Subgroups — #34077639

Order 19328.

726 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#18012786	C₂	1	1	yes	maximal, characteristic
2	C₂×#18012745	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752692	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752220	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752219	C₂	1	2	yes	maximal, complemented
6	#23752213	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752210	C₂	1	2	yes	maximal, complemented
8	#23752205	C₂	1	2	yes	maximal, complemented
9	C₂×#18012664	C₂	1	1	yes	maximal, characteristic, complemented
10	#103×D₁₅₁	C₂	1	2	yes	maximal, complemented
11	#416×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#18012786	—	2	4	no	—
13	#18012786	C₄	1	8	yes	complemented
14	C₂×#14559004	—	2	1	no	—
15	C₂×#14559004	C₂²	1	1	yes	characteristic
16	C₂×#14559003	—	2	2	no	—
17	C₂×#14559003	C₂²	1	2	yes	—
18	C₂²×Dic₃₀₂	—	2	1	no	—
19	C₂²×Dic₃₀₂	C₂²	1	1	yes	characteristic
20	C₂×Q₈×D₁₅₁	C₂²	1	2	yes	—
21	#18012745	—	2	4	no	—
22	C₂×#14558988	C₂²	1	1	yes	characteristic
23	C₂×#14558987	C₂²	1	2	yes	—
24	C₂×C₈×D₁₅₁	C₂²	1	2	yes	—
25	C₄×Dic₃₀₂	—	2	1	no	—
26	C₄×D₆₀₄	—	2	1	no	—
27	(C₂²⋊D₁₅₁)⋊C₄	—	2	2	no	—
28	#18012697	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₁₅₁	C₂²	1	2	yes	—
31	#18012664	C₂²	1	2	yes	—
32	#18012664	C₂²	1	2	yes	—
33	#18012663	C₂²	1	2	yes	—
34	#18012662	C₂²	1	2	yes	—
35	#18012662	C₂²	1	2	yes	—
36	#18012660	C₂²	1	2	yes	—
37	C₂×#41×C₁₅₁	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
39	#117×C₁₅₁	C₂²	1	1	yes	characteristic
40	#103×C₁₅₁	C₂²	1	1	yes	characteristic
41	#103×C₁₅₁	C₂²	1	1	yes	characteristic
42	#102×C₁₅₁	C₂²	1	1	yes	characteristic
43	#102×C₁₅₁	C₂²	1	1	yes	characteristic
44	#14559004	—	2	8	no	—
45	#14559004	C₂×C₄	1	4	yes	—
46	#14559004	D₄	1	4	yes	—
47	#14559003	—	2	16	no	—
48	#14559003	—	2	4	no	—
49	#14559003	C₂×C₄	1	8	yes	complemented
50	C₂×Dic₃₀₂	—	2	8	no	—
51	C₂×Dic₃₀₂	—	2	2	no	—
52	C₂×Dic₃₀₂	—	2	1	no	—
53	C₂×Dic₃₀₂	—	2	2	no	—
54	C₂×Dic₃₀₂	C₂×C₄	1	4	yes	complemented
55	C₂×Dic₃₀₂	D₄	1	1	yes	characteristic
56	C₂×Dic₃₀₂	C₂³	1	2	yes	—
57	Q₈×D₁₅₁	—	2	4	no	—
58	Q₈×D₁₅₁	C₂×C₄	1	8	yes	complemented
59	C₂×D₆₀₄	D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₅₁	—	2	2	no	—
61	C₂×C₂²⋊D₁₅₁	D₄	1	2	yes	—
62	C₂²×Dic₁₅₁	—	2	2	no	—
63	C₂²×Dic₁₅₁	—	2	1	no	—
64	C₂²×Dic₁₅₁	—	2	1	no	—
65	C₂×C₄×D₁₅₁	—	2	2	no	—
66	C₂×C₄×D₁₅₁	—	2	2	no	—
67	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
68	#14558988	—	2	2	no	—
69	#14558987	—	2	4	no	—
