Subgroups — #418×D₁₀₁ #27179630

Order 12928.

1008 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#117×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265183	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265182	C₂	1	1	yes	maximal, characteristic, complemented
6	#20265181	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265180	C₂	1	4	yes	maximal, complemented
8	#102×D₁₀₁	C₂	1	4	yes	maximal, complemented
9	#418×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₁₀₁	—	2	1	no	—
11	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₁₀₁	—	2	1	no	—
13	C₂×D₄×D₁₀₁	—	2	1	no	—
14	C₂×D₄×D₁₀₁	—	2	1	no	—
15	C₂×D₄×D₁₀₁	—	2	4	no	—
16	C₂×D₄×D₁₀₁	C₄	1	8	yes	complemented
17	C₂×D₄×D₁₀₁	C₂²	1	2	yes	—
18	C₂×D₄×D₁₀₁	C₂²	1	2	yes	—
19	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
20	C₂×#13108058	C₂²	1	1	yes	characteristic, complemented
21	C₂×#13108057	C₂²	1	2	yes	complemented
22	M₁₆×D₁₀₁	—	2	4	no	—
23	C₂×C₈×D₁₀₁	C₂²	1	2	yes	complemented
24	#15692214	C₂²	1	1	yes	characteristic, complemented
25	#15692213	C₂²	1	2	yes	complemented
26	#15692212	—	2	1	no	—
27	#15692211	—	2	1	no	—
28	C₄⋊C₄×D₁₀₁	C₂²	1	2	yes	complemented
29	C₂²⋊C₄×D₁₀₁	—	2	1	no	—
30	C₄²×D₁₀₁	—	2	1	no	—
31	#15692177	C₂²	1	4	yes	complemented
32	#15692176	C₂²	1	4	yes	complemented
33	#15692175	C₂²	1	4	yes	complemented
34	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
36	#117×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
37	#102×C₁₀₁	C₂²	1	4	yes	complemented
38	C₂³×D₁₀₁	—	2	1	no	—
39	C₂³×D₁₀₁	—	2	8	no	—
40	C₂³×D₁₀₁	—	2	2	no	—
41	C₂³×D₁₀₁	—	2	2	no	—
42	C₂³×D₁₀₁	—	4	8	no	—
43	C₂³×D₁₀₁	D₄	1	1	yes	characteristic
44	C₂×D₄₀₄	—	2	1	no	—
45	C₂×D₄₀₄	—	2	2	no	—
46	C₂×D₄₀₄	C₂×C₄	1	4	yes	complemented
47	C₂×D₄₀₄	C₂³	1	2	yes	—
48	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
49	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
50	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
51	D₄×D₁₀₁	—	2	8	no	—
52	D₄×D₁₀₁	—	2	8	no	—
53	D₄×D₁₀₁	—	2	4	no	—
54	D₄×D₁₀₁	—	2	4	no	—
55	D₄×D₁₀₁	C₂×C₄	1	8	yes	—
56	D₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
57	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
58	C₂×C₄×D₁₀₁	—	2	2	no	—
59	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	—
60	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
61	C₂×C₄×D₁₀₁	D₄	1	1	yes	characteristic
62	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
63	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
65	#13108058	—	2	2	no	—
66	#13108057	—	2	4	no	—
67	C₂×#11717579	C₂³	1	2	yes	complemented
68	C₈×D₁₀₁	—	2	4	no	—
69	C₄⋊Dic₁₀₁	C₂³	1	2	yes	complemented
70	#13108053	—	2	2	no	—
71	C₂²⋊Dic₁₀₁	—	2	1	no	—
72	#13108051	—	2	2	no	—
73	C₄×Dic₁₀₁	—	2	1	no	—
74	C₄×Dic₁₀₁	—	2	2	no	—
75	C₂⁴×C₁₀₁	—	2	1	no	—
76	C₂×D₄×C₁₀₁	—	2	1	no	—
77	C₂×D₄×C₁₀₁	—	2	2	no	—
