Subgroups — #533×D₁₅₁ #34082395

Order 19328.

1168 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752769	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752768	C₂	1	2	yes	maximal, complemented
3	#23752767	C₂	1	2	yes	maximal, complemented
4	#23752766	C₂	1	4	yes	maximal, complemented
5	#23752765	C₂	1	2	yes	maximal, complemented
6	#23752764	C₂	1	4	yes	maximal, complemented
7	C₄⋊Q₈×D₁₅₁	C₂	1	2	yes	maximal, complemented
8	#125×D₁₅₁	C₂	1	2	yes	maximal, complemented
9	C₂²⋊Q₈×D₁₅₁	C₂	1	4	yes	maximal
10	C₄×Q₈×D₁₅₁	C₂	1	4	yes	maximal, complemented
11	#117×D₁₅₁	C₂	1	2	yes	maximal
12	C₂×C₄²×D₁₅₁	C₂	1	1	yes	maximal, characteristic
13	#533×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₁₅₁	C₂²	1	4	yes	—
15	C₂²×C₄×D₁₅₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
17	#18012739	C₂²	1	2	yes	complemented
18	#18012738	C₂²	1	4	yes	complemented
19	#18012738	C₂²	1	4	yes	complemented
20	#18012737	C₂²	1	4	yes	complemented
21	#18012731	C₂²	1	2	yes	complemented
22	#18012730	C₂²	1	4	yes	complemented
23	#18012729	C₂²	1	8	yes	complemented
24	#18012719	C₂²	1	4	yes	—
25	#18012718	C₂²	1	4	yes	—
26	#18012717	C₂²	1	8	yes	—
27	#18012716	C₂²	1	8	yes	—
28	#18012715	C₂²	1	4	yes	—
29	Q₈×Dic₁₅₁	C₂²	1	4	yes	complemented
30	C₄×Dic₃₀₂	C₂²	1	8	yes	complemented
31	C₄×Dic₃₀₂	C₂²	1	4	yes	complemented
32	#18012704	C₂²	1	4	yes	complemented
33	#18012704	C₂²	1	8	yes	complemented
34	#18012698	C₂²	1	2	yes	—
35	#18012697	C₂²	1	4	yes	—
36	#18012696	C₂²	1	4	yes	—
37	#18012695	C₂²	1	4	yes	—
38	C₄⋊C₄×D₁₅₁	C₂²	1	8	yes	—
39	C₄⋊C₄×D₁₅₁	C₂²	1	4	yes	—
40	C₄⋊C₄×D₁₅₁	C₂²	1	4	yes	—
41	C₂²⋊C₄×D₁₅₁	C₂²	1	4	yes	—
42	C₂×C₄×Dic₁₅₁	C₂²	1	2	yes	—
43	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
44	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
45	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
46	C₄²×D₁₅₁	C₂²	1	2	yes	—
47	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
48	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
49	C₄²×D₁₅₁	C₂²	1	2	yes	—
50	C₄²×D₁₅₁	C₂²	1	4	yes	—
51	C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	complemented
52	#125×C₁₅₁	C₂²	1	2	yes	complemented
53	C₂²⋊Q₈×C₁₅₁	C₂²	1	4	yes	—
54	C₄×Q₈×C₁₅₁	C₂²	1	4	yes	complemented
55	#117×C₁₅₁	C₂²	1	2	yes	—
56	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
