Subgroups — C₂×C₁₀₁×#1298 #38790204

Order 25856.

938 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#407×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₁₀₁	C₂	1	2	yes	maximal
3	C₁₀₁×#1298	C₂	1	4	yes	maximal, direct product factor
4	C₂⁴×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₀₁	C₄	1	2	yes	—
6	C₂³×C₈×C₁₀₁	—	2	2	no	—
7	#407×C₁₀₁	—	2	2	no	—
8	#407×C₁₀₁	C₄	1	2	yes	—
9	#407×C₁₀₁	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₈×C₁₀₁	—	2	2	no	—
11	C₂×C₂²⋊C₈×C₁₀₁	—	2	2	no	—
12	C₂×C₂²⋊C₈×C₁₀₁	—	2	2	no	—
13	C₂×C₂²⋊C₈×C₁₀₁	—	2	2	no	—
14	C₂×C₂²⋊C₈×C₁₀₁	C₄	1	4	yes	—
15	C₂×C₂²⋊C₈×C₁₀₁	C₄	1	4	yes	—
16	C₂×C₂²⋊C₈×C₁₀₁	C₂²	1	4	yes	—
17	C₂⁵×C₁₀₁	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₁	—	4	4	no	—
19	C₂³×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
24	C₂³×C₄×C₁₀₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₀₁	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
27	C₂³×C₄×C₁₀₁	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₁₀₁	—	2	2	no	—
30	C₂×M₁₆×C₁₀₁	—	4	4	no	—
31	C₂×M₁₆×C₁₀₁	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₀₁	—	2	4	no	—
33	C₂²×C₈×C₁₀₁	—	2	4	no	—
34	C₂²×C₈×C₁₀₁	—	2	2	no	—
35	C₂²×C₈×C₁₀₁	—	2	4	no	—
36	C₂²×C₈×C₁₀₁	—	4	8	no	—
37	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
38	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
39	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
40	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
41	C₂²⋊C₈×C₁₀₁	—	2	4	no	—
42	C₂²⋊C₈×C₁₀₁	C₂×C₄	1	8	yes	—
43	C₂⁴×C₁₀₁	—	2	8	no	—
44	C₂⁴×C₁₀₁	—	4	2	no	—
45	C₂⁴×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂⁴×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂⁴×C₁₀₁	C₄⋊C₄	1	2	yes	—
48	C₂⁴×C₁₀₁	C₄⋊C₄	1	2	yes	—
49	C₂⁴×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₁	—	2	8	no	—
51	C₂²×C₄×C₁₀₁	—	2	4	no	—
52	C₂²×C₄×C₁₀₁	—	4	4	no	—
53	C₂²×C₄×C₁₀₁	—	4	2	no	—
54	C₂²×C₄×C₁₀₁	—	4	2	no	—
55	C₂²×C₄×C₁₀₁	—	4	4	no	—
56	C₂²×C₄×C₁₀₁	—	4	2	no	—
57	C₂²×C₄×C₁₀₁	—	4	2	no	—
58	C₂²×C₄×C₁₀₁	—	4	4	no	—
59	C₂²×C₄×C₁₀₁	—	4	2	no	—
60	C₂²×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₁	C₄²	1	2	yes	—
63	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
78	M₁₆×C₁₀₁	—	4	4	no	—
79	M₁₆×C₁₀₁	—	4	4	no	—
80	C₂×C₈×C₁₀₁	—	2	4	no	—
81	C₂×C₈×C₁₀₁	—	2	8	no	—
82	C₂×C₈×C₁₀₁	—	4	8	no	—
83	C₂×C₈×C₁₀₁	—	4	8	no	—
84	C₂×C₈×C₁₀₁	—	4	8	no	—
85	C₂×C₈×C₁₀₁	—	4	4	no	—
86	C₂³×C₁₀₁	—	2	4	no	—
87	C₂³×C₁₀₁	—	2	4	no	—
88	C₂³×C₁₀₁	—	2	4	no	—
89	C₂³×C₁₀₁	—	2	4	no	—
90	C₂³×C₁₀₁	—	2	4	no	—
91	C₂³×C₁₀₁	—	2	2	no	—
92	C₂³×C₁₀₁	—	2	4	no	—
93	C₂³×C₁₀₁	—	2	4	no	—
94	C₂³×C₁₀₁	—	4	1	no	—
95	C₂³×C₁₀₁	—	4	2	no	—
96	C₂³×C₁₀₁	—	4	1	no	—
97	C₂³×C₁₀₁	—	4	16	no	—
98	C₂³×C₁₀₁	#95	1	1	yes	characteristic
99	C₂³×C₁₀₁	#95	1	2	yes	—
100	C₂³×C₁₀₁	#95	1	1	yes	characteristic
101	C₂³×C₁₀₁	#108	1	8	yes	—
102	C₂³×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₀₁	—	2	4	no	—
105	C₂×C₄×C₁₀₁	—	2	2	no	—
106	C₂×C₄×C₁₀₁	—	2	4	no	—
107	C₂×C₄×C₁₀₁	—	2	2	no	—
108	C₂×C₄×C₁₀₁	—	2	4	no	—
109	C₂×C₄×C₁₀₁	—	2	2	no	—
110	C₂×C₄×C₁₀₁	—	2	4	no	—
