Subgroups — C₄×D₄⋊C₈×C₁₀₁ #38790319

Order 25856.

774 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊C₈×C₁₀₁	C₂	1	2	yes	maximal
5	C₁₀₁×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₁₀₁	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₀₁	—	2	1	no	—
9	C₂×C₄⋊C₈×C₁₀₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₀₁	—	2	2	no	—
11	C₂×C₄×C₈×C₁₀₁	—	2	1	no	—
12	C₂×C₄×C₈×C₁₀₁	C₂²	1	2	yes	—
13	C₄×C₄⋊C₄×C₁₀₁	C₄	1	2	yes	—
14	C₄×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊C₄×C₁₀₁	—	2	1	no	—
16	C₄×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
17	C₄³×C₁₀₁	C₂²	1	1	yes	characteristic
18	#336×C₁₀₁	—	2	1	no	—
19	D₄⋊C₈×C₁₀₁	C₄	1	8	yes	direct product factor
20	C₂²×D₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₀₁	—	2	2	no	—
22	C₂³×C₄×C₁₀₁	—	2	1	no	—
23	C₂²×C₈×C₁₀₁	—	2	2	no	—
24	C₂²×C₈×C₁₀₁	—	2	2	no	—
25	C₄×D₄×C₁₀₁	—	2	4	no	—
26	C₄×D₄×C₁₀₁	—	2	2	no	—
27	C₄×D₄×C₁₀₁	C₈	1	8	yes	complemented
28	C₄×D₄×C₁₀₁	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₁₀₁	—	2	1	no	—
33	C₂×C₄²×C₁₀₁	—	2	2	no	—
34	C₂×C₄²×C₁₀₁	—	2	1	no	—
35	C₂×C₄²×C₁₀₁	—	4	2	no	—
36	C₂×C₄²×C₁₀₁	—	4	2	no	—
37	C₂×C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
39	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₁₀₁	—	2	2	no	—
43	C₄⋊C₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
44	C₄×C₈×C₁₀₁	—	2	8	no	—
45	C₄×C₈×C₁₀₁	—	2	2	no	—
46	C₄×C₈×C₁₀₁	—	4	2	no	—
47	C₄×C₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
48	#95×C₁₀₁	—	2	2	no	—
49	#95×C₁₀₁	—	2	1	no	—
50	C₂⁴×C₁₀₁	—	2	1	no	—
51	C₂×D₄×C₁₀₁	—	2	1	no	—
52	C₂×D₄×C₁₀₁	—	2	2	no	—
53	C₂×D₄×C₁₀₁	C₄²	1	2	yes	—
54	C₂×D₄×C₁₀₁	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₀₁	—	2	1	no	—
56	C₂²×C₄×C₁₀₁	—	2	2	no	—
57	C₂²×C₄×C₁₀₁	—	2	8	no	—
58	C₂²×C₄×C₁₀₁	—	2	4	no	—
59	C₂²×C₄×C₁₀₁	—	4	4	no	—
60	C₂²×C₄×C₁₀₁	—	4	4	no	—
61	C₂²×C₄×C₁₀₁	—	4	1	no	—
62	C₂²×C₄×C₁₀₁	—	4	1	no	—
63	C₂²×C₄×C₁₀₁	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₀₁	—	2	8	no	—
70	C₂×C₈×C₁₀₁	—	2	4	no	—
71	C₂×C₈×C₁₀₁	—	2	4	no	—
72	C₂×C₈×C₁₀₁	—	4	2	no	—
73	C₂×C₈×C₁₀₁	—	4	2	no	—
74	C₄⋊C₄×C₁₀₁	—	2	1	no	—
75	C₄⋊C₄×C₁₀₁	—	2	2	no	—
76	C₄⋊C₄×C₁₀₁	C₄²	1	2	yes	—
77	C₄⋊C₄×C₁₀₁	C₂×C₈	1	4	yes	—
78	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
79	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
80	C₄²×C₁₀₁	—	2	4	no	—
81	C₄²×C₁₀₁	—	2	2	no	—
82	C₄²×C₁₀₁	—	4	4	no	—
83	C₄²×C₁₀₁	—	4	1	no	—
