Subgroups — #3035×C₁₅₁ #34066367

Order 19328.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₁₅₁	C₂	1	2	yes	maximal, complemented
2	#532×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#41×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#464×C₁₅₁	C₂	1	2	yes	maximal, complemented
5	#463×C₁₅₁	C₂	1	2	yes	maximal, complemented
6	#462×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#461×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#460×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#459×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₅₁	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₁₅₁	C₂²	1	2	yes	—
13	C₂×#41×C₁₅₁	C₂²	1	1	yes	characteristic
14	#135×C₁₅₁	—	2	8	no	—
15	C₂×Q₁₆×C₁₅₁	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₁₅₁	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₁₅₁	C₂²	1	2	yes	—
18	C₂×D₈×C₁₅₁	C₂²	1	2	yes	—
19	C₂²×C₈×C₁₅₁	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	#127×C₁₅₁	C₂²	1	1	yes	characteristic
22	#124×C₁₅₁	C₂²	1	2	yes	—
23	#121×C₁₅₁	—	2	1	no	—
24	#121×C₁₅₁	—	2	1	no	—
25	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₅₁	—	2	1	no	—
28	C₄×D₄×C₁₅₁	—	2	1	no	—
29	C₄×D₄×C₁₅₁	—	2	1	no	—
30	C₄×D₄×C₁₅₁	—	2	1	no	—
31	C₄×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
33	#117×C₁₅₁	—	2	1	no	—
34	#117×C₁₅₁	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₁₅₁	—	2	1	no	—
36	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₅₁	C₂²	1	2	yes	—
40	#103×C₁₅₁	C₂²	1	2	yes	—
41	#103×C₁₅₁	C₂²	1	2	yes	—
42	#102×C₁₅₁	C₂²	1	2	yes	—
43	#102×C₁₅₁	C₂²	1	2	yes	—
44	#41×C₁₅₁	—	2	2	no	—
45	#41×C₁₅₁	—	2	4	no	—
46	#41×C₁₅₁	—	2	4	no	—
47	#41×C₁₅₁	D₄	1	4	yes	complemented
48	C₂×Q₈×C₁₅₁	C₂³	1	2	yes	—
49	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₅₁	—	2	1	no	—
51	C₂×D₄×C₁₅₁	—	2	2	no	—
52	C₂×D₄×C₁₅₁	—	2	1	no	—
53	C₂×D₄×C₁₅₁	—	2	1	no	—
54	C₂×D₄×C₁₅₁	C₂³	1	2	yes	—
55	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₅₁	—	2	1	no	—
57	C₂²×C₄×C₁₅₁	—	2	1	no	—
58	C₂²×C₄×C₁₅₁	—	2	1	no	—
59	C₂²×C₄×C₁₅₁	—	2	2	no	—
60	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₅₁	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
63	Q₁₆×C₁₅₁	—	2	4	no	—
64	SD₁₆×C₁₅₁	—	2	4	no	—
65	SD₁₆×C₁₅₁	—	2	4	no	—
66	D₈×C₁₅₁	—	2	4	no	—
67	C₂×C₈×C₁₅₁	—	2	4	no	—
68	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
69	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
70	C₄⋊C₄×C₁₅₁	—	2	1	no	—
71	C₄⋊C₄×C₁₅₁	—	2	1	no	—
72	C₄⋊C₄×C₁₅₁	—	2	1	no	—
73	C₄⋊C₄×C₁₅₁	—	2	1	no	—
74	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
77	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
78	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
79	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
80	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
81	C₄²×C₁₅₁	—	2	1	no	—
82	C₄²×C₁₅₁	—	2	1	no	—
83	C₄²×C₁₅₁	—	2	1	no	—
84	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
85	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
86	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
87	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
88	C₂³×C₁₅₁	—	2	1	no	—
89	C₂³×C₁₅₁	—	2	2	no	—
90	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₁₅₁	—	2	1	no	—
92	Q₈×C₁₅₁	—	2	2	no	—
93	Q₈×C₁₅₁	—	2	2	no	—