78	C₂×D₄×C₁₀₁	C₂×C₄	1	4	yes	complemented
79	C₂×D₄×C₁₀₁	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
81	M₁₆×C₁₀₁	—	2	2	no	—
82	C₂×C₈×C₁₀₁	C₂³	1	2	yes	complemented
83	C₄⋊C₄×C₁₀₁	C₂³	1	2	yes	complemented
84	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
85	C₄²×C₁₀₁	—	2	1	no	—
86	C₂³×C₁₀₁	—	2	4	no	—
87	C₂³×C₁₀₁	—	2	2	no	—
88	C₂³×C₁₀₁	—	4	4	no	—
89	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₀₁	—	2	4	no	—
91	C₂²×D₁₀₁	—	2	4	no	—
92	C₂²×D₁₀₁	—	2	2	no	—
93	C₂²×D₁₀₁	—	2	8	no	—
94	C₂²×D₁₀₁	—	2	8	no	—
95	C₂²×D₁₀₁	—	4	8	no	—
96	C₂²×D₁₀₁	—	4	32	no	—
97	C₂²×D₁₀₁	—	4	4	no	—
98	C₂²×D₁₀₁	—	4	8	no	—
99	C₂²×D₁₀₁	C₂²⋊C₄	1	4	yes	—
100	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
102	D₄×C₁₀₁	—	2	4	no	—
103	D₄×C₁₀₁	—	2	2	no	—
104	D₄×C₁₀₁	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₁₀₁	—	2	2	no	—
106	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₁₀₁	—	2	8	no	—
111	C₂²⋊D₁₀₁	—	2	8	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	C₂²⋊C₄	1	4	yes	—
114	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
116	D₄₀₄	—	2	4	no	—
117	D₄₀₄	—	2	2	no	—
118	D₄₀₄	C₂²×C₄	1	4	yes	—
119	C₄×D₁₀₁	—	4	4	no	—
120	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
121	C₄×D₁₀₁	C₂²⋊C₄	1	4	yes	—
122	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
123	C₄×D₁₀₁	C₂²×C₄	1	2	yes	—
124	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
125	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
126	C₈×C₁₀₁	—	2	2	no	—
127	#11717579	—	2	2	no	—
128	C₂²×C₁₀₁	—	2	2	no	—
129	C₂²×C₁₀₁	—	2	4	no	—
130	C₂²×C₁₀₁	—	4	8	no	—
131	C₂²×C₁₀₁	—	4	4	no	—
132	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×D₁₀₁	—	2	2	no	—
135	C₂×D₁₀₁	—	2	8	no	—
136	C₂×D₁₀₁	—	2	4	no	—
137	C₂×D₁₀₁	—	4	16	no	—
138	C₂×D₁₀₁	—	4	8	no	—
139	C₂×D₁₀₁	—	4	4	no	—
140	C₂×D₁₀₁	—	4	16	no	—
141	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
143	C₂×D₁₀₁	AGL(1,ℤ/8)	1	4	yes	—
144	C₄×C₁₀₁	—	4	2	no	—
145	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
146	C₄×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
147	C₄×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
148	Dic₁₀₁	—	4	2	no	—
149	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
150	Dic₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₁₀₁	—	2	2	no	—
152	C₂×C₁₀₁	—	4	4	no	—
153	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₁₀₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
155	D₁₀₁	—	2	2	no	—
156	D₁₀₁	—	4	4	no	—
157	D₁₀₁	#418	1	4	yes	direct product factor
158	C₂×#418	—	101	1	no	maximal, Sylow
159	C₁₀₁	C₂×#418	1	1	yes	characteristic, complemented, Sylow
160	C₂³×D₄	—	101	1	no	—
161	C₂²×M₁₆	—	101	1	no	—
162	C₂×#117	—	101	1	no	—
163	#418	—	101	1	no	—
164	#418	—	101	1	no	—
165	#418	—	101	1	no	—
166	#418	—	101	4	no	—
167	#418	D₁₀₁	1	1	yes	characteristic, direct product factor