70	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
71	C₈×D₁₅₁	—	2	4	no	—
72	C₄⋊Dic₁₅₁	—	2	1	no	—
73	C₄⋊Dic₁₅₁	C₂³	1	2	yes	—
74	Dic₁₅₁⋊C₄	—	2	2	no	—
75	(C₂×D₁₅₁)⋊C₄	—	2	2	no	—
76	C₄×Dic₁₅₁	—	2	2	no	—
77	#41×C₁₅₁	—	2	2	no	—
78	#41×C₁₅₁	C₂×C₄	1	4	yes	complemented
79	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
80	C₂×D₄×C₁₅₁	—	2	1	no	—
81	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₅₁	—	2	1	no	—
83	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₁₅₁	—	2	2	no	—
85	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
88	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
89	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
90	C₄²×C₁₅₁	—	2	1	no	—
91	C₂²×D₁₅₁	C₂×D₄	1	2	yes	—
92	Dic₃₀₂	—	2	4	no	—
93	Dic₃₀₂	—	2	2	no	—
94	Dic₃₀₂	—	2	8	no	—
95	Dic₃₀₂	C₂²⋊C₄	1	4	yes	—
96	Dic₃₀₂	C₂²×C₄	1	4	yes	—
97	D₆₀₄	C₂²⋊C₄	1	4	yes	—
98	C₂²⋊D₁₅₁	—	2	8	no	—
99	C₂²⋊D₁₅₁	C₂²⋊C₄	1	8	yes	—
100	C₂×Dic₁₅₁	—	2	4	no	—
101	C₂×Dic₁₅₁	—	2	2	no	—
102	C₂×Dic₁₅₁	—	2	8	no	—
103	C₂×Dic₁₅₁	—	2	2	no	—
104	C₂×Dic₁₅₁	—	8	1	no	—
105	C₂×Dic₁₅₁	C₂×D₄	1	2	yes	—
106	C₄×D₁₅₁	—	2	8	no	—
107	C₄×D₁₅₁	—	8	2	no	—
108	C₄×D₁₅₁	C₂²×C₄	1	4	yes	—
109	C₄×D₁₅₁	C₂×D₄	1	4	yes	—
110	C₁₅₁⋊C₈	—	2	2	no	—
111	C₂³×C₁₅₁	—	2	1	no	—
112	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₁₅₁	—	2	1	no	—
114	Q₈×C₁₅₁	C₂²×C₄	1	2	yes	—
115	D₄×C₁₅₁	—	2	4	no	—
116	D₄×C₁₅₁	—	2	1	no	—
117	D₄×C₁₅₁	C₂²×C₄	1	2	yes	—
118	C₂×C₄×C₁₅₁	—	2	1	no	—
119	C₂×C₄×C₁₅₁	—	2	4	no	—
120	C₂×C₄×C₁₅₁	—	2	1	no	—
121	C₂×C₄×C₁₅₁	—	2	1	no	—
122	C₂×C₄×C₁₅₁	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
125	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
126	C₈×C₁₅₁	—	2	1	no	—
127	C₈×C₁₅₁	—	2	1	no	—
128	C₂²×C₁₅₁	—	2	1	no	—
129	C₂²×C₁₅₁	—	2	1	no	—
130	C₂²×C₁₅₁	—	2	2	no	—
131	C₂²×C₁₅₁	—	4	1	no	—
132	C₂²×C₁₅₁	—	4	1	no	—
133	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×D₁₅₁	—	2	2	no	—
136	C₂×D₁₅₁	—	2	2	no	—
137	C₂×D₁₅₁	C₂×C₂²⋊C₄	1	4	yes	—
138	C₄×C₁₅₁	—	2	2	no	—
139	C₄×C₁₅₁	—	4	1	no	—
140	C₄×C₁₅₁	—	4	1	no	—
141	C₄×C₁₅₁	C₂×C₂²⋊C₄	1	2	yes	—
142	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
143	C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
144	Dic₁₅₁	—	2	4	no	—
145	Dic₁₅₁	—	4	2	no	—
146	Dic₁₅₁	C₂×C₂²⋊C₄	1	4	yes	—
147	C₂×C₁₅₁	—	2	2	no	—
148	C₂×C₁₅₁	—	4	2	no	—
149	C₂×C₁₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₁₅₁	#577	1	1	yes	characteristic
151	C₂×C₁₅₁	#578	1	1	yes	characteristic
152	D₁₅₁	—	2	4	no	—
