Subgroups — C₁₀₁×#17228 #38799379

Order 25856.

1010 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	C₂×#400×C₁₀₁	C₂	1	2	yes	maximal
2	C₁₀₁×#1445	C₂	1	2	yes	maximal
3	C₁₀₁×#1445	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#1444	C₂	1	4	yes	maximal
5	C₁₀₁×#1443	C₂	1	4	yes	maximal
6	C₁₀₁×#1442	C₂	1	2	yes	maximal
7	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
11	#400×C₁₀₁	—	2	8	no	—
12	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
13	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
14	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
15	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
16	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
17	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
18	C₂×#95×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×#95×C₁₀₁	C₂²	1	4	yes	—
20	C₂×#95×C₁₀₁	C₂²	1	2	yes	—
21	C₂×#95×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₁₀₁	C₂³	1	4	yes	—
24	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
27	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₁₀₁	—	4	1	no	—
31	C₂×C₄⋊C₄×C₁₀₁	—	4	1	no	—
32	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
33	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
34	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
35	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
36	C₂×C₄⋊C₄×C₁₀₁	—	4	4	no	—
37	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
38	C₂×C₄⋊C₄×C₁₀₁	D₄	1	4	yes	—
39	#95×C₁₀₁	—	2	4	no	—
40	#95×C₁₀₁	—	2	8	no	—
41	#95×C₁₀₁	—	2	8	no	—
42	#95×C₁₀₁	—	2	8	no	—
43	#95×C₁₀₁	—	2	4	no	—
44	#95×C₁₀₁	—	2	8	no	—
45	#95×C₁₀₁	—	4	1	no	—
46	#95×C₁₀₁	—	4	1	no	—
47	#95×C₁₀₁	—	4	2	no	—
48	#95×C₁₀₁	—	4	4	no	—
49	C₂⁴×C₁₀₁	C₂⁴	1	1	yes	characteristic
50	C₂²×C₄×C₁₀₁	—	2	4	no	—
51	C₂²×C₄×C₁₀₁	—	2	4	no	—
52	C₂²×C₄×C₁₀₁	—	2	1	no	—
53	C₂²×C₄×C₁₀₁	—	2	1	no	—
54	C₂²×C₄×C₁₀₁	—	2	2	no	—
55	C₂²×C₄×C₁₀₁	—	2	2	no	—
56	C₂²×C₄×C₁₀₁	—	2	2	no	—
57	C₂²×C₄×C₁₀₁	—	2	4	no	—
58	C₂²×C₄×C₁₀₁	—	2	2	no	—
59	C₂²×C₄×C₁₀₁	—	2	2	no	—
60	C₂²×C₄×C₁₀₁	—	2	4	no	—
61	C₂²×C₄×C₁₀₁	—	2	4	no	—
62	C₂²×C₄×C₁₀₁	—	2	2	no	—
63	C₂²×C₄×C₁₀₁	—	2	4	no	—
64	C₂²×C₄×C₁₀₁	—	2	4	no	—
65	C₂²×C₄×C₁₀₁	—	2	4	no	—
66	C₂²×C₄×C₁₀₁	—	2	4	no	—
67	C₂²×C₄×C₁₀₁	—	2	4	no	—
68	C₂²×C₄×C₁₀₁	—	2	4	no	—
69	C₂²×C₄×C₁₀₁	—	2	4	no	—
70	C₂²×C₄×C₁₀₁	—	2	4	no	—
71	C₂²×C₄×C₁₀₁	—	2	4	no	—
72	C₂²×C₄×C₁₀₁	—	2	4	no	—
73	C₂²×C₄×C₁₀₁	—	2	4	no	—
74	C₂²×C₄×C₁₀₁	—	2	4	no	—
75	C₂²×C₄×C₁₀₁	—	2	4	no	—
76	C₂²×C₄×C₁₀₁	—	2	4	no	—
77	C₂²×C₄×C₁₀₁	—	2	4	no	—
78	C₂²×C₄×C₁₀₁	—	2	4	no	—
79	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
