—	2	1	no	—
58	C₄⋊Dic₁₅₁	C₂³	1	2	yes	—
59	Dic₁₅₁⋊C₄	—	2	2	no	—
60	(C₂×D₁₅₁)⋊C₄	—	2	2	no	—
61	C₄×Dic₁₅₁	—	2	2	no	—
62	C₂⁴×C₁₅₁	—	2	1	no	—
63	C₂×D₄×C₁₅₁	—	2	1	no	—
64	C₂×D₄×C₁₅₁	—	2	2	no	—
65	C₂×D₄×C₁₅₁	C₂×C₄	1	4	yes	complemented
66	C₂×D₄×C₁₅₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₅₁	—	2	2	no	—
68	C₂²×C₄×C₁₅₁	—	2	2	no	—
69	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₅₁	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₅₁	D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
75	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
76	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
77	C₂²⋊C₄×C₁₅₁	D₄	1	2	yes	—
78	C₄²×C₁₅₁	—	2	1	no	—
79	C₄²×C₁₅₁	C₂³	1	2	yes	—
80	C₂²×D₁₅₁	—	2	1	no	—
81	Dic₃₀₂	—	2	1	no	—
82	Dic₃₀₂	C₂²×C₄	1	2	yes	—
83	D₆₀₄	—	2	1	no	—
84	D₆₀₄	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₁₅₁	—	2	4	no	—
86	C₂×Dic₁₅₁	—	2	1	no	—
87	C₂×Dic₁₅₁	—	2	2	no	—
88	C₂×Dic₁₅₁	—	8	1	no	—
89	C₄×D₁₅₁	—	2	4	no	—
90	C₄×D₁₅₁	—	8	2	no	—
91	C₁₅₁⋊C₈	—	2	1	no	—
92	C₁₅₁⋊C₈	—	2	1	no	—
93	C₁₅₁⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂³×C₁₅₁	—	2	2	no	—
95	C₂³×C₁₅₁	—	2	4	no	—
96	C₂³×C₁₅₁	—	8	1	no	—
97	C₂³×C₁₅₁	—	8	1	no	—
98	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
99	D₄×C₁₅₁	—	2	4	no	—
100	D₄×C₁₅₁	—	2	2	no	—
101	D₄×C₁₅₁	C₂²×C₄	1	4	yes	—
102	C₂×C₄×C₁₅₁	—	2	1	no	—
103	C₂×C₄×C₁₅₁	—	2	1	no	—
104	C₂×C₄×C₁₅₁	—	8	2	no	—
105	C₂×C₄×C₁₅₁	—	8	2	no	—
106	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₅₁	#41	1	2	yes	—
111	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
112	C₂²×C₁₅₁	—	2	1	no	—
113	C₂²×C₁₅₁	—	2	1	no	—
114	C₂²×C₁₅₁	—	2	4	no	—
115	C₂²×C₁₅₁	—	4	2	no	—
116	C₂²×C₁₅₁	—	4	2	no	—
117	C₂²×C₁₅₁	—	8	4	no	—
118	C₂²×C₁₅₁	C₄×D₄	1	2	yes	—
119	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₁₅₁	—	2	2	no	—
121	C₂×D₁₅₁	—	4	1	no	—
122	C₂×D₁₅₁	—	4	1	no	—
123	C₄×C₁₅₁	—	2	2	no	—
124	C₄×C₁₅₁	—	2	2	no	—
125	C₄×C₁₅₁	—	4	1	no	—
126	C₄×C₁₅₁	—	4	1	no	—
127	C₄×C₁₅₁	C₄×D₄	1	2	yes	—
128	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
130	Dic₁₅₁	—	2	2	no	—
131	Dic₁₅₁	—	4	2	no	—
132	C₂×C₁₅₁	—	2	2	no	—
133	C₂×C₁₅₁	—	4	4	no	—
134	C₂×C₁₅₁	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₁₅₁	#576	1	1	yes	characteristic
136	C₂×C₁₅₁	#577	1	1	yes	characteristic
137	D₁₅₁	—	4	2	no	—
138	C₁₅₁	#2946	1	1	yes	characteristic, complemented, Sylow
139	#2946	—	151	1	no	maximal, Sylow
140	C₂×AGL(1,ℤ/8)	—	151	1	no	—
141	#520	—	151	1	no	—
142	#518	D₁₅₁	1	1	yes	characteristic, complemented
143	#442	—	151	2	no	—
144	#440	—	151	2	no	—
145	C₄×SD₁₆	—	151	2	no	—
146	C₄×D₈	—	151	2	no	—
147	#428	—	151	1	no	—
148	#416	—	151	1	no	—
149	C₂×#102	—	151	1	no	—
150	#405	—	151	1	no	—
151	C₂×#41	—	151	1	no	—
152	C₂²×D₄	C₂×D₁₅₁	1	1	yes	characteristic
153	AGL(1,ℤ/8)	—	151	8	no	—
154	C₂×SD₁₆	—	151	2	no	—
155	C₂×D₈	—	151	2	no	—
156	C₂×M₁₆	—	151	1	no	—
157	C₂²×C₈	—	151	1	no	—
158	C₄×Q₈	—	151	2	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	302	2	no	—
162	C₄×D₄	—	151	2	no	—
163	C₄×D₄	C₂×D₁₅₁	1	2	yes	—
164	C₄×D₄	C₂×D₁₅₁	1	2	yes	—
165	#117	—	302	1	no	—
166	#117	C₂×D₁₅₁	1	1	yes	characteristic
167	#117	C₂×D₁₅₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	302	1	no	—
169	C₂×C₄⋊C₄	—	151	1	no	—
170	C₂×C₂²⋊C₄	—	2	2	no	—
171	#107	—	151	2	no	—
172	#106	—	151	2	no	—
173	#103	—	151	1	no	—
174	#103	—	151	1	no	—
175	#102	—	151	2	no	—
176	#102	—	151	2	no	—
177	#102	—	151	1	no	—
178	#102	—	151	1	no	—
179	#97	—	151	2	no	—
