Subgroups — #34079267

Order 19328.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752764	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752680	C₂	1	1	yes	maximal, characteristic
3	#23752553	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752506	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752501	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752421	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752416	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752376	C₂	1	1	yes	maximal, characteristic
9	#23752372	C₂	1	1	yes	maximal, characteristic
10	#23752318	C₂	1	1	yes	maximal, characteristic, complemented
11	#23752316	C₂	1	1	yes	maximal, characteristic, complemented
12	#23752315	C₂	1	1	yes	maximal, characteristic, complemented
13	#23752311	C₂	1	1	yes	maximal, characteristic, complemented
14	#23752266	C₂	1	1	yes	maximal, characteristic, complemented
15	#485×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14559004	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₀₄	C₂²	1	1	yes	characteristic
18	C₂×#14558994	C₂²	1	1	yes	characteristic
19	C₂×#14558993	C₂²	1	1	yes	characteristic
20	C₂×#14558991	C₂²	1	1	yes	characteristic
21	#18012738	C₂²	1	1	yes	characteristic
22	#18012737	C₂²	1	1	yes	characteristic
23	#18012729	—	2	1	no	—
24	#18012718	C₂²	1	1	yes	characteristic
25	#18012716	—	2	1	no	—
26	Q₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
27	#18012704	—	2	1	no	—
28	#18012704	C₂²	1	1	yes	characteristic
29	#18012704	C₂²	1	1	yes	characteristic
30	#18012704	C₂²	1	1	yes	characteristic
31	#18012701	C₂²	1	1	yes	characteristic
32	#18012701	C₂²	1	1	yes	characteristic
33	(C₂²⋊D₁₅₁)⋊C₄	—	2	1	no	—
34	(C₂²⋊D₁₅₁)⋊C₄	—	2	1	no	—
35	#18012696	—	2	1	no	—
36	#18012695	—	2	1	no	—
37	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
38	C₄²×D₁₅₁	—	2	1	no	—
39	#18012672	C₂²	1	1	yes	characteristic
40	#18012670	C₂²	1	1	yes	characteristic
41	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
43	#18012664	C₂²	1	1	yes	characteristic
44	#18012662	C₂²	1	1	yes	characteristic
45	#18012662	C₂²	1	1	yes	characteristic
46	#18012662	C₂²	1	1	yes	characteristic
47	#18012661	C₂²	1	1	yes	characteristic
48	#18012659	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
50	#18012649	C₂²	1	1	yes	characteristic
51	#18012649	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
53	#117×C₁₅₁	C₂²	1	1	yes	characteristic
54	#107×C₁₅₁	C₂²	1	1	yes	characteristic
55	#106×C₁₅₁	C₂²	1	1	yes	characteristic
56	#103×C₁₅₁	C₂²	1	1	yes	characteristic
57	#103×C₁₅₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
59	#14559004	—	2	2	no	—
60	#14559004	—	2	2	no	—
61	C₂×Dic₃₀₂	—	2	1	no	—
62	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
64	C₂×D₆₀₄	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₅₁	—	2	1	no	—
66	C₂²×Dic₁₅₁	D₄	1	1	yes	characteristic
67	C₂²×Dic₁₅₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₅₁	—	2	1	no	—
69	C₂×C₄×D₁₅₁	—	2	1	no	—
70	C₂×C₄×D₁₅₁	—	2	1	no	—
71	Dic₆₀₄	—	2	2	no	—
72	#14558994	—	2	2	no	—
73	#14558993	—	2	2	no	—
74	#14558991	—	2	2	no	—
75	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₅₁	—	2	1	no	—
78	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
79	Dic₁₅₁⋊C₄	—	2	1	no	—
80	Dic₁₅₁⋊C₄	—	2	1	no	—
81	Dic₁₅₁⋊C₄	—	2	1	no	—
82	Dic₁₅₁⋊C₄	—	2	1	no	—
83	Dic₁₅₁⋊C₄	—	2	1	no	—
84	Dic₁₅₁⋊C₄	—	2	1	no	—
85	Dic₁₅₁⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₁₅₁	—	2	1	no	—