84	C₄²×C₁₀₁	—	4	1	no	—
85	C₄²×C₁₀₁	C₂×C₈	1	4	yes	—
86	C₄²×C₁₀₁	M₁₆	1	4	yes	—
87	C₄²×C₁₀₁	D₈	1	2	yes	—
88	C₄²×C₁₀₁	SD₁₆	1	2	yes	—
89	C₄²×C₁₀₁	C₂²×C₄	1	2	yes	—
90	C₄²×C₁₀₁	#41	1	2	yes	—
91	C₂³×C₁₀₁	—	2	2	no	—
92	C₂³×C₁₀₁	—	2	4	no	—
93	C₂³×C₁₀₁	—	4	1	no	—
94	C₂³×C₁₀₁	—	4	1	no	—
95	C₂³×C₁₀₁	—	4	2	no	—
96	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	D₄×C₁₀₁	—	2	2	no	—
98	D₄×C₁₀₁	—	2	4	no	—
99	D₄×C₁₀₁	C₄×C₈	1	4	yes	complemented
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	—	2	2	no	—
104	C₂×C₄×C₁₀₁	—	2	4	no	—
105	C₂×C₄×C₁₀₁	—	4	2	no	—
106	C₂×C₄×C₁₀₁	—	4	1	no	—
107	C₂×C₄×C₁₀₁	—	4	1	no	—
108	C₂×C₄×C₁₀₁	—	4	2	no	—
109	C₂×C₄×C₁₀₁	—	4	16	no	—
110	C₂×C₄×C₁₀₁	—	4	4	no	—
111	C₂×C₄×C₁₀₁	—	4	4	no	—
112	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₁₀₁	C₂²⋊C₈	1	4	yes	—
114	C₂×C₄×C₁₀₁	#102	1	4	yes	—
115	C₂×C₄×C₁₀₁	#104	1	4	yes	—
116	C₂×C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₀₁	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₀₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₀₁	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₁₀₁	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₁₀₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₁₀₁	#135	1	2	yes	—
125	C₈×C₁₀₁	—	2	8	no	—
126	C₈×C₁₀₁	—	4	4	no	—
127	C₂²×C₁₀₁	—	2	4	no	—
128	C₂²×C₁₀₁	—	4	4	no	—
129	C₂²×C₁₀₁	—	4	4	no	—
130	C₂²×C₁₀₁	—	4	2	no	—
131	C₂²×C₁₀₁	—	4	2	no	—
132	C₂²×C₁₀₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₁₀₁	C₂×C₂²⋊C₈	1	2	yes	—
134	C₂²×C₁₀₁	C₂×#102	1	1	yes	characteristic
135	C₂²×C₁₀₁	#416	1	1	yes	characteristic
136	C₂²×C₁₀₁	C₂×#104	1	2	yes	—
137	C₄×C₁₀₁	—	2	4	no	—
138	C₄×C₁₀₁	—	2	2	no	—
139	C₄×C₁₀₁	—	4	4	no	—
140	C₄×C₁₀₁	—	4	2	no	—
141	C₄×C₁₀₁	D₄⋊C₈	1	8	yes	direct product factor
142	C₄×C₁₀₁	C₂×C₄×C₈	1	1	yes	characteristic
143	C₄×C₁₀₁	C₄×M₁₆	1	1	yes	characteristic
144	C₄×C₁₀₁	#431	1	2	yes	—
145	C₄×C₁₀₁	C₄×D₈	1	2	yes	—
146	C₄×C₁₀₁	C₄×SD₁₆	1	2	yes	—
147	C₄×D₄⋊C₈	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂×C₁₀₁	—	4	4	no	—
149	C₂×C₁₀₁	C₂×D₄⋊C₈	1	2	yes	—
150	C₂×C₁₀₁	#1476	1	2	yes	—
151	C₂×C₁₀₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
152	C₂×C₁₀₁	C₄×#104	1	1	yes	characteristic
153	C₂×C₁₀₁	C₄×#102	1	1	yes	characteristic
154	C₄²×D₄	C₂×C₁₀₁	1	1	yes	characteristic
155	C₄×C₄⋊C₈	C₂×C₁₀₁	1	1	yes	characteristic, complemented
156	C₄²×C₈	C₂×C₁₀₁	1	1	yes	characteristic, complemented