57	C₂³×D₁₅₁	C₂³	1	1	yes	characteristic
58	C₂×Dic₃₀₂	C₂³	1	8	yes	—
59	C₂×Dic₃₀₂	C₂³	1	4	yes	—
60	Q₈×D₁₅₁	—	2	16	no	—
61	C₂²×Dic₁₅₁	C₂³	1	2	yes	—
62	C₂²×Dic₁₅₁	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₅₁	—	2	8	no	—
64	C₂×C₄×D₁₅₁	Q₈	1	8	yes	complemented
65	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
66	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
67	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
68	C₂×C₄×D₁₅₁	C₂³	1	8	yes	—
69	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
74	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
75	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
76	C₄⋊Dic₁₅₁	C₂³	1	8	yes	—
77	Dic₁₅₁⋊C₄	C₂³	1	8	yes	—
78	Dic₁₅₁⋊C₄	C₂³	1	8	yes	—
79	Dic₁₅₁⋊C₄	C₂³	1	8	yes	—
80	Dic₁₅₁⋊C₄	C₂³	1	8	yes	—
81	C₂²⋊Dic₁₅₁	C₂³	1	4	yes	—
82	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
83	C₄×Dic₁₅₁	C₂³	1	2	yes	—
84	C₄×Dic₁₅₁	C₂³	1	2	yes	—
85	C₄×Dic₁₅₁	C₂³	1	2	yes	—
86	C₄×Dic₁₅₁	C₂³	1	2	yes	—
87	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
88	C₄×Dic₁₅₁	C₂³	1	2	yes	—
89	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
90	C₄×Dic₁₅₁	C₂³	1	8	yes	—
91	C₄×Dic₁₅₁	C₂³	1	4	yes	—
92	C₂×Q₈×C₁₅₁	C₂³	1	4	yes	—
93	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
96	C₄⋊C₄×C₁₅₁	C₂³	1	8	yes	—
97	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
98	C₂²⋊C₄×C₁₅₁	C₂³	1	4	yes	—
99	C₄²×C₁₅₁	C₂³	1	2	yes	—
100	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
101	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
102	C₄²×C₁₅₁	C₂³	1	4	yes	—
103	C₂²×D₁₅₁	—	2	4	no	—
104	C₂²×D₁₅₁	C₂×Q₈	1	4	yes	—
105	C₂²×D₁₅₁	C₂⁴	1	1	yes	characteristic
106	C₂²×D₁₅₁	C₂⁴	1	1	yes	characteristic
107	Dic₃₀₂	—	2	8	no	—
108	Dic₃₀₂	—	2	16	no	—
109	C₂×Dic₁₅₁	—	2	4	no	—
110	C₂×Dic₁₅₁	C₂×Q₈	1	4	yes	complemented
111	C₂×Dic₁₅₁	C₂⁴	1	2	yes	—
112	C₂×Dic₁₅₁	C₂⁴	1	8	yes	—
113	C₂×Dic₁₅₁	C₂⁴	1	1	yes	characteristic
114	C₂×Dic₁₅₁	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₅₁	C₂⁴	1	2	yes	—
116	C₄×D₁₅₁	—	2	4	no	—
117	C₄×D₁₅₁	—	2	16	no	—
118	C₄×D₁₅₁	—	2	4	no	—
119	C₄×D₁₅₁	—	2	4	no	—
120	C₄×D₁₅₁	C₂×Q₈	1	4	yes	—
121	C₄×D₁₅₁	C₂×Q₈	1	4	yes	—
122	C₄×D₁₅₁	C₂×Q₈	1	4	yes	complemented
123	C₄×D₁₅₁	C₂×Q₈	1	4	yes	complemented
124	C₄×D₁₅₁	#41	1	8	yes	—
125	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
126	Q₈×C₁₅₁	—	2	8	no	—