87	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
89	C₄×Dic₁₅₁	—	2	1	no	—
90	C₄×Dic₁₅₁	—	2	1	no	—
91	C₄×Dic₁₅₁	—	2	1	no	—
92	C₄×Dic₁₅₁	—	2	1	no	—
93	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
95	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₅₁	—	2	1	no	—
102	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
106	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
107	C₄²×C₁₅₁	—	2	1	no	—
108	C₂²×D₁₅₁	—	2	1	no	—
109	Dic₃₀₂	—	2	1	no	—
110	Dic₃₀₂	—	2	1	no	—
111	Dic₃₀₂	—	2	1	no	—
112	Dic₃₀₂	—	4	2	no	—
113	Dic₃₀₂	#41	1	2	yes	—
114	D₆₀₄	—	2	1	no	—
115	D₆₀₄	#41	1	2	yes	—
116	C₂²⋊D₁₅₁	—	2	4	no	—
117	C₂×Dic₁₅₁	—	2	1	no	—
118	C₂×Dic₁₅₁	—	2	2	no	—
119	C₂×Dic₁₅₁	—	2	1	no	—
120	C₂×Dic₁₅₁	—	2	1	no	—
121	C₂×Dic₁₅₁	—	4	1	no	—
122	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
126	C₄×D₁₅₁	—	2	4	no	—
127	C₄×D₁₅₁	—	4	2	no	—
128	C₄×D₁₅₁	—	4	2	no	—
129	C₁₅₁⋊C₈	—	2	1	no	—
130	C₁₅₁⋊C₈	—	2	1	no	—
131	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
132	Q₈×C₁₅₁	—	2	1	no	—
133	Q₈×C₁₅₁	—	2	1	no	—
134	C₂×C₄×C₁₅₁	—	2	1	no	—
135	C₂×C₄×C₁₅₁	—	2	1	no	—
136	C₂×C₄×C₁₅₁	—	2	1	no	—
137	C₂×C₄×C₁₅₁	—	2	1	no	—
138	C₂×C₄×C₁₅₁	—	2	1	no	—
139	C₂×C₄×C₁₅₁	—	2	1	no	—
140	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₅₁	—	2	1	no	—
143	C₈×C₁₅₁	—	2	1	no	—
144	C₂²×C₁₅₁	—	2	1	no	—
145	C₂²×C₁₅₁	—	2	1	no	—
146	C₂²×C₁₅₁	—	2	1	no	—
147	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₁₅₁	—	2	2	no	—
149	C₂×D₁₅₁	—	4	1	no	—
150	C₂×D₁₅₁	—	4	1	no	—
151	C₄×C₁₅₁	—	2	1	no	—
152	C₄×C₁₅₁	—	4	1	no	—
153	C₄×C₁₅₁	—	4	1	no	—
154	C₄×C₁₅₁	—	4	1	no	—
155	C₄×C₁₅₁	—	4	1	no	—
156	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
157	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
158	Dic₁₅₁	—	2	2	no	—
159	Dic₁₅₁	—	2	1	no	—
160	Dic₁₅₁	—	2	1	no	—
161	Dic₁₅₁	—	2	1	no	—
162	Dic₁₅₁	—	4	1	no	—
163	Dic₁₅₁	—	4	1	no	—
164	Dic₁₅₁	#137	1	2	yes	—
165	C₂×C₁₅₁	—	4	1	no	—
166	C₂×C₁₅₁	#525	1	1	yes	characteristic
167	C₂×C₁₅₁	C₂×#137	1	1	yes	characteristic
168	C₂×C₁₅₁	#575	1	1	yes	characteristic
169	D₁₅₁	—	4	2	no	—
170	C₁₅₁	#3116	1	1	yes	characteristic, complemented, Sylow
171	#3116	—	151	1	no	maximal, Sylow
172	#533	—	151	1	no	—
173	C₄×#41	—	151	1	no	—
174	#485	—	151	1	no	—
175	#485	D₁₅₁	1	1	yes	characteristic, complemented
176	#477	—	151	1	no	—
177	#476	—	151	1	no	—
178	#462	—	151	1	no	—
179	#461	—	151	1	no	—
180	#452	—	151	1	no	—
181	#452	—	151	1	no	—
182	#441	—	151	1	no	—
183	#441	—	151	1	no	—
184	#440	—	151	1	no	—
185	#440	—	151	1	no	—
186	#432	—	151	1	no	—
187	C₂×#41	—	151	1	no	—
188	C₂×Q₁₆	—	151	1	no	—
189	C₂×Q₁₆	—	151	1	no	—
190	C₂×SD₁₆	—	151	1	no	—
191	C₂×SD₁₆	—	151	1	no	—
192	C₄⋊Q₈	—	151	1	no	—
193	C₄⋊Q₈	—	151	1	no	—
194	#125	—	302	1	no	—
195	C₂²⋊Q₈	—	302	1	no	—
196	C₂²⋊Q₈	—	151	1	no	—
197	C₂²⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
198	C₄×Q₈	—	302	1	no	—
199	C₄×Q₈	—	151	1	no	—
200	C₄×Q₈	—	151	1	no	—
201	C₄×Q₈	—	151	1	no	—
202	C₄×Q₈	—	151	1	no	—
203	C₄×D₄	—	302	1	no	—
204	C₄×D₄	—	302	1	no	—
205	C₄×D₄	—	151	1	no	—
206	C₄×D₄	—	151	1	no	—
207	#117	—	302	1	no	—
208	#117	—	302	1	no	—
209	#117	C₂×D₁₅₁	1	1	yes	characteristic
210	C₂×C₄²	—	302	1	no	—
211	C₂×C₄²	—	151	1	no	—
212	#107	—	151	1	no	—
213	#107	C₂×D₁₅₁	1	1	yes	characteristic
214	#106	—	151	1	no	—
215	#106	C₂×D₁₅₁	1	1	yes	characteristic
216	C₄⋊C₈	—	151	1	no	—
217	C₄⋊C₈	—	151	1	no	—
218	#103	—	151	1	no	—
219	#103	—	151	1	no	—
