Subgroups — #3377×C₁₅₁ #34066709

Order 19328.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#557×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#556×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#491×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#489×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#488×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	#479×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#478×C₁₅₁	C₂	1	1	yes	maximal, characteristic
8	#477×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#464×C₁₅₁	C₂	1	1	yes	maximal, characteristic
10	#464×C₁₅₁	C₂	1	1	yes	maximal, characteristic
11	#462×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#446×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#441×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
14	#440×C₁₅₁	C₂	1	1	yes	maximal, characteristic
15	C₄×Q₁₆×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
20	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
22	#126×C₁₅₁	—	2	1	no	—
23	#126×C₁₅₁	—	2	1	no	—
24	#125×C₁₅₁	—	2	1	no	—
25	#125×C₁₅₁	—	2	1	no	—
26	#125×C₁₅₁	C₂²	1	1	yes	characteristic
27	#125×C₁₅₁	C₂²	1	1	yes	characteristic
28	#124×C₁₅₁	C₂²	1	1	yes	characteristic
29	#124×C₁₅₁	C₂²	1	1	yes	characteristic
30	C₂²⋊Q₈×C₁₅₁	—	2	1	no	—
31	C₄×Q₈×C₁₅₁	—	2	1	no	—
32	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
37	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
39	#117×C₁₅₁	—	2	1	no	—
40	#107×C₁₅₁	C₂²	1	1	yes	characteristic
41	#107×C₁₅₁	C₂²	1	1	yes	characteristic
42	#106×C₁₅₁	C₂²	1	1	yes	characteristic
43	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
44	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
45	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
46	#103×C₁₅₁	C₂²	1	1	yes	characteristic
47	#103×C₁₅₁	C₂²	1	1	yes	characteristic
48	#103×C₁₅₁	C₂²	1	1	yes	characteristic
49	#103×C₁₅₁	C₂²	1	1	yes	characteristic
50	#103×C₁₅₁	C₂²	1	1	yes	characteristic
51	#103×C₁₅₁	C₂²	1	1	yes	characteristic
52	#103×C₁₅₁	C₂²	1	1	yes	characteristic
53	#102×C₁₅₁	C₂²	1	1	yes	characteristic
54	#102×C₁₅₁	C₂²	1	1	yes	characteristic
55	#102×C₁₅₁	C₂²	1	1	yes	characteristic
56	#97×C₁₅₁	C₂²	1	1	yes	characteristic
57	#97×C₁₅₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₅₁	—	2	1	no	—
60	C₂×Q₈×C₁₅₁	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₅₁	—	2	1	no	—
66	Q₁₆×C₁₅₁	—	2	2	no	—
67	Q₁₆×C₁₅₁	—	2	2	no	—
68	SD₁₆×C₁₅₁	—	2	2	no	—
69	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₅₁	—	2	1	no	—
74	C₄⋊C₄×C₁₅₁	—	2	1	no	—
75	C₄⋊C₄×C₁₅₁	—	2	1	no	—
76	C₄⋊C₄×C₁₅₁	—	2	1	no	—
77	C₄⋊C₄×C₁₅₁	—	2	1	no	—
78	C₄⋊C₄×C₁₅₁	—	2	1	no	—
79	C₄⋊C₄×C₁₅₁	—	2	1	no	—
80	C₄⋊C₄×C₁₅₁	—	2	1	no	—
81	C₄⋊C₄×C₁₅₁	—	2	1	no	—
82	C₄⋊C₄×C₁₅₁	—	2	1	no	—
83	C₄⋊C₄×C₁₅₁	—	2	1	no	—
84	C₄⋊C₄×C₁₅₁	—	2	1	no	—
85	C₄⋊C₄×C₁₅₁	D₄	1	1	yes	characteristic
86	C₄⋊C₄×C₁₅₁	D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₅₁	D₄	1	1	yes	characteristic
88	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
90	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
91	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
92	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
93	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
94	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
95	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
96	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
97	C₄²×C₁₅₁	—	2	1	no	—
98	C₄²×C₁₅₁	—	2	1	no	—