94	Q₈×C₁₅₁	C₂×D₄	1	2	yes	—
95	D₄×C₁₅₁	—	2	4	no	—
96	D₄×C₁₅₁	—	2	2	no	—
97	D₄×C₁₅₁	—	2	1	no	—
98	D₄×C₁₅₁	—	2	2	no	—
99	D₄×C₁₅₁	—	4	2	no	—
100	D₄×C₁₅₁	—	4	4	no	—
101	D₄×C₁₅₁	—	4	2	no	—
102	D₄×C₁₅₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₅₁	—	2	1	no	—
104	C₂×C₄×C₁₅₁	—	2	4	no	—
105	C₂×C₄×C₁₅₁	—	2	2	no	—
106	C₂×C₄×C₁₅₁	—	2	2	no	—
107	C₂×C₄×C₁₅₁	—	2	1	no	—
108	C₂×C₄×C₁₅₁	—	2	1	no	—
109	C₂×C₄×C₁₅₁	—	4	1	no	—
110	C₂×C₄×C₁₅₁	—	4	2	no	—
111	C₂×C₄×C₁₅₁	—	4	4	no	—
112	C₂×C₄×C₁₅₁	—	4	2	no	—
113	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₅₁	#41	1	2	yes	—
120	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
121	C₈×C₁₅₁	—	2	2	no	—
122	C₈×C₁₅₁	—	2	2	no	—
123	C₂²×C₁₅₁	—	2	1	no	—
124	C₂²×C₁₅₁	—	2	1	no	—
125	C₂²×C₁₅₁	—	2	2	no	—
126	C₂²×C₁₅₁	—	4	1	no	—
127	C₂²×C₁₅₁	—	4	1	no	—
128	C₂²×C₁₅₁	—	4	2	no	—
129	C₂²×C₁₅₁	—	4	2	no	—
130	C₂²×C₁₅₁	—	4	2	no	—
131	C₂²×C₁₅₁	#121	1	2	yes	—
132	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₁₅₁	—	2	2	no	—
134	C₄×C₁₅₁	—	2	1	no	—
135	C₄×C₁₅₁	—	2	1	no	—
136	C₄×C₁₅₁	—	2	1	no	—
137	C₄×C₁₅₁	—	4	2	no	—
138	C₄×C₁₅₁	—	4	1	no	—
139	C₄×C₁₅₁	—	4	1	no	—
140	C₄×C₁₅₁	#121	1	2	yes	—
141	C₄×C₁₅₁	#135	1	2	yes	complemented
142	C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
144	C₂×C₁₅₁	—	2	2	no	—
145	C₂×C₁₅₁	—	4	2	no	—
146	C₂×C₁₅₁	—	8	2	no	—
147	C₂×C₁₅₁	C₂×#121	1	1	yes	characteristic
148	C₂×C₁₅₁	C₂×#135	1	1	yes	characteristic
149	C₂×C₁₅₁	#575	1	1	yes	characteristic
150	C₁₅₁	#3035	1	1	yes	characteristic, direct product factor, Sylow
151	#3035	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×#135	C₂×C₁₅₁	1	2	yes	complemented
153	#532	C₂×C₁₅₁	1	1	yes	characteristic, complemented
154	C₄×#41	C₂×C₁₅₁	1	1	yes	characteristic, complemented
155	#464	C₂×C₁₅₁	1	2	yes	complemented
156	#463	C₂×C₁₅₁	1	2	yes	complemented
157	#462	C₂×C₁₅₁	1	1	yes	characteristic, complemented
158	#461	C₂×C₁₅₁	1	1	yes	characteristic, complemented
159	#460	C₂×C₁₅₁	1	1	yes	characteristic, complemented
160	#459	C₂×C₁₅₁	1	1	yes	characteristic, complemented
161	C₂×C₄⋊C₈	C₂×C₁₅₁	1	1	yes	characteristic, complemented
162	#416	C₂×C₁₅₁	1	2	yes	complemented
163	C₂×#41	C₂²×C₁₅₁	1	2	yes	—
164	C₂×#41	C₂²×C₁₅₁	1	1	yes	characteristic
165	#135	—	2	8	no	—
166	C₂×Q₁₆	C₂²×C₁₅₁	1	2	yes	—
167	C₂×SD₁₆	C₂²×C₁₅₁	1	2	yes	—
168	C₂×SD₁₆	C₂²×C₁₅₁	1	2	yes	—
169	C₂×D₈	C₂²×C₁₅₁	1	2	yes	—
170	C₂²×C₈	C₂²×C₁₅₁	1	2	yes	—
171	C₄⋊Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
172	#127	C₂²×C₁₅₁	1	1	yes	characteristic
173	#124	C₂²×C₁₅₁	1	2	yes	—
174	#121	—	2	1	no	—
175	#121	—	2	1	no	—
176	C₄×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
177	C₄×Q₈	C₂²×C₁₅₁	1	1	yes	characteristic
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	—	2	1	no	—
182	C₄×D₄	C₂²×C₁₅₁	1	1	yes	characteristic
183	C₄×D₄	C₂²×C₁₅₁	1	1	yes	characteristic
184	#117	—	2	1	no	—
185	#117	C₂²×C₁₅₁	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂²×C₁₅₁	1	1	yes	characteristic
188	C₄⋊C₈	C₂²×C₁₅₁	1	1	yes	characteristic
189	C₄⋊C₈	C₂²×C₁₅₁	1	1	yes	characteristic
190	C₄⋊C₈	C₂²×C₁₅₁	1	2	yes	—
191	#103	C₂²×C₁₅₁	1	2	yes	—
192	#103	C₂²×C₁₅₁	1	2	yes	—
193	#102	C₂²×C₁₅₁	1	2	yes	—
194	#102	C₂²×C₁₅₁	1	2	yes	—
195	#41	—	2	2	no	—
196	#41	—	2	4	no	—
197	#41	—	2	4	no	—
198	#41	D₄×C₁₅₁	1	4	yes	complemented