127	C₂×C₄×C₁₅₁	—	2	4	no	—
128	C₂×C₄×C₁₅₁	C₂×Q₈	1	4	yes	complemented
129	C₂×C₄×C₁₅₁	C₂⁴	1	8	yes	—
130	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
131	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
132	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
133	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
134	C₂²×C₁₅₁	—	2	2	no	—
135	C₂²×C₁₅₁	C₂²×Q₈	1	2	yes	—
136	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
137	C₂×D₁₅₁	—	2	2	no	—
138	C₂×D₁₅₁	—	2	8	no	—
139	C₂×D₁₅₁	C₂²×Q₈	1	2	yes	—
140	C₂×D₁₅₁	C₂²×Q₈	1	2	yes	—
141	C₂×D₁₅₁	C₂×#41	1	4	yes	—
142	C₄×C₁₅₁	—	2	2	no	—
143	C₄×C₁₅₁	—	2	8	no	—
144	C₄×C₁₅₁	C₂²×Q₈	1	2	yes	—
145	C₄×C₁₅₁	C₂²×Q₈	1	2	yes	—
146	C₄×C₁₅₁	C₂×#41	1	4	yes	—
147	Dic₁₅₁	—	2	2	no	—
148	Dic₁₅₁	—	2	8	no	—
149	Dic₁₅₁	C₂²×Q₈	1	2	yes	—
150	Dic₁₅₁	C₂²×Q₈	1	2	yes	—
151	Dic₁₅₁	C₂×#41	1	4	yes	—
152	C₂×C₁₅₁	—	2	2	no	—
153	C₂×C₁₅₁	C₂³×Q₈	1	1	yes	characteristic
154	C₂×C₁₅₁	C₂²×#41	1	2	yes	—
155	D₁₅₁	—	2	2	no	—
156	D₁₅₁	#533	1	4	yes	direct product factor
157	C₁₅₁	C₂×#533	1	1	yes	characteristic, complemented, Sylow
158	C₂×#533	—	151	1	no	maximal, Sylow
159	#533	—	151	2	no	—
160	#533	—	151	2	no	—
161	#533	—	151	4	no	—
162	#533	—	151	4	no	—
163	#533	—	151	2	no	—
164	#533	—	151	1	no	—
165	#533	D₁₅₁	1	1	yes	characteristic, direct product factor
166	C₂×C₄⋊Q₈	—	151	2	no	—
167	C₂×#125	—	151	2	no	—
168	C₂×C₂²⋊Q₈	—	151	4	no	—
169	C₂×C₄×Q₈	—	151	4	no	—
170	C₂×#117	—	151	2	no	—
171	C₂²×C₄²	—	151	1	no	—
172	C₂²×Q₈	—	151	4	no	—
173	C₂³×C₄	—	151	2	no	—
174	C₂³×C₄	—	151	1	no	—
175	C₄⋊Q₈	—	151	4	no	—
176	C₄⋊Q₈	—	151	4	no	—
177	C₄⋊Q₈	—	151	4	no	—
178	C₄⋊Q₈	—	151	2	no	—
179	C₄⋊Q₈	C₂×D₁₅₁	1	2	yes	complemented
180	#125	—	151	8	no	—
181	#125	—	151	4	no	—
182	#125	—	151	2	no	—
183	#125	C₂×D₁₅₁	1	2	yes	complemented
184	C₂²⋊Q₈	—	151	4	no	—
185	C₂²⋊Q₈	—	151	4	no	—
186	C₂²⋊Q₈	—	151	8	no	—
187	C₂²⋊Q₈	—	151	4	no	—
188	C₂²⋊Q₈	—	151	8	no	—
189	C₂²⋊Q₈	C₂×D₁₅₁	1	4	yes	—
190	C₄×Q₈	—	151	4	no	—
191	C₄×Q₈	—	151	4	no	—
192	C₄×Q₈	—	151	8	no	—
193	C₄×Q₈	—	151	8	no	—
194	C₄×Q₈	—	151	4	no	—
195	C₄×Q₈	C₂×D₁₅₁	1	4	yes	complemented
196	#117	—	151	2	no	—
197	#117	—	151	4	no	—
198	#117	—	151	4	no	—
199	#117	—	151	4	no	—
200	#117	C₂×D₁₅₁	1	2	yes	—
201	C₂×C₄⋊C₄	—	151	4	no	—
202	C₂×C₄⋊C₄	—	151	8	no	—
203	C₂×C₄⋊C₄	—	151	4	no	—