99	C₄²×C₁₅₁	—	2	1	no	—
100	C₄²×C₁₅₁	—	2	1	no	—
101	C₄²×C₁₅₁	—	2	1	no	—
102	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
103	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
104	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
105	C₂³×C₁₅₁	—	2	1	no	—
106	Q₈×C₁₅₁	—	2	1	no	—
107	Q₈×C₁₅₁	—	2	1	no	—
108	Q₈×C₁₅₁	—	2	1	no	—
109	Q₈×C₁₅₁	—	2	1	no	—
110	Q₈×C₁₅₁	—	2	1	no	—
111	Q₈×C₁₅₁	—	4	1	no	—
112	Q₈×C₁₅₁	—	4	2	no	—
113	Q₈×C₁₅₁	#41	1	2	yes	—
114	D₄×C₁₅₁	—	2	1	no	—
115	D₄×C₁₅₁	—	2	1	no	—
116	C₂×C₄×C₁₅₁	—	2	1	no	—
117	C₂×C₄×C₁₅₁	—	2	1	no	—
118	C₂×C₄×C₁₅₁	—	2	1	no	—
119	C₂×C₄×C₁₅₁	—	2	1	no	—
120	C₂×C₄×C₁₅₁	—	2	1	no	—
121	C₂×C₄×C₁₅₁	—	2	1	no	—
122	C₂×C₄×C₁₅₁	—	2	1	no	—
123	C₂×C₄×C₁₅₁	—	4	2	no	—
124	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
131	C₈×C₁₅₁	—	2	1	no	—
132	C₈×C₁₅₁	—	2	1	no	—
133	C₈×C₁₅₁	—	2	1	no	—
134	C₈×C₁₅₁	—	2	1	no	—
135	C₂²×C₁₅₁	—	4	1	no	—
136	C₂²×C₁₅₁	—	4	1	no	—
137	C₂²×C₁₅₁	—	4	1	no	—
138	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₁₅₁	—	2	2	no	—
140	C₄×C₁₅₁	—	2	1	no	—
141	C₄×C₁₅₁	—	2	1	no	—
142	C₄×C₁₅₁	—	2	1	no	—
143	C₄×C₁₅₁	—	2	1	no	—
144	C₄×C₁₅₁	—	2	1	no	—
145	C₄×C₁₅₁	—	2	1	no	—
146	C₄×C₁₅₁	—	4	1	no	—
147	C₄×C₁₅₁	—	4	1	no	—
148	C₄×C₁₅₁	—	4	1	no	—
149	C₄×C₁₅₁	—	4	1	no	—
150	C₄×C₁₅₁	—	4	1	no	—
151	C₄×C₁₅₁	—	4	1	no	—
152	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
153	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×C₁₅₁	—	8	1	no	—
155	C₂×C₁₅₁	#548	1	1	yes	characteristic
156	C₂×C₁₅₁	#575	1	1	yes	characteristic
157	C₂×C₁₅₁	#578	1	1	yes	characteristic
158	C₁₅₁	#3377	1	1	yes	characteristic, direct product factor, Sylow
159	#3377	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	#557	C₂×C₁₅₁	1	1	yes	characteristic, complemented
161	#556	C₂×C₁₅₁	1	1	yes	characteristic
162	#491	C₂×C₁₅₁	1	1	yes	characteristic, complemented
163	#489	C₂×C₁₅₁	1	1	yes	characteristic
164	#488	C₂×C₁₅₁	1	1	yes	characteristic
165	#479	C₂×C₁₅₁	1	1	yes	characteristic, complemented
166	#478	C₂×C₁₅₁	1	1	yes	characteristic
167	#477	C₂×C₁₅₁	1	1	yes	characteristic, complemented
168	#464	C₂×C₁₅₁	1	1	yes	characteristic
169	#464	C₂×C₁₅₁	1	1	yes	characteristic
170	#462	C₂×C₁₅₁	1	1	yes	characteristic, complemented
171	#446	C₂×C₁₅₁	1	1	yes	characteristic, complemented
172	#441	C₂×C₁₅₁	1	1	yes	characteristic, complemented
173	#440	C₂×C₁₅₁	1	1	yes	characteristic
174	C₄×Q₁₆	C₂×C₁₅₁	1	1	yes	characteristic, complemented
175	C₂×Q₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
176	C₂×Q₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
177	C₂×SD₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	1	no	—
179	C₄⋊Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
180	C₄⋊Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
181	#126	—	2	1	no	—
182	#126	—	2	1	no	—
183	#125	—	2	1	no	—
184	#125	—	2	1	no	—
185	#125	C₂²×C₁₅₁	1	1	yes	characteristic
186	#125	C₂²×C₁₅₁	1	1	yes	characteristic
187	#124	C₂²×C₁₅₁	1	1	yes	characteristic
188	#124	C₂²×C₁₅₁	1	1	yes	characteristic
189	C₂²⋊Q₈	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
192	C₄×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
193	C₄×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
194	C₄×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
195	C₄×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
196	C₄×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
197	C₄×D₄	C₂²×C₁₅₁	1	1	yes	characteristic
198	#117	—	2	1	no	—
