Subgroups — C₂×#2102×C₁₅₁ #50328675

Order 38656.

710 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₁₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#2102×C₁₅₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#344×C₁₅₁	C₂	1	2	yes	maximal
5	C₂×#339×C₁₅₁	C₂	1	2	yes	maximal
6	C₂×H(3,ℤ/4)×C₁₅₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₁₅₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₅₁	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
12	#430×C₁₅₁	—	2	2	no	—
13	#430×C₁₅₁	C₂²	1	4	yes	—
14	C₂×#108×C₁₅₁	—	2	2	no	—
15	#423×C₁₅₁	—	2	2	no	—
16	#423×C₁₅₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₁₅₁	—	2	2	no	—
18	#344×C₁₅₁	C₂²	1	8	yes	—
19	#339×C₁₅₁	C₂²	1	8	yes	—
20	H(3,ℤ/4)×C₁₅₁	C₂²	1	4	yes	—
21	C₂³×C₄×C₁₅₁	—	2	1	no	—
22	C₂³×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₁₅₁	—	2	2	no	—
24	C₂×M₁₆×C₁₅₁	—	2	2	no	—
25	C₂×M₁₆×C₁₅₁	—	2	1	no	—
26	C₂×M₁₆×C₁₅₁	—	2	2	no	—
27	C₂×M₁₆×C₁₅₁	—	2	2	no	—
28	C₂×M₁₆×C₁₅₁	—	2	2	no	—
29	C₂×M₁₆×C₁₅₁	—	2	2	no	—
30	C₂×M₁₆×C₁₅₁	—	2	2	no	—
31	C₂×M₁₆×C₁₅₁	—	2	2	no	—
32	C₂×M₁₆×C₁₅₁	—	2	2	no	—
33	C₂×M₁₆×C₁₅₁	D₄	1	2	yes	—
34	C₂×M₁₆×C₁₅₁	C₂³	1	4	yes	—
35	C₂×M₁₆×C₁₅₁	C₂³	1	2	yes	—
36	C₂×M₁₆×C₁₅₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₅₁	—	2	1	no	—
38	C₂²×C₈×C₁₅₁	—	2	2	no	—
39	C₂²×C₈×C₁₅₁	D₄	1	2	yes	—
40	#117×C₁₅₁	—	2	2	no	—
41	#117×C₁₅₁	—	2	2	no	—
42	#117×C₁₅₁	—	2	2	no	—
43	#117×C₁₅₁	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₁₅₁	—	2	2	no	—
46	C₂×C₄²×C₁₅₁	—	2	1	no	—
47	C₂×C₄²×C₁₅₁	—	2	1	no	—
48	C₂×C₄²×C₁₅₁	—	2	1	no	—
49	C₂×C₄²×C₁₅₁	—	2	2	no	—
50	C₂×C₄²×C₁₅₁	Q₈	1	1	yes	characteristic
51	C₂×C₄²×C₁₅₁	Q₈	1	2	yes	—
52	C₂×C₄²×C₁₅₁	Q₈	1	1	yes	characteristic
53	C₂×C₄²×C₁₅₁	C₂³	1	2	yes	—
54	#108×C₁₅₁	—	2	8	no	—
55	C₄⋊C₈×C₁₅₁	—	2	8	no	—
56	C₂⁴×C₁₅₁	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₅₁	—	2	1	no	—
58	C₂²×C₄×C₁₅₁	—	2	2	no	—
59	C₂²×C₄×C₁₅₁	—	2	1	no	—
60	C₂²×C₄×C₁₅₁	—	2	2	no	—
61	C₂²×C₄×C₁₅₁	—	4	4	no	—
62	C₂²×C₄×C₁₅₁	—	4	2	no	—
63	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₅₁	#41	1	1	yes	characteristic
68	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
69	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
70	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
71	C₂²×C₄×C₁₅₁	#41	1	2	yes	—
72	C₂²×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₁₅₁	—	2	4	no	—
74	M₁₆×C₁₅₁	—	2	4	no	—
75	M₁₆×C₁₅₁	—	2	4	no	—
76	M₁₆×C₁₅₁	—	2	4	no	—