199	C₂×Q₈	C₂³×C₁₅₁	1	2	yes	—
200	C₂×Q₈	C₂³×C₁₅₁	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	C₂³×C₁₅₁	1	2	yes	—
206	C₂×D₄	C₂³×C₁₅₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	D₄×C₁₅₁	1	1	yes	characteristic
212	C₂²×C₄	D₄×C₁₅₁	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₁₅₁	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	SD₁₆	—	2	4	no	—
216	SD₁₆	—	2	4	no	—
217	D₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂³×C₁₅₁	1	2	yes	—
220	C₂×C₈	C₂³×C₁₅₁	1	2	yes	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	C₂³×C₁₅₁	1	1	yes	characteristic
226	C₄⋊C₄	C₂³×C₁₅₁	1	1	yes	characteristic
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	D₄×C₁₅₁	1	1	yes	characteristic
236	C₄²	D₄×C₁₅₁	1	1	yes	characteristic
237	C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
238	C₄²	C₂³×C₁₅₁	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	2	2	no	—
241	C₂³	C₂×D₄×C₁₅₁	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	Q₈	C₂×D₄×C₁₅₁	1	2	yes	—
246	D₄	—	2	4	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	1	no	—
249	D₄	—	2	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	2	no	—
253	D₄	C₂×D₄×C₁₅₁	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
265	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
267	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₁₅₁	1	2	yes	—
269	C₂×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
270	C₂×C₄	#41×C₁₅₁	1	2	yes	—
271	C₂×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#121×C₁₅₁	1	2	yes	—
283	C₂²	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	#121×C₁₅₁	1	2	yes	—
292	C₄	#135×C₁₅₁	1	2	yes	complemented
293	C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
294	C₄	C₂×#41×C₁₅₁	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	4	2	no	—
297	C₂	—	8	2	no	—
298	C₂	C₂×#121×C₁₅₁	1	1	yes	characteristic
299	C₂	C₂×#135×C₁₅₁	1	1	yes	characteristic
300	C₂	#575×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₅₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₅₁
socle	C₂²×C₁₅₁	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂×C₄²×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#135×C₁₅₁	C₂	semidirect
2. #532×C₁₅₁	C₂	semidirect
3. C₄×#41×C₁₅₁	C₂	semidirect
4. #464×C₁₅₁	C₂	semidirect
5. #463×C₁₅₁	C₂	semidirect
6. #462×C₁₅₁	C₂	semidirect
7. #461×C₁₅₁	C₂	semidirect
8. #460×C₁₅₁	C₂	semidirect
9. #459×C₁₅₁	C₂	semidirect
10. C₂×C₄⋊C₈×C₁₅₁	C₂	semidirect
11. #416×C₁₅₁	C₂	semidirect
47. #41×C₁₅₁	D₄	semidirect
141. C₄×C₁₅₁	#135	semidirect
150. C₁₅₁	#3035	direct (N × H)
151. #3035	C₁₅₁	direct (N × H)
152. C₂×#135	C₂×C₁₅₁	semidirect
153. #532	C₂×C₁₅₁	semidirect
154. C₄×#41	C₂×C₁₅₁	semidirect
155. #464	C₂×C₁₅₁	semidirect
156. #463	C₂×C₁₅₁	semidirect
157. #462	C₂×C₁₅₁	semidirect
158. #461	C₂×C₁₅₁	semidirect
159. #460	C₂×C₁₅₁	semidirect
160. #459	C₂×C₁₅₁	semidirect
161. C₂×C₄⋊C₈	C₂×C₁₅₁	semidirect
162. #416	C₂×C₁₅₁	semidirect
198. #41	D₄×C₁₅₁	semidirect
292. C₄	#135×C₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135×C₁₅₁	2
#532×C₁₅₁	2
C₄×#41×C₁₅₁	2
#464×C₁₅₁	2
#463×C₁₅₁	2
#462×C₁₅₁	2
#461×C₁₅₁	2
#460×C₁₅₁	2
#459×C₁₅₁	2
C₂×C₄⋊C₈×C₁₅₁	2
#416×C₁₅₁	2
#3035	151

Subgroups — #3035×C151 #34066367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3035×C₁₅₁ #34066367