220	#103	—	151	1	no	—
221	#103	—	151	1	no	—
222	#103	C₂×D₁₅₁	1	1	yes	characteristic
223	#103	C₂×D₁₅₁	1	1	yes	characteristic
224	#102	—	151	1	no	—
225	#102	—	151	1	no	—
226	C₂²⋊C₈	—	151	1	no	—
227	C₂²⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
228	#97	—	151	1	no	—
229	#97	—	151	1	no	—
230	#41	—	302	2	no	—
231	#41	—	302	2	no	—
232	C₂×Q₈	—	302	1	no	—
233	C₂×Q₈	—	151	1	no	—
234	C₂×Q₈	—	151	1	no	—
235	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
236	C₂×D₄	—	302	1	no	—
237	C₂×D₄	—	151	1	no	—
238	C₂²×C₄	—	302	1	no	—
239	C₂²×C₄	—	302	1	no	—
240	C₂²×C₄	—	302	1	no	—
241	C₂²×C₄	—	151	1	no	—
242	C₂²×C₄	—	151	1	no	—
243	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
244	Q₁₆	—	302	2	no	—
245	Q₁₆	—	302	2	no	—
246	SD₁₆	—	302	2	no	—
247	SD₁₆	—	302	2	no	—
248	C₂×C₈	—	151	1	no	—
249	C₂×C₈	—	151	1	no	—
250	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	302	1	no	—
254	C₄⋊C₄	—	302	1	no	—
255	C₄⋊C₄	—	302	1	no	—
256	C₄⋊C₄	—	302	1	no	—
257	C₄⋊C₄	—	302	1	no	—
258	C₄⋊C₄	—	302	1	no	—
259	C₄⋊C₄	—	302	1	no	—
260	C₄⋊C₄	—	302	1	no	—
261	C₄⋊C₄	—	151	1	no	—
262	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	302	1	no	—
268	C₂²⋊C₄	—	302	1	no	—
269	C₂²⋊C₄	—	302	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	302	1	no	—
272	C₄²	—	302	1	no	—
273	C₄²	—	302	1	no	—
274	C₄²	—	302	1	no	—
275	C₄²	—	151	1	no	—
276	C₄²	—	151	1	no	—
277	C₄²	—	151	1	no	—
278	C₄²	—	151	1	no	—
279	C₂³	—	302	1	no	—
280	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	302	1	no	—
284	Q₈	—	302	1	no	—
285	Q₈	—	302	1	no	—
286	Q₈	—	604	2	no	—
287	Q₈	—	151	2	no	—
288	D₄	—	302	4	no	—
289	D₄	—	302	1	no	—
290	D₄	—	151	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	302	4	no	—
298	C₂×C₄	—	302	1	no	—
299	C₂×C₄	—	302	2	no	—
300	C₂×C₄	—	302	1	no	—
301	C₂×C₄	—	302	1	no	—
302	C₂×C₄	—	604	2	no	—
303	C₂×C₄	—	604	2	no	—
304	C₂×C₄	—	604	1	no	—
305	C₂×C₄	—	151	1	no	—
306	C₂×C₄	—	151	1	no	—
307	C₂×C₄	—	151	1	no	—
308	C₂×C₄	—	151	1	no	—
309	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	302	1	no	—
314	C₈	—	302	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	302	2	no	—
319	C₂²	—	604	1	no	—
320	C₂²	—	604	1	no	—
321	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	302	2	no	—
328	C₄	—	302	1	no	—
329	C₄	—	302	1	no	—
330	C₄	—	302	1	no	—
331	C₄	—	604	1	no	—
332	C₄	—	604	1	no	—
333	C₄	—	151	2	no	—
334	C₄	#41×D₁₅₁	1	1	yes	characteristic
335	C₄	#41×D₁₅₁	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	604	2	no	—
338	C₂	#23752720	1	1	yes	characteristic
339	C₂	#137×D₁₅₁	1	1	yes	characteristic
340	C₂	#23753009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#485×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×#14559004	C₂²
hypercenter	#485	D₁₅₁
hypocenter	C₁₅₁	#3116
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
1. #23752764	C₂	semidirect
3. #23752553	C₂	semidirect
4. #23752506	C₂	semidirect
5. #23752501	C₂	semidirect
6. #23752421	C₂	semidirect
7. #23752416	C₂	semidirect
10. #23752318	C₂	semidirect
11. #23752316	C₂	semidirect
12. #23752315	C₂	semidirect
13. #23752311	C₂	semidirect
14. #23752266	C₂	semidirect
15. #485×C₁₅₁	C₂	semidirect
170. C₁₅₁	#3116	semidirect
175. #485	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752764	2
#23752680	2
#23752553	2
#23752506	2
#23752501	2
#23752421	2
#23752416	2
#23752376	2
#23752372	2
#23752318	2
#23752316	2
#23752315	2
#23752311	2
#23752266	2
#485×C₁₅₁	2
#3116	151