77	M₁₆×C₁₅₁	—	2	4	no	—
78	M₁₆×C₁₅₁	—	4	2	no	—
79	M₁₆×C₁₅₁	—	4	2	no	—
80	M₁₆×C₁₅₁	—	4	2	no	—
81	C₂×C₈×C₁₅₁	—	2	2	no	—
82	C₂×C₈×C₁₅₁	—	2	2	no	—
83	C₂×C₈×C₁₅₁	—	2	2	no	—
84	C₂×C₈×C₁₅₁	—	2	4	no	—
85	C₂×C₈×C₁₅₁	—	4	1	no	—
86	C₂×C₈×C₁₅₁	—	4	1	no	—
87	C₂×C₈×C₁₅₁	—	4	2	no	—
88	C₂×C₈×C₁₅₁	—	4	2	no	—
89	C₂×C₈×C₁₅₁	C₂×D₄	1	4	yes	—
90	C₄⋊C₄×C₁₅₁	—	2	4	no	—
91	C₄⋊C₄×C₁₅₁	—	4	2	no	—
92	C₂²⋊C₄×C₁₅₁	—	2	4	no	—
93	C₂²⋊C₄×C₁₅₁	—	4	2	no	—
94	C₄²×C₁₅₁	—	2	1	no	—
95	C₄²×C₁₅₁	—	2	2	no	—
96	C₄²×C₁₅₁	—	2	1	no	—
97	C₄²×C₁₅₁	—	2	2	no	—
98	C₄²×C₁₅₁	—	2	4	no	—
99	C₄²×C₁₅₁	—	4	2	no	—
100	C₄²×C₁₅₁	C₂×Q₈	1	2	yes	—
101	C₄²×C₁₅₁	C₂×Q₈	1	2	yes	—
102	C₂³×C₁₅₁	—	4	2	no	—
103	C₂³×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₁₅₁	#126	1	2	yes	—
106	C₂³×C₁₅₁	#126	1	2	yes	—
107	C₂³×C₁₅₁	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₁₅₁	—	2	4	no	—
109	C₂×C₄×C₁₅₁	—	2	2	no	—
110	C₂×C₄×C₁₅₁	—	2	2	no	—
111	C₂×C₄×C₁₅₁	—	4	4	no	—
112	C₂×C₄×C₁₅₁	—	4	4	no	—
113	C₂×C₄×C₁₅₁	—	4	2	no	—
114	C₂×C₄×C₁₅₁	—	4	4	no	—
115	C₂×C₄×C₁₅₁	—	8	4	no	—
116	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₅₁	#123	1	1	yes	characteristic
119	C₂×C₄×C₁₅₁	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₅₁	#123	1	2	yes	—
121	C₂×C₄×C₁₅₁	#124	1	2	yes	—
122	C₂×C₄×C₁₅₁	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₅₁	#124	1	1	yes	characteristic
124	C₂×C₄×C₁₅₁	#125	1	2	yes	—
125	C₂×C₄×C₁₅₁	#125	1	2	yes	—
126	C₂×C₄×C₁₅₁	#125	1	2	yes	—
127	C₂×C₄×C₁₅₁	#125	1	2	yes	—
128	C₂×C₄×C₁₅₁	#126	1	2	yes	—
129	C₂×C₄×C₁₅₁	#126	1	2	yes	—
130	C₂×C₄×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₅₁	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₅₁	C₂×#41	1	2	yes	—
134	C₈×C₁₅₁	—	2	4	no	—
135	C₈×C₁₅₁	—	4	2	no	—
136	C₈×C₁₅₁	—	4	4	no	—
137	C₂²×C₁₅₁	—	2	2	no	—
138	C₂²×C₁₅₁	—	2	4	no	—
139	C₂²×C₁₅₁	—	4	1	no	—
140	C₂²×C₁₅₁	—	4	1	no	—
141	C₂²×C₁₅₁	—	4	2	no	—
142	C₂²×C₁₅₁	#400	1	1	yes	characteristic
143	C₂²×C₁₅₁	#400	1	2	yes	—
144	C₂²×C₁₅₁	#400	1	1	yes	characteristic
145	C₂²×C₁₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₁₅₁	C₂×#126	1	2	yes	—
147	C₄×C₁₅₁	—	2	4	no	—
148	C₄×C₁₅₁	—	4	2	no	—
149	C₄×C₁₅₁	—	8	4	no	—
150	C₄×C₁₅₁	#400	1	1	yes	characteristic
151	C₄×C₁₅₁	#400	1	2	yes	—
152	C₄×C₁₅₁	#400	1	1	yes	characteristic
153	C₄×C₁₅₁	C₂×#123	1	1	yes	characteristic
154	C₄×C₁₅₁	C₂×#124	1	1	yes	characteristic
155	C₄×C₁₅₁	C₂×#125	1	2	yes	—
156	C₂×C₁₅₁	—	2	1	no	—
157	C₂×C₁₅₁	—	2	2	no	—
158	C₂×C₁₅₁	—	2	1	no	—