199	#107	C₂²×C₁₅₁	1	1	yes	characteristic
200	#107	C₂²×C₁₅₁	1	1	yes	characteristic
201	#106	C₂²×C₁₅₁	1	1	yes	characteristic
202	C₄⋊C₈	C₂²×C₁₅₁	1	1	yes	characteristic
203	C₄⋊C₈	C₂²×C₁₅₁	1	1	yes	characteristic
204	C₄⋊C₈	C₂²×C₁₅₁	1	1	yes	characteristic
205	#103	C₂²×C₁₅₁	1	1	yes	characteristic
206	#103	C₂²×C₁₅₁	1	1	yes	characteristic
207	#103	C₂²×C₁₅₁	1	1	yes	characteristic
208	#103	C₂²×C₁₅₁	1	1	yes	characteristic
209	#103	C₂²×C₁₅₁	1	1	yes	characteristic
210	#103	C₂²×C₁₅₁	1	1	yes	characteristic
211	#103	C₂²×C₁₅₁	1	1	yes	characteristic
212	#102	C₂²×C₁₅₁	1	1	yes	characteristic
213	#102	C₂²×C₁₅₁	1	1	yes	characteristic
214	#102	C₂²×C₁₅₁	1	1	yes	characteristic
215	#97	C₂²×C₁₅₁	1	1	yes	characteristic
216	#97	C₂²×C₁₅₁	1	1	yes	characteristic
217	C₄×C₈	C₂²×C₁₅₁	1	1	yes	characteristic
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	D₄×C₁₅₁	1	1	yes	characteristic
220	C₂×Q₈	C₂³×C₁₅₁	1	1	yes	characteristic
221	C₂×Q₈	C₂³×C₁₅₁	1	1	yes	characteristic
222	C₂×Q₈	C₂³×C₁₅₁	1	1	yes	characteristic
223	C₂×D₄	C₂³×C₁₅₁	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	2	2	no	—
228	C₂×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
229	C₂×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
230	C₂×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
231	C₂×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	D₄×C₁₅₁	1	1	yes	characteristic
245	C₄⋊C₄	D₄×C₁₅₁	1	1	yes	characteristic
246	C₄⋊C₄	D₄×C₁₅₁	1	1	yes	characteristic
247	C₄⋊C₄	C₂³×C₁₅₁	1	1	yes	characteristic
248	C₄⋊C₄	C₂³×C₁₅₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂³×C₁₅₁	1	1	yes	characteristic
250	C₄⋊C₄	C₂³×C₁₅₁	1	1	yes	characteristic
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
262	C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
263	C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
264	C₂³	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	4	1	no	—
271	Q₈	—	4	2	no	—
272	Q₈	#41×C₁₅₁	1	2	yes	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
284	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
285	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
287	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
289	C₂×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
312	C₄	2₋¹⁺⁴×C₁₅₁	1	1	yes	characteristic
313	C₂	—	8	1	no	—
314	C₂	#548×C₁₅₁	1	1	yes	characteristic
315	C₂	#575×C₁₅₁	1	1	yes	characteristic
316	C₂	#578×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₅₁
prime radical	C₂×D₄×C₁₅₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₄×Q₈×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #557×C₁₅₁	C₂	semidirect
3. #491×C₁₅₁	C₂	semidirect
6. #479×C₁₅₁	C₂	semidirect
8. #477×C₁₅₁	C₂	semidirect
11. #462×C₁₅₁	C₂	semidirect
12. #446×C₁₅₁	C₂	semidirect
13. #441×C₁₅₁	C₂	semidirect
15. C₄×Q₁₆×C₁₅₁	C₂	semidirect
158. C₁₅₁	#3377	direct (N × H)
159. #3377	C₁₅₁	direct (N × H)
160. #557	C₂×C₁₅₁	semidirect
162. #491	C₂×C₁₅₁	semidirect
165. #479	C₂×C₁₅₁	semidirect
167. #477	C₂×C₁₅₁	semidirect
170. #462	C₂×C₁₅₁	semidirect
171. #446	C₂×C₁₅₁	semidirect
172. #441	C₂×C₁₅₁	semidirect
174. C₄×Q₁₆	C₂×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#557×C₁₅₁	2
#556×C₁₅₁	2
#491×C₁₅₁	2
#489×C₁₅₁	2
#488×C₁₅₁	2
#479×C₁₅₁	2
#478×C₁₅₁	2
#477×C₁₅₁	2
#464×C₁₅₁	2
#462×C₁₅₁	2
#446×C₁₅₁	2
#441×C₁₅₁	2
#440×C₁₅₁	2
C₄×Q₁₆×C₁₅₁	2
#3377	151

Subgroups — #3377×C151 #34066709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3377×C₁₅₁ #34066709