157	C₂×D₄⋊C₈	C₂×C₁₀₁	1	2	yes	—
158	#1279	C₂×C₁₀₁	1	2	yes	complemented
159	C₁₀₁	C₄×D₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×C₄×D₄	C₄×C₁₀₁	1	2	yes	—
161	C₂×C₄×D₄	C₂²×C₁₀₁	1	1	yes	characteristic
162	C₂²×C₄²	—	2	1	no	—
163	C₂×C₄⋊C₈	C₂²×C₁₀₁	1	2	yes	—
164	C₂×C₄×C₈	—	2	2	no	—
165	C₂×C₄×C₈	—	2	1	no	—
166	C₂×C₄×C₈	C₂²×C₁₀₁	1	2	yes	—
167	C₄×C₄⋊C₄	C₄×C₁₀₁	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
169	C₄×C₂²⋊C₄	—	2	1	no	—
170	C₄×C₂²⋊C₄	—	2	2	no	—
171	C₄³	C₂²×C₁₀₁	1	1	yes	characteristic
172	#336	—	2	1	no	—
173	D₄⋊C₈	C₄×C₁₀₁	1	8	yes	direct product factor
174	C₂²×D₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	—	2	1	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₄×D₄	—	2	4	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	C₈×C₁₀₁	1	8	yes	complemented
182	C₄×D₄	C₂×C₄×C₁₀₁	1	4	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	—	2	1	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	—	4	2	no	—
190	C₂×C₄²	—	4	2	no	—
191	C₂×C₄²	C₂×C₄×C₁₀₁	1	2	yes	—
192	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
194	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	C₂×C₄×C₁₀₁	1	4	yes	complemented
198	C₄×C₈	—	2	8	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	4	2	no	—
201	C₄×C₈	C₂×C₄×C₁₀₁	1	4	yes	complemented
202	#95	—	2	2	no	—
203	#95	—	2	1	no	—
204	C₂⁴	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	C₄²×C₁₀₁	1	2	yes	—
208	C₂×D₄	C₂×C₈×C₁₀₁	1	4	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
219	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
223	C₂×C₈	—	2	8	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₄²×C₁₀₁	1	2	yes	—
231	C₄⋊C₄	C₂×C₈×C₁₀₁	1	4	yes	—
232	C₂²⋊C₄	—	2	8	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	4	4	no	—
237	C₄²	—	4	1	no	—
238	C₄²	—	4	1	no	—
239	C₄²	C₂×C₈×C₁₀₁	1	4	yes	—
240	C₄²	M₁₆×C₁₀₁	1	4	yes	—
241	C₄²	D₈×C₁₀₁	1	2	yes	—
242	C₄²	SD₁₆×C₁₀₁	1	2	yes	—
243	C₄²	C₂²×C₄×C₁₀₁	1	2	yes	—
244	C₄²	#41×C₁₀₁	1	2	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	4	1	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
251	D₄	—	2	2	no	—
252	D₄	—	2	4	no	—
253	D₄	C₄×C₈×C₁₀₁	1	4	yes	complemented
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	16	no	—
266	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	4	yes	—
267	C₂×C₄	C₂²⋊C₈×C₁₀₁	1	4	yes	—
268	C₂×C₄	#102×C₁₀₁	1	4	yes	—
269	C₂×C₄	#104×C₁₀₁	1	4	yes	—