159	C₂×C₁₅₁	—	2	2	no	—
160	C₂×C₁₅₁	#2102	1	2	yes	direct product factor
161	C₂×C₁₅₁	C₂×#400	1	1	yes	characteristic
162	C₂×#2102	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₁₅₁	C₂×#2102	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×#430	C₂×C₁₅₁	1	1	yes	characteristic
165	C₂×#423	C₂×C₁₅₁	1	1	yes	characteristic
166	#2102	C₂×C₁₅₁	1	8	yes	direct product factor
167	C₂×#344	C₂×C₁₅₁	1	2	yes	—
168	C₂×#339	C₂×C₁₅₁	1	2	yes	—
169	C₂×H(3,ℤ/4)	C₂×C₁₅₁	1	1	yes	characteristic
170	C₂²×M₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
171	C₂²×M₁₆	C₂²×C₁₅₁	1	2	yes	—
172	C₂²×M₁₆	C₂²×C₁₅₁	1	1	yes	characteristic
173	C₂×#117	C₂²×C₁₅₁	1	2	yes	—
174	C₂²×C₄²	C₂²×C₁₅₁	1	1	yes	characteristic
175	#430	—	2	2	no	—
176	#430	C₂²×C₁₅₁	1	4	yes	—
177	C₂×#108	—	2	2	no	—
178	#423	—	2	2	no	—
179	#423	C₂²×C₁₅₁	1	4	yes	—
180	C₂×C₄⋊C₈	—	2	2	no	—
181	#344	C₂²×C₁₅₁	1	8	yes	—
182	#339	C₂²×C₁₅₁	1	8	yes	—
183	H(3,ℤ/4)	C₂²×C₁₅₁	1	4	yes	—
184	C₂³×C₄	—	2	1	no	—
185	C₂³×C₄	C₂³×C₁₅₁	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	1	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	D₄×C₁₅₁	1	2	yes	—
197	C₂×M₁₆	C₂³×C₁₅₁	1	4	yes	—
198	C₂×M₁₆	C₂³×C₁₅₁	1	2	yes	—
199	C₂×M₁₆	C₂³×C₁₅₁	1	2	yes	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	D₄×C₁₅₁	1	2	yes	—
203	#117	—	2	2	no	—
204	#117	—	2	2	no	—
205	#117	—	2	2	no	—
206	#117	C₂³×C₁₅₁	1	4	yes	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	1	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	Q₈×C₁₅₁	1	1	yes	characteristic
214	C₂×C₄²	Q₈×C₁₅₁	1	2	yes	—
215	C₂×C₄²	Q₈×C₁₅₁	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₁₅₁	1	2	yes	—
217	#108	—	2	8	no	—
218	C₄⋊C₈	—	2	8	no	—
219	C₂⁴	#41×C₁₅₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₅₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×Q₈×C₁₅₁	1	1	yes	characteristic
230	C₂²×C₄	#41×C₁₅₁	1	1	yes	characteristic
231	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
232	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
233	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
234	C₂²×C₄	#41×C₁₅₁	1	2	yes	—
235	C₂²×C₄	C₂⁴×C₁₅₁	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	M₁₆	—	4	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	C₂×D₄×C₁₅₁	1	4	yes	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	4	no	—
262	C₄²	—	4	2	no	—
263	C₄²	C₂×Q₈×C₁₅₁	1	2	yes	—
264	C₄²	C₂×Q₈×C₁₅₁	1	2	yes	—
265	C₂³	—	4	2	no	—