153	C₁₅₁	#2898	1	1	yes	characteristic, complemented, Sylow
154	#2898	—	151	1	no	maximal, Sylow
155	C₂×2₋¹⁺⁴	—	151	1	no	—
156	C₂×#131	—	151	1	no	—
157	#520	—	151	1	no	—
158	#419	—	151	1	no	—
159	#419	—	151	2	no	—
160	#417	—	151	1	no	—
161	#417	—	151	2	no	—
162	#416	—	151	2	no	—
163	#416	D₁₅₁	1	1	yes	characteristic, complemented
164	C₂×#103	—	151	1	no	—
165	C₂×#103	—	151	2	no	—
166	2₋¹⁺⁴	—	302	4	no	—
167	2₋¹⁺⁴	—	151	8	no	—
168	C₂×#41	—	302	1	no	—
169	C₂×#41	—	302	2	no	—
170	C₂×#41	—	151	2	no	—
171	C₂×#41	—	151	1	no	—
172	C₂×#41	C₂×D₁₅₁	1	1	yes	characteristic
173	C₂²×Q₈	—	302	1	no	—
174	C₂²×Q₈	—	151	2	no	—
175	C₂²×Q₈	—	151	1	no	—
176	#131	—	302	4	no	—
177	C₂×M₁₆	—	151	1	no	—
178	C₂×M₁₆	—	151	2	no	—
179	C₂²×C₈	—	151	2	no	—
180	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
181	C₄×Q₈	—	302	1	no	—
182	C₄×D₄	—	302	1	no	—
183	C₄×D₄	—	302	2	no	—
184	#117	—	151	2	no	—
185	#117	C₂×D₁₅₁	1	1	yes	characteristic
186	C₂×C₄⋊C₄	—	151	1	no	—
187	C₂×C₄⋊C₄	—	151	2	no	—
188	#103	—	151	2	no	—
189	#103	—	151	2	no	—
190	#103	—	151	2	no	—
191	#103	—	151	2	no	—
192	#103	—	151	2	no	—
193	#103	C₂×D₁₅₁	1	1	yes	characteristic
194	#103	C₂×D₁₅₁	1	1	yes	characteristic
195	#102	—	151	2	no	—
196	#102	C₂×D₁₅₁	1	1	yes	characteristic
197	#102	C₂×D₁₅₁	1	1	yes	characteristic
198	#41	—	2	2	no	—
199	#41	—	302	8	no	—
200	#41	—	302	16	no	—
201	#41	—	302	4	no	—
202	#41	—	151	8	no	—
203	#41	—	151	4	no	—
204	#41	—	151	4	no	—
205	#41	C₄×D₁₅₁	1	4	yes	complemented
206	C₂×Q₈	—	302	8	no	—
207	C₂×Q₈	—	302	2	no	—
208	C₂×Q₈	—	302	4	no	—
209	C₂×Q₈	—	302	1	no	—
210	C₂×Q₈	—	302	2	no	—
211	C₂×Q₈	—	151	8	no	—
212	C₂×Q₈	—	151	4	no	—
213	C₂×Q₈	—	151	2	no	—
214	C₂×Q₈	—	151	1	no	—
215	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	302	2	no	—
218	C₂×D₄	—	151	2	no	—
219	C₂×D₄	—	151	1	no	—
220	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	302	2	no	—
223	C₂²×C₄	—	302	2	no	—
224	C₂²×C₄	—	302	1	no	—
225	C₂²×C₄	—	302	2	no	—
226	C₂²×C₄	—	302	1	no	—
227	C₂²×C₄	—	151	2	no	—
228	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
229	M₁₆	—	302	4	no	—
230	M₁₆	—	302	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	302	4	no	—
233	C₂×C₈	—	151	2	no	—
234	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
236	C₄⋊C₄	—	302	2	no	—
237	C₄⋊C₄	—	302	1	no	—
238	C₄⋊C₄	—	151	2	no	—
239	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
240	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
241	C₂²⋊C₄	—	2	1	no	—