270	C₂×C₄	C₂×C₄²×C₁₀₁	1	1	yes	characteristic
271	C₂×C₄	#117×C₁₀₁	1	1	yes	characteristic
272	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₁₀₁	1	2	yes	—
274	C₂×C₄	C₂²×C₈×C₁₀₁	1	2	yes	—
275	C₂×C₄	C₂×M₁₆×C₁₀₁	1	2	yes	—
276	C₂×C₄	C₂×D₈×C₁₀₁	1	1	yes	characteristic
277	C₂×C₄	C₂×SD₁₆×C₁₀₁	1	1	yes	characteristic
278	C₂×C₄	#135×C₁₀₁	1	2	yes	—
279	C₈	—	2	8	no	—
280	C₈	—	4	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	4	4	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₄×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
287	C₂²	C₂×C₂²⋊C₈×C₁₀₁	1	2	yes	—
288	C₂²	C₂×#102×C₁₀₁	1	1	yes	characteristic
289	C₂²	#416×C₁₀₁	1	1	yes	characteristic
290	C₂²	C₂×#104×C₁₀₁	1	2	yes	—
291	C₄	—	2	4	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	4	no	—
294	C₄	—	4	2	no	—
295	C₄	D₄⋊C₈×C₁₀₁	1	8	yes	direct product factor
296	C₄	C₂×C₄×C₈×C₁₀₁	1	1	yes	characteristic
297	C₄	C₄×M₁₆×C₁₀₁	1	1	yes	characteristic
298	C₄	#431×C₁₀₁	1	2	yes	—
299	C₄	C₄×D₈×C₁₀₁	1	2	yes	—
300	C₄	C₄×SD₁₆×C₁₀₁	1	2	yes	—
301	C₂	—	4	4	no	—
302	C₂	C₂×D₄⋊C₈×C₁₀₁	1	2	yes	—
303	C₂	C₁₀₁×#1476	1	2	yes	—
304	C₂	C₄×C₂²⋊C₈×C₁₀₁	1	1	yes	characteristic
305	C₂	C₄×#104×C₁₀₁	1	1	yes	characteristic
306	C₂	C₄×#102×C₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₁₀₁	D₄
commutator	C₄	C₂×C₄×C₈×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
socle	C₂³×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₀₁	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₀₁ ◁ C₄³×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊C₈×C₁₀₁	C₂	semidirect
3. C₄²×C₈×C₁₀₁	C₂	semidirect
5. C₁₀₁×#1279	C₂	semidirect
19. D₄⋊C₈×C₁₀₁	C₄	direct (N × H)
27. C₄×D₄×C₁₀₁	C₈	semidirect
43. C₄⋊C₈×C₁₀₁	C₂×C₄	semidirect
47. C₄×C₈×C₁₀₁	C₂×C₄	semidirect
99. D₄×C₁₀₁	C₄×C₈	semidirect
141. C₄×C₁₀₁	D₄⋊C₈	direct (N × H)
147. C₄×D₄⋊C₈	C₁₀₁	direct (N × H)
155. C₄×C₄⋊C₈	C₂×C₁₀₁	semidirect
156. C₄²×C₈	C₂×C₁₀₁	semidirect
158. #1279	C₂×C₁₀₁	semidirect
159. C₁₀₁	C₄×D₄⋊C₈	direct (N × H)
173. D₄⋊C₈	C₄×C₁₀₁	direct (N × H)
181. C₄×D₄	C₈×C₁₀₁	semidirect
197. C₄⋊C₈	C₂×C₄×C₁₀₁	semidirect
201. C₄×C₈	C₂×C₄×C₁₀₁	semidirect
253. D₄	C₄×C₈×C₁₀₁	semidirect
295. C₄	D₄⋊C₈×C₁₀₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₁₀₁	2
C₄×C₄⋊C₈×C₁₀₁	2
C₄²×C₈×C₁₀₁	2
C₂×D₄⋊C₈×C₁₀₁	2
C₁₀₁×#1279	2
C₄×D₄⋊C₈	101

Subgroups — C4×D4⋊C8×C101 #38790319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₄⋊C₈×C₁₀₁ #38790319