111	C₂×C₄×C₁₀₁	—	2	2	no	—
112	C₂×C₄×C₁₀₁	—	2	2	no	—
113	C₂×C₄×C₁₀₁	—	2	4	no	—
114	C₂×C₄×C₁₀₁	—	4	2	no	—
115	C₂×C₄×C₁₀₁	—	4	2	no	—
116	C₂×C₄×C₁₀₁	—	4	2	no	—
117	C₂×C₄×C₁₀₁	—	4	2	no	—
118	C₂×C₄×C₁₀₁	—	4	1	no	—
119	C₂×C₄×C₁₀₁	—	4	2	no	—
120	C₂×C₄×C₁₀₁	—	4	1	no	—
121	C₂×C₄×C₁₀₁	#95	1	1	yes	characteristic
122	C₂×C₄×C₁₀₁	#95	1	2	yes	—
123	C₂×C₄×C₁₀₁	#95	1	1	yes	characteristic
124	C₂×C₄×C₁₀₁	#99	1	4	yes	—
125	C₂×C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₈×C₁₀₁	—	4	8	no	—
129	C₈×C₁₀₁	—	4	4	no	—
130	C₂²×C₁₀₁	—	2	16	no	—
131	C₂²×C₁₀₁	—	2	2	no	—
132	C₂²×C₁₀₁	—	2	2	no	—
133	C₂²×C₁₀₁	—	2	2	no	—
134	C₂²×C₁₀₁	—	2	2	no	—
135	C₂²×C₁₀₁	—	2	2	no	—
136	C₂²×C₁₀₁	—	2	2	no	—
137	C₂²×C₁₀₁	—	2	1	no	—
138	C₂²×C₁₀₁	—	2	1	no	—
139	C₂²×C₁₀₁	—	4	4	no	—
140	C₂²×C₁₀₁	—	4	8	no	—
141	C₂²×C₁₀₁	—	4	4	no	—
142	C₂²×C₁₀₁	—	4	4	no	—
143	C₂²×C₁₀₁	#341	1	4	yes	—
144	C₂²×C₁₀₁	#341	1	4	yes	—
145	C₂²×C₁₀₁	#342	1	2	yes	—
146	C₂²×C₁₀₁	C₂×#95	1	1	yes	characteristic
147	C₂²×C₁₀₁	C₂×#108	1	4	yes	—
148	C₄×C₁₀₁	—	2	2	no	—
149	C₄×C₁₀₁	—	2	2	no	—
150	C₄×C₁₀₁	—	2	1	no	—
151	C₄×C₁₀₁	—	2	1	no	—
152	C₄×C₁₀₁	#342	1	2	yes	—
153	C₄×C₁₀₁	C₂×#99	1	2	yes	—
154	C₂×#1298	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₁₀₁	—	2	4	no	—
156	C₂×C₁₀₁	—	2	4	no	—
157	C₂×C₁₀₁	—	4	1	no	—
158	C₂×C₁₀₁	—	4	1	no	—
159	C₂×C₁₀₁	#1298	1	4	yes	direct product factor
160	C₂×C₁₀₁	C₂×#341	1	2	yes	—
161	C₂×C₁₀₁	C₂×#342	1	1	yes	characteristic
162	C₂×#407	C₂×C₁₀₁	1	1	yes	characteristic
163	C₂²×C₂²⋊C₈	C₂×C₁₀₁	1	2	yes	—
164	#1298	C₂×C₁₀₁	1	4	yes	direct product factor
165	C₁₀₁	C₂×#1298	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁴×C₄	C₂²×C₁₀₁	1	1	yes	characteristic
167	C₂²×M₁₆	C₄×C₁₀₁	1	2	yes	—
168	C₂³×C₈	—	2	2	no	—
169	#407	—	2	2	no	—
170	#407	C₄×C₁₀₁	1	2	yes	—
171	#407	C₂²×C₁₀₁	1	2	yes	—
172	C₂×C₂²⋊C₈	—	2	2	no	—
173	C₂×C₂²⋊C₈	—	2	2	no	—
174	C₂×C₂²⋊C₈	—	2	2	no	—
175	C₂×C₂²⋊C₈	—	2	2	no	—
176	C₂×C₂²⋊C₈	C₄×C₁₀₁	1	4	yes	—
177	C₂×C₂²⋊C₈	C₄×C₁₀₁	1	4	yes	—
178	C₂×C₂²⋊C₈	C₂²×C₁₀₁	1	4	yes	—
179	C₂⁵	Q₈×C₁₀₁	1	1	yes	characteristic
180	C₂³×C₄	—	4	4	no	—
181	C₂³×C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
182	C₂³×C₄	C₂×C₄×C₁₀₁	1	2	yes	—
183	C₂³×C₄	C₂×C₄×C₁₀₁	1	2	yes	—
184	C₂³×C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
185	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
186	C₂³×C₄	D₄×C₁₀₁	1	1	yes	characteristic
187	C₂³×C₄	D₄×C₁₀₁	1	1	yes	characteristic
188	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
189	C₂³×C₄	Q₈×C₁₀₁	1	1	yes	characteristic
190	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	4	4	no	—
193	C₂×M₁₆	C₂×C₄×C₁₀₁	1	4	yes	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	—	2	4	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	4	no	—
198	C₂²×C₈	—	4	8	no	—
199	C₂²⋊C₈	—	2	4	no	—
200	C₂²⋊C₈	—	2	4	no	—
201	C₂²⋊C₈	—	2	4	no	—
202	C₂²⋊C₈	—	2	4	no	—
203	C₂²⋊C₈	—	2	4	no	—
204	C₂²⋊C₈	C₂×C₄×C₁₀₁	1	8	yes	—
205	C₂⁴	—	2	8	no	—
206	C₂⁴	—	4	2	no	—
207	C₂⁴	C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
208	C₂⁴	C₄⋊C₄×C₁₀₁	1	2	yes	—