266	C₂³	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
267	C₂³	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
268	C₂³	#126×C₁₅₁	1	2	yes	—
269	C₂³	#126×C₁₅₁	1	2	yes	—
270	C₂³	C₂×#41×C₁₅₁	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	8	4	no	—
279	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
280	C₂×C₄	C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
281	C₂×C₄	#123×C₁₅₁	1	1	yes	characteristic
282	C₂×C₄	#123×C₁₅₁	1	1	yes	characteristic
283	C₂×C₄	#123×C₁₅₁	1	2	yes	—
284	C₂×C₄	#124×C₁₅₁	1	1	yes	characteristic
285	C₂×C₄	#124×C₁₅₁	1	2	yes	—
286	C₂×C₄	#124×C₁₅₁	1	1	yes	characteristic
287	C₂×C₄	#125×C₁₅₁	1	2	yes	—
288	C₂×C₄	#125×C₁₅₁	1	2	yes	—
289	C₂×C₄	#125×C₁₅₁	1	2	yes	—
290	C₂×C₄	#125×C₁₅₁	1	2	yes	—
291	C₂×C₄	#126×C₁₅₁	1	2	yes	—
292	C₂×C₄	#126×C₁₅₁	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₁₅₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₁₅₁	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₁₅₁	1	2	yes	—
296	C₂×C₄	C₂×#41×C₁₅₁	1	2	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	4	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#400×C₁₅₁	1	1	yes	characteristic
306	C₂²	#400×C₁₅₁	1	2	yes	—
307	C₂²	#400×C₁₅₁	1	1	yes	characteristic
308	C₂²	C₂×C₂²⋊Q₈×C₁₅₁	1	1	yes	characteristic
309	C₂²	C₂×#126×C₁₅₁	1	2	yes	—
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	#400×C₁₅₁	1	1	yes	characteristic
314	C₄	#400×C₁₅₁	1	1	yes	characteristic
315	C₄	#400×C₁₅₁	1	2	yes	—
316	C₄	C₂×#123×C₁₅₁	1	1	yes	characteristic
317	C₄	C₂×#124×C₁₅₁	1	1	yes	characteristic
318	C₄	C₂×#125×C₁₅₁	1	2	yes	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	2	no	—
323	C₂	#2102×C₁₅₁	1	2	yes	direct product factor
324	C₂	C₂×#400×C₁₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₅₁	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₅₁
socle	C₂²×C₁₅₁	#400
prime radical	C₂⁴×C₁₅₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₅₁ ◁ C₂³×C₄×C₁₅₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #2102×C₁₅₁	C₂	direct (N × H)
160. C₂×C₁₅₁	#2102	direct (N × H)
162. C₂×#2102	C₁₅₁	direct (N × H)
163. C₁₅₁	C₂×#2102	direct (N × H)
166. #2102	C₂×C₁₅₁	direct (N × H)
323. C₂	#2102×C₁₅₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#430×C₁₅₁	2
C₂×#423×C₁₅₁	2
#2102×C₁₅₁	2
C₂×#344×C₁₅₁	2
C₂×#339×C₁₅₁	2
C₂×H(3,ℤ/4)×C₁₅₁	2
C₂×#2102	151

Subgroups — C2×#2102×C151 #50328675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2102×C₁₅₁ #50328675