82	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
83	C₄⋊C₄×C₁₀₁	—	4	4	no	—
84	C₄⋊C₄×C₁₀₁	—	4	8	no	—
85	C₄⋊C₄×C₁₀₁	—	4	8	no	—
86	C₄⋊C₄×C₁₀₁	—	4	8	no	—
87	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
88	C₂³×C₁₀₁	C₂×#41	1	2	yes	—
89	C₂³×C₁₀₁	C₂×#41	1	2	yes	—
90	C₂³×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
91	C₂³×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂³×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
93	C₂³×C₁₀₁	2₋¹⁺⁴	1	4	yes	—
94	C₂³×C₁₀₁	2₋¹⁺⁴	1	2	yes	—
95	C₂×C₄×C₁₀₁	—	2	4	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	4	no	—
98	C₂×C₄×C₁₀₁	—	2	4	no	—
99	C₂×C₄×C₁₀₁	—	2	8	no	—
100	C₂×C₄×C₁₀₁	—	2	4	no	—
101	C₂×C₄×C₁₀₁	—	2	8	no	—
102	C₂×C₄×C₁₀₁	—	2	8	no	—
103	C₂×C₄×C₁₀₁	—	4	4	no	—
104	C₂×C₄×C₁₀₁	—	4	4	no	—
105	C₂×C₄×C₁₀₁	—	4	2	no	—
106	C₂×C₄×C₁₀₁	—	4	2	no	—
107	C₂×C₄×C₁₀₁	—	4	2	no	—
108	C₂×C₄×C₁₀₁	—	4	2	no	—
109	C₂×C₄×C₁₀₁	—	4	2	no	—
110	C₂×C₄×C₁₀₁	—	4	2	no	—
111	C₂×C₄×C₁₀₁	—	4	4	no	—
112	C₂×C₄×C₁₀₁	—	4	2	no	—
113	C₂×C₄×C₁₀₁	—	4	2	no	—
114	C₂×C₄×C₁₀₁	—	4	4	no	—
115	C₂×C₄×C₁₀₁	—	4	4	no	—
116	C₂×C₄×C₁₀₁	—	4	4	no	—
117	C₂×C₄×C₁₀₁	—	4	4	no	—
118	C₂×C₄×C₁₀₁	—	4	4	no	—
119	C₂×C₄×C₁₀₁	—	4	4	no	—
120	C₂×C₄×C₁₀₁	—	4	4	no	—
121	C₂×C₄×C₁₀₁	—	4	4	no	—
122	C₂×C₄×C₁₀₁	—	4	4	no	—
123	C₂×C₄×C₁₀₁	—	4	4	no	—
124	C₂×C₄×C₁₀₁	—	4	4	no	—
125	C₂×C₄×C₁₀₁	—	4	4	no	—
126	C₂×C₄×C₁₀₁	—	4	4	no	—
127	C₂×C₄×C₁₀₁	—	4	4	no	—
128	C₂²×C₁₀₁	#534	1	1	yes	characteristic
129	C₂²×C₁₀₁	#536	1	2	yes	—
130	C₂²×C₁₀₁	#538	1	2	yes	—
131	C₂²×C₁₀₁	#539	1	4	yes	—
132	C₂²×C₁₀₁	#539	1	2	yes	—
133	C₂²×C₁₀₁	#541	1	4	yes	—
134	C₂²×C₁₀₁	#547	1	2	yes	—
135	C₂²×C₁₀₁	#550	1	2	yes	—
136	C₂²×C₁₀₁	#552	1	4	yes	—
137	C₂²×C₁₀₁	#555	1	1	yes	characteristic
138	C₂²×C₁₀₁	#559	1	1	yes	characteristic
139	C₂²×C₁₀₁	H(3,𝔽₄)	1	1	yes	characteristic
140	C₂²×C₁₀₁	#563	1	2	yes	—
141	C₂²×C₁₀₁	#563	1	2	yes	—
142	C₂²×C₁₀₁	#563	1	1	yes	characteristic
143	C₂²×C₁₀₁	#563	1	4	yes	—
144	C₄×C₁₀₁	—	4	2	no	—
145	C₄×C₁₀₁	—	4	4	no	—
146	C₄×C₁₀₁	—	4	4	no	—
147	C₄×C₁₀₁	—	4	4	no	—
148	C₄×C₁₀₁	—	8	4	no	—
149	C₄×C₁₀₁	—	8	4	no	—
150	#17228	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₁₀₁	#2721	1	4	yes	—
152	C₂×C₁₀₁	#2724	1	1	yes	characteristic
153	C₂×C₁₀₁	#2733	1	1	yes	characteristic
154	C₂×C₁₀₁	#2769	1	2	yes	—
155	C₂×C₁₀₁	#2771	1	2	yes	—
156	C₂×C₁₀₁	#2817	1	1	yes	characteristic
157	C₂×C₁₀₁	#2833	1	2	yes	—
158	C₂×C₁₀₁	#2835	1	2	yes	—
159	C₂×#400	C₂×C₁₀₁	1	2	yes	—
160	#1445	C₂×C₁₀₁	1	2	yes	—
161	#1445	C₂×C₁₀₁	1	1	yes	characteristic
162	#1444	C₂×C₁₀₁	1	4	yes	—
163	#1443	C₂×C₁₀₁	1	4	yes	—