168	C₂×#102	—	101	4	no	—
169	C₂⁵	—	202	1	no	—
170	C₂²×D₄	—	101	8	no	—
171	C₂²×D₄	—	101	2	no	—
172	C₂²×D₄	—	101	2	no	—
173	C₂²×D₄	—	101	1	no	—
174	C₂²×D₄	—	202	1	no	—
175	C₂²×D₄	—	202	1	no	—
176	C₂²×D₄	—	202	4	no	—
177	C₂²×D₄	—	202	1	no	—
178	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
179	C₂³×C₄	—	101	1	no	—
180	C₂×M₁₆	—	101	1	no	—
181	C₂×M₁₆	—	101	2	no	—
182	C₂×M₁₆	—	202	4	no	—
183	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic, complemented
184	C₂²×C₈	—	101	2	no	—
185	#117	—	101	1	no	—
186	#117	—	101	2	no	—
187	#117	—	202	1	no	—
188	#117	—	202	1	no	—
189	#117	C₂×D₁₀₁	1	1	yes	characteristic, complemented
190	C₂×C₄⋊C₄	—	101	2	no	—
191	C₂×C₂²⋊C₄	—	202	1	no	—
192	C₂×C₄²	—	202	1	no	—
193	#102	—	101	4	no	—
194	#102	—	101	4	no	—
195	#102	—	101	4	no	—
196	#102	C₂×D₁₀₁	1	4	yes	complemented
197	C₂⁴	—	2	1	no	—
198	C₂⁴	—	101	1	no	—
199	C₂⁴	—	404	8	no	—
200	C₂⁴	—	202	2	no	—
201	C₂⁴	—	202	2	no	—
202	C₂⁴	—	202	8	no	—
203	C₂⁴	—	202	1	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	101	8	no	—
207	C₂×D₄	—	101	8	no	—
208	C₂×D₄	—	101	4	no	—
209	C₂×D₄	—	101	2	no	—
210	C₂×D₄	—	202	4	no	—
211	C₂×D₄	—	202	4	no	—
212	C₂×D₄	—	202	8	no	—
213	C₂×D₄	—	202	8	no	—
214	C₂×D₄	—	202	2	no	—
215	C₂×D₄	—	202	2	no	—
216	C₂×D₄	—	202	8	no	—
217	C₂×D₄	—	202	2	no	—
218	C₂×D₄	—	202	1	no	—
219	C₂×D₄	C₄×D₁₀₁	1	4	yes	complemented
220	C₂×D₄	C₂²×D₁₀₁	1	2	yes	—
221	C₂²×C₄	—	101	1	no	—
222	C₂²×C₄	—	101	4	no	—
223	C₂²×C₄	—	101	1	no	—
224	C₂²×C₄	—	101	1	no	—
225	C₂²×C₄	—	101	4	no	—
226	C₂²×C₄	—	101	1	no	—
227	C₂²×C₄	—	101	1	no	—
228	C₂²×C₄	—	202	2	no	—
229	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
230	M₁₆	—	2	2	no	—
231	M₁₆	—	202	4	no	—
232	M₁₆	—	202	2	no	—
233	C₂×C₈	—	101	2	no	—
234	C₂×C₈	—	202	4	no	—
235	C₂×C₈	C₂²×D₁₀₁	1	2	yes	complemented
236	C₄⋊C₄	—	101	2	no	—
237	C₄⋊C₄	—	202	2	no	—
238	C₄⋊C₄	C₂²×D₁₀₁	1	2	yes	complemented
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	202	2	no	—
241	C₂²⋊C₄	—	202	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	202	2	no	—
244	C₄²	—	202	1	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	101	1	no	—
249	C₂³	—	101	1	no	—
250	C₂³	—	101	4	no	—
251	C₂³	—	404	4	no	—
252	C₂³	—	404	8	no	—
253	C₂³	—	404	32	no	—
254	C₂³	—	404	8	no	—
255	C₂³	—	202	8	no	—
256	C₂³	—	202	8	no	—
257	C₂³	—	202	4	no	—
258	C₂³	—	202	2	no	—
259	C₂³	—	202	4	no	—
260	C₂³	D₄×D₁₀₁	1	1	yes	characteristic
261	D₄	—	2	2	no	—
262	D₄	—	2	4	no	—
263	D₄	—	101	4	no	—
264	D₄	—	202	8	no	—
265	D₄	—	202	8	no	—
266	D₄	—	202	2	no	—
267	D₄	—	202	4	no	—
268	D₄	C₂×C₄×D₁₀₁	1	4	yes	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	101	1	no	—
271	C₂×C₄	—	101	1	no	—
272	C₂×C₄	—	101	4	no	—
273	C₂×C₄	—	101	2	no	—
274	C₂×C₄	—	101	2	no	—
275	C₂×C₄	—	101	2	no	—
276	C₂×C₄	—	101	2	no	—
277	C₂×C₄	—	101	4	no	—
278	C₂×C₄	—	101	4	no	—