204	C₂×C₂²⋊C₄	—	151	4	no	—
205	C₂×C₄²	—	151	2	no	—
206	C₂×C₄²	—	151	1	no	—
207	C₂×C₄²	—	151	4	no	—
208	C₂×C₄²	—	151	2	no	—
209	C₂×C₄²	—	151	1	no	—
210	C₂×C₄²	—	151	1	no	—
211	C₂×C₄²	—	151	1	no	—
212	C₂×C₄²	—	151	1	no	—
213	C₂×C₄²	—	151	2	no	—
214	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
215	C₂⁴	—	151	1	no	—
216	C₂×Q₈	—	302	16	no	—
217	C₂×Q₈	—	151	4	no	—
218	C₂×Q₈	—	151	8	no	—
219	C₂×Q₈	C₂²×D₁₅₁	1	4	yes	—
220	C₂²×C₄	—	302	8	no	—
221	C₂²×C₄	—	151	2	no	—
222	C₂²×C₄	—	151	2	no	—
223	C₂²×C₄	—	151	2	no	—
224	C₂²×C₄	—	151	2	no	—
225	C₂²×C₄	—	151	8	no	—
226	C₂²×C₄	—	151	1	no	—
227	C₂²×C₄	—	151	8	no	—
228	C₂²×C₄	—	151	1	no	—
229	C₂²×C₄	—	151	2	no	—
230	C₂²×C₄	—	151	1	no	—
231	C₂²×C₄	—	151	1	no	—
232	C₂²×C₄	—	151	1	no	—
233	C₂²×C₄	C₂²×D₁₅₁	1	2	yes	—
234	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
235	C₄⋊C₄	—	151	8	no	—
236	C₄⋊C₄	—	151	8	no	—
237	C₄⋊C₄	—	151	8	no	—
238	C₄⋊C₄	—	151	8	no	—
239	C₄⋊C₄	—	151	4	no	—
240	C₄⋊C₄	—	151	8	no	—
241	C₄⋊C₄	—	151	4	no	—
242	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
243	C₄⋊C₄	C₂²×D₁₅₁	1	8	yes	—
244	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
245	C₂²⋊C₄	—	151	4	no	—
246	C₂²⋊C₄	—	151	8	no	—
247	C₂²⋊C₄	C₂²×D₁₅₁	1	4	yes	—
248	C₄²	—	151	2	no	—
249	C₄²	—	151	2	no	—
250	C₄²	—	151	1	no	—
251	C₄²	—	151	2	no	—
252	C₄²	—	151	2	no	—
253	C₄²	—	151	1	no	—
254	C₄²	—	151	8	no	—
255	C₄²	—	151	2	no	—
256	C₄²	—	151	4	no	—
257	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
258	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
259	C₄²	C₂²×D₁₅₁	1	2	yes	—
260	C₄²	C₂²×D₁₅₁	1	4	yes	—
261	C₂³	—	302	4	no	—
262	C₂³	—	151	4	no	—
263	C₂³	—	151	1	no	—
264	C₂³	—	151	1	no	—
265	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
266	Q₈	—	2	8	no	—
267	Q₈	—	302	16	no	—
268	Q₈	—	302	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	302	4	no	—
271	C₂×C₄	—	302	16	no	—
272	C₂×C₄	—	302	4	no	—
273	C₂×C₄	—	302	4	no	—
274	C₂×C₄	—	302	4	no	—
275	C₂×C₄	—	151	2	no	—
276	C₂×C₄	—	151	8	no	—
277	C₂×C₄	—	151	4	no	—
278	C₂×C₄	—	151	1	no	—
279	C₂×C₄	—	151	1	no	—
280	C₂×C₄	—	151	2	no	—
281	C₂×C₄	—	151	4	no	—
282	C₂×C₄	—	151	4	no	—
283	C₂×C₄	—	151	4	no	—
284	C₂×C₄	—	151	4	no	—
285	C₂×C₄	—	151	8	no	—
286	C₂×C₄	Q₈×D₁₅₁	1	4	yes	complemented
287	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