242	C₂²⋊C₄	—	302	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	302	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	151	2	no	—
247	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
248	Q₈	—	2	1	no	—
249	Q₈	—	302	8	no	—
250	Q₈	—	302	4	no	—
251	Q₈	—	302	2	no	—
252	Q₈	—	151	4	no	—
253	Q₈	—	151	4	no	—
254	Q₈	C₂×C₄×D₁₅₁	1	2	yes	—
255	D₄	—	2	4	no	—
256	D₄	—	2	1	no	—
257	D₄	—	302	8	no	—
258	D₄	—	151	8	no	—
259	D₄	—	151	4	no	—
260	D₄	C₂×C₄×D₁₅₁	1	2	yes	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	302	8	no	—
266	C₂×C₄	—	302	4	no	—
267	C₂×C₄	—	302	2	no	—
268	C₂×C₄	—	302	8	no	—
269	C₂×C₄	—	302	2	no	—
270	C₂×C₄	—	151	2	no	—
271	C₂×C₄	—	151	4	no	—
272	C₂×C₄	—	151	4	no	—
273	C₂×C₄	—	1208	2	no	—
274	C₂×C₄	—	1208	1	no	—
275	C₂×C₄	C₂×C₄×D₁₅₁	1	2	yes	—
276	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
277	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
278	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	302	2	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	302	2	no	—
288	C₂²	—	302	2	no	—
289	C₂²	—	151	4	no	—
290	C₂²	C₂²⋊C₄×D₁₅₁	1	2	yes	—
291	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	302	4	no	—
296	C₄	—	604	2	no	—
297	C₄	—	151	4	no	—
298	C₄	C₂²⋊C₄×D₁₅₁	1	2	yes	—
299	C₄	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
300	C₄	C₂×D₄×D₁₅₁	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	2	no	—
303	C₂	—	302	4	no	—
304	C₂	C₂×C₂²⋊C₄×D₁₅₁	1	1	yes	characteristic
305	C₂	#23753019	1	1	yes	characteristic
306	C₂	#23753026	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator	C₄×C₁₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#416×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×#18012786	C₂
cosocle	C₂×C₄×C₁₅₁	C₂⁴
hypercenter	#416	D₁₅₁
hypocenter	C₁₅₁	#2898
LCS	G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. C₂×#18012745	C₂	semidirect
3. #23752692	C₂	semidirect
4. #23752220	C₂	semidirect
5. #23752219	C₂	semidirect
6. #23752213	C₂	semidirect
7. #23752210	C₂	semidirect
8. #23752205	C₂	semidirect
9. C₂×#18012664	C₂	semidirect
10. #103×D₁₅₁	C₂	semidirect
11. #416×C₁₅₁	C₂	semidirect
13. #18012786	C₄	semidirect
49. #14559003	C₂×C₄	semidirect
54. C₂×Dic₃₀₂	C₂×C₄	semidirect
58. Q₈×D₁₅₁	C₂×C₄	semidirect
78. #41×C₁₅₁	C₂×C₄	semidirect
153. C₁₅₁	#2898	semidirect
163. #416	D₁₅₁	semidirect
205. #41	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×#18012786	2
C₂×#18012745	2
#23752692	2
#23752220	2
#23752219	2
#23752213	2
#23752210	2
#23752205	2
C₂×#18012664	2
#103×D₁₅₁	2
#416×C₁₅₁	2
#2898	151