279	C₂×C₄	—	404	4	no	—
280	C₂×C₄	—	202	2	no	—
281	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
282	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
283	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
284	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
285	C₈	—	2	2	no	—
286	C₈	—	202	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	8	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	101	2	no	—
292	C₂²	—	101	4	no	—
293	C₂²	—	101	2	no	—
294	C₂²	—	404	8	no	—
295	C₂²	—	404	16	no	—
296	C₂²	—	404	16	no	—
297	C₂²	—	404	4	no	—
298	C₂²	—	202	8	no	—
299	C₂²	—	202	4	no	—
300	C₂²	—	202	2	no	—
301	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
302	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
303	C₄	—	4	2	no	—
304	C₄	—	101	2	no	—
305	C₄	—	101	2	no	—
306	C₄	—	404	2	no	—
307	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
308	C₄	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
309	C₄	C₂×D₄×D₁₀₁	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	4	4	no	—
312	C₂	—	101	4	no	—
313	C₂	—	404	4	no	—
314	C₂	—	202	2	no	—
315	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
316	C₂	AGL(1,ℤ/8)×D₁₀₁	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#418×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂²×D₄×D₁₀₁	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#418	D₁₀₁
hypocenter	C₁₀₁	C₂×#418
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×D₁₀₁	C₂	semidirect
3. #117×D₁₀₁	C₂	semidirect
4. #20265183	C₂	semidirect
5. #20265182	C₂	semidirect
6. #20265181	C₂	semidirect
7. #20265180	C₂	semidirect
8. #102×D₁₀₁	C₂	semidirect
9. #418×C₁₀₁	C₂	semidirect
16. C₂×D₄×D₁₀₁	C₄	semidirect
20. C₂×#13108058	C₂²	semidirect
21. C₂×#13108057	C₂²	semidirect
23. C₂×C₈×D₁₀₁	C₂²	semidirect
24. #15692214	C₂²	semidirect
25. #15692213	C₂²	semidirect
28. C₄⋊C₄×D₁₀₁	C₂²	semidirect
31. #15692177	C₂²	semidirect
32. #15692176	C₂²	semidirect
33. #15692175	C₂²	semidirect
35. C₂×M₁₆×C₁₀₁	C₂²	semidirect
36. #117×C₁₀₁	C₂²	semidirect
37. #102×C₁₀₁	C₂²	semidirect
46. C₂×D₄₀₄	C₂×C₄	semidirect
56. D₄×D₁₀₁	C₂×C₄	semidirect
67. C₂×#11717579	C₂³	semidirect
69. C₄⋊Dic₁₀₁	C₂³	semidirect
78. C₂×D₄×C₁₀₁	C₂×C₄	semidirect
82. C₂×C₈×C₁₀₁	C₂³	semidirect
83. C₄⋊C₄×C₁₀₁	C₂³	semidirect
157. D₁₀₁	#418	direct (N × H)
159. C₁₀₁	C₂×#418	semidirect
167. #418	D₁₀₁	direct (N × H)
183. C₂×M₁₆	C₂×D₁₀₁	semidirect
189. #117	C₂×D₁₀₁	semidirect
196. #102	C₂×D₁₀₁	semidirect
219. C₂×D₄	C₄×D₁₀₁	semidirect
235. C₂×C₈	C₂²×D₁₀₁	semidirect
238. C₄⋊C₄	C₂²×D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×D₁₀₁	2
C₂×M₁₆×D₁₀₁	2
#117×D₁₀₁	2
#20265183	2
#20265182	2
#20265181	2
#20265180	2
#102×D₁₀₁	2
#418×C₁₀₁	2
C₂×#418	101

Subgroups — #418×D101 #27179630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #418×D₁₀₁ #27179630