209	C₂⁴	C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
210	C₂⁴	C₄⋊C₄×C₁₀₁	1	2	yes	—
211	C₂⁴	C₂×Q₈×C₁₀₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₄²×C₁₀₁	1	1	yes	characteristic
223	C₂²×C₄	C₄²×C₁₀₁	1	1	yes	characteristic
224	C₂²×C₄	C₄²×C₁₀₁	1	2	yes	—
225	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
226	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
227	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
228	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
230	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
231	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
232	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
233	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
234	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
235	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
237	C₂²×C₄	C₂²×C₄×C₁₀₁	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
240	M₁₆	—	4	4	no	—
241	M₁₆	—	4	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	8	no	—
244	C₂×C₈	—	4	8	no	—
245	C₂×C₈	—	4	8	no	—
246	C₂×C₈	—	4	8	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	4	1	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	16	no	—
260	C₂³	#95×C₁₀₁	1	1	yes	characteristic
261	C₂³	#95×C₁₀₁	1	2	yes	—
262	C₂³	#95×C₁₀₁	1	1	yes	characteristic
263	C₂³	#108×C₁₀₁	1	8	yes	—
264	C₂³	C₂×C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
265	C₂³	C₂×C₄⋊C₄×C₁₀₁	1	2	yes	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	#95×C₁₀₁	1	1	yes	characteristic
284	C₂×C₄	#95×C₁₀₁	1	2	yes	—
285	C₂×C₄	#95×C₁₀₁	1	1	yes	characteristic
286	C₂×C₄	#99×C₁₀₁	1	4	yes	—
287	C₂×C₄	C₂×C₄²×C₁₀₁	1	1	yes	characteristic
288	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	2	yes	—
289	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
290	C₈	—	4	8	no	—
291	C₈	—	4	4	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	4	8	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	#341×C₁₀₁	1	4	yes	—
306	C₂²	#341×C₁₀₁	1	4	yes	—
307	C₂²	#342×C₁₀₁	1	2	yes	—
308	C₂²	C₂×#95×C₁₀₁	1	1	yes	characteristic
309	C₂²	C₂×#108×C₁₀₁	1	4	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	#342×C₁₀₁	1	2	yes	—
315	C₄	C₂×#99×C₁₀₁	1	2	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	2	4	no	—
318	C₂	—	4	1	no	—
319	C₂	—	4	1	no	—
320	C₂	C₁₀₁×#1298	1	4	yes	direct product factor
321	C₂	C₂×#341×C₁₀₁	1	2	yes	—
322	C₂	C₂×#342×C₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₁
Frattini, cosocle	C₂³×C₄	C₂³×C₁₀₁
socle	C₂³×C₁₀₁	#95
prime radical	C₂⁵×C₁₀₁	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂⁴×C₄×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₁×#1298	C₂	direct (N × H)
154. C₂×#1298	C₁₀₁	direct (N × H)
159. C₂×C₁₀₁	#1298	direct (N × H)
164. #1298	C₂×C₁₀₁	direct (N × H)
165. C₁₀₁	C₂×#1298	direct (N × H)
320. C₂	C₁₀₁×#1298	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×#407×C₁₀₁	2
C₂²×C₂²⋊C₈×C₁₀₁	2
C₁₀₁×#1298	2
C₂×#1298	101

Subgroups — C2×C101×#1298 #38790204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₁×#1298 #38790204