288	C₂×C₄	C₂³×D₁₅₁	1	8	yes	—
289	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
291	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
292	C₂²	—	2	2	no	—
293	C₂²	—	302	8	no	—
294	C₂²	—	302	2	no	—
295	C₂²	—	151	2	no	—
296	C₂²	—	151	2	no	—
297	C₂²	—	151	4	no	—
298	C₂²	C₂×Q₈×D₁₅₁	1	2	yes	—
299	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	8	no	—
302	C₄	—	302	2	no	—
303	C₄	—	302	8	no	—
304	C₄	—	151	2	no	—
305	C₄	—	151	2	no	—
306	C₄	—	151	4	no	—
307	C₄	C₂×Q₈×D₁₅₁	1	2	yes	—
308	C₄	C₂×Q₈×D₁₅₁	1	2	yes	—
309	C₄	#41×D₁₅₁	1	4	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	302	2	no	—
312	C₂	—	151	4	no	—
313	C₂	C₂²×Q₈×D₁₅₁	1	1	yes	characteristic
314	C₂	C₂×#41×D₁₅₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅₁
Fitting	#533×C₁₅₁	C₂
prime radical	C₂³×D₁₅₁	C₂³
hypercenter	#533	D₁₅₁
hypocenter	C₁₅₁	C₂×#533
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #23752769	C₂	semidirect
2. #23752768	C₂	semidirect
3. #23752767	C₂	semidirect
4. #23752766	C₂	semidirect
5. #23752765	C₂	semidirect
6. #23752764	C₂	semidirect
7. C₄⋊Q₈×D₁₅₁	C₂	semidirect
8. #125×D₁₅₁	C₂	semidirect
10. C₄×Q₈×D₁₅₁	C₂	semidirect
13. #533×C₁₅₁	C₂	semidirect
17. #18012739	C₂²	semidirect
18. #18012738	C₂²	semidirect
19. #18012738	C₂²	semidirect
20. #18012737	C₂²	semidirect
21. #18012731	C₂²	semidirect
22. #18012730	C₂²	semidirect
23. #18012729	C₂²	semidirect
29. Q₈×Dic₁₅₁	C₂²	semidirect
30. C₄×Dic₃₀₂	C₂²	semidirect
31. C₄×Dic₃₀₂	C₂²	semidirect
32. #18012704	C₂²	semidirect
33. #18012704	C₂²	semidirect
51. C₄⋊Q₈×C₁₅₁	C₂²	semidirect
52. #125×C₁₅₁	C₂²	semidirect
54. C₄×Q₈×C₁₅₁	C₂²	semidirect
64. C₂×C₄×D₁₅₁	Q₈	semidirect
110. C₂×Dic₁₅₁	C₂×Q₈	semidirect
122. C₄×D₁₅₁	C₂×Q₈	semidirect
123. C₄×D₁₅₁	C₂×Q₈	semidirect
128. C₂×C₄×C₁₅₁	C₂×Q₈	semidirect
156. D₁₅₁	#533	direct (N × H)
157. C₁₅₁	C₂×#533	semidirect
165. #533	D₁₅₁	direct (N × H)
179. C₄⋊Q₈	C₂×D₁₅₁	semidirect
183. #125	C₂×D₁₅₁	semidirect
195. C₄×Q₈	C₂×D₁₅₁	semidirect
286. C₂×C₄	Q₈×D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752769	2
#23752768	2
#23752767	2
#23752766	2
#23752765	2
#23752764	2
C₄⋊Q₈×D₁₅₁	2
#125×D₁₅₁	2
C₂²⋊Q₈×D₁₅₁	2
C₄×Q₈×D₁₅₁	2
#117×D₁₅₁	2
C₂×C₄²×D₁₅₁	2
#533×C₁₅₁	2
C₂×#533	151

Subgroups — #533×D151 #34082395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #533×D₁₅₁ #34082395