180	C₄×C₈	—	151	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	302	2	no	—
183	#41	—	151	4	no	—
184	C₂×Q₈	—	151	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	302	1	no	—
188	C₂×D₄	—	151	1	no	—
189	C₂×D₄	C₄×D₁₅₁	1	4	yes	complemented
190	C₂×D₄	C₂²×D₁₅₁	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	302	1	no	—
194	C₂²×C₄	—	302	1	no	—
195	C₂²×C₄	—	302	2	no	—
196	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
197	SD₁₆	—	151	8	no	—
198	D₈	—	151	8	no	—
199	M₁₆	—	151	4	no	—
200	C₂×C₈	—	302	2	no	—
201	C₂×C₈	—	151	1	no	—
202	C₂×C₈	—	151	1	no	—
203	C₂×C₈	—	151	2	no	—
204	C₄⋊C₄	—	302	2	no	—
205	C₄⋊C₄	—	302	1	no	—
206	C₄⋊C₄	—	151	2	no	—
207	C₄⋊C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₁₅₁	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	302	2	no	—
215	C₂²⋊C₄	C₂²⋊D₁₅₁	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	302	2	no	—
218	C₄²	C₂²×D₁₅₁	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	302	1	no	—
224	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
225	Q₈	—	302	1	no	—
226	Q₈	—	151	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	302	4	no	—
230	D₄	—	302	1	no	—
231	D₄	—	151	2	no	—
232	D₄	C₂×C₄×D₁₅₁	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	302	1	no	—
238	C₂×C₄	—	302	4	no	—
239	C₂×C₄	—	302	2	no	—
240	C₂×C₄	—	1208	2	no	—
241	C₂×C₄	—	1208	1	no	—
242	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
246	C₂×C₄	#14559004	1	2	yes	—
247	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
248	C₈	—	302	1	no	—
249	C₈	—	302	1	no	—
250	C₈	—	151	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	302	2	no	—
258	C₂²	—	604	1	no	—
259	C₂²	—	604	1	no	—
260	C₂²	C₄×C₂²⋊D₁₅₁	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	302	2	no	—
267	C₄	—	604	2	no	—
268	C₄	C₄×C₂²⋊D₁₅₁	1	2	yes	—
269	C₄	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
270	C₄	C₂×#14559004	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	604	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₁₅₁	1	1	yes	characteristic
275	C₂	#23753015	1	1	yes	characteristic
276	C₂	#23753023	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₅₁
commutator	C₄×C₁₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#518×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×#18012765	C₂
cosocle	C₂×C₄×C₁₅₁	C₂⁴
hypercenter	#518	D₁₅₁
hypocenter	C₁₅₁	#2946
LCS	G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #23752692	C₂	semidirect
3. #23752322	C₂	semidirect
4. #23752312	C₂	semidirect
5. C₄×#14558992	C₂	semidirect
6. C₄×#14558989	C₂	semidirect
7. #23752255	C₂	semidirect
8. #23752206	C₂	semidirect
9. C₂×#18012660	C₂	semidirect
10. #23752168	C₂	semidirect
11. #518×C₁₅₁	C₂	semidirect
13. #18012765	C₄	semidirect
18. C₄×Dic₃₀₂	C₂²	semidirect
19. C₄×D₆₀₄	C₂²	semidirect
24. #18012672	C₂²	semidirect
25. #18012670	C₂²	semidirect
26. #18012662	C₂²	semidirect
27. #18012662	C₂²	semidirect
30. #18012659	C₂²	semidirect
31. #18012659	C₂²	semidirect
32. #18012650	C₂²	semidirect
33. C₄×C₁₅₁⋊C₈	C₂²	semidirect
43. #14559004	C₂×C₄	semidirect
50. #14558992	C₂×C₄	semidirect
51. #14558989	C₂×C₄	semidirect
52. #14558988	C₂×C₄	semidirect
65. C₂×D₄×C₁₅₁	C₂×C₄	semidirect
138. C₁₅₁	#2946	semidirect
142. #518	D₁₅₁	semidirect
189. C₂×D₄	C₄×D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18012765	2
#23752692	2
#23752322	2
#23752312	2
C₄×#14558992	2
C₄×#14558989	2
#23752255	2
#23752206	2
C₂×#18012660	2
#23752168	2
#518×C₁₅₁	2
#2946	151