164	#1442	C₂×C₁₀₁	1	2	yes	—
165	C₁₀₁	#17228	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
167	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
168	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	2	yes	—
169	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
170	#400	—	2	8	no	—
171	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
172	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
173	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
174	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
175	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
176	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
177	C₂×#95	C₂²×C₁₀₁	1	1	yes	characteristic
178	C₂×#95	C₂²×C₁₀₁	1	4	yes	—
179	C₂×#95	C₂²×C₁₀₁	1	2	yes	—
180	C₂×#95	C₂²×C₁₀₁	1	1	yes	characteristic
181	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
182	C₂³×C₄	C₂³×C₁₀₁	1	4	yes	—
183	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
184	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
186	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	4	1	no	—
190	C₂×C₄⋊C₄	—	4	1	no	—
191	C₂×C₄⋊C₄	—	4	2	no	—
192	C₂×C₄⋊C₄	—	4	2	no	—
193	C₂×C₄⋊C₄	—	4	2	no	—
194	C₂×C₄⋊C₄	—	4	2	no	—
195	C₂×C₄⋊C₄	—	4	4	no	—
196	C₂×C₄⋊C₄	—	4	2	no	—
197	C₂×C₄⋊C₄	D₄×C₁₀₁	1	4	yes	—
198	#95	—	2	4	no	—
199	#95	—	2	8	no	—
200	#95	—	2	8	no	—
201	#95	—	2	8	no	—
202	#95	—	2	4	no	—
203	#95	—	2	8	no	—
204	#95	—	4	1	no	—
205	#95	—	4	1	no	—
206	#95	—	4	2	no	—
207	#95	—	4	4	no	—
208	C₂⁴	C₂⁴×C₁₀₁	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
240	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
241	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	8	no	—
244	C₄⋊C₄	—	4	8	no	—
245	C₄⋊C₄	—	4	8	no	—
246	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
247	C₂³	C₂×#41×C₁₀₁	1	2	yes	—
248	C₂³	C₂×#41×C₁₀₁	1	2	yes	—
249	C₂³	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
250	C₂³	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
251	C₂³	H(5,𝔽₂)×C₁₀₁	1	2	yes	—
252	C₂³	2₋¹⁺⁴×C₁₀₁	1	4	yes	—
253	C₂³	2₋¹⁺⁴×C₁₀₁	1	2	yes	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂²	#534×C₁₀₁	1	1	yes	characteristic
288	C₂²	#536×C₁₀₁	1	2	yes	—
289	C₂²	#538×C₁₀₁	1	2	yes	—
290	C₂²	#539×C₁₀₁	1	4	yes	—
291	C₂²	#539×C₁₀₁	1	2	yes	—
292	C₂²	#541×C₁₀₁	1	4	yes	—
293	C₂²	#547×C₁₀₁	1	2	yes	—
294	C₂²	#550×C₁₀₁	1	2	yes	—
295	C₂²	#552×C₁₀₁	1	4	yes	—
296	C₂²	#555×C₁₀₁	1	1	yes	characteristic
297	C₂²	#559×C₁₀₁	1	1	yes	characteristic
298	C₂²	H(3,𝔽₄)×C₁₀₁	1	1	yes	characteristic
299	C₂²	#563×C₁₀₁	1	4	yes	—
300	C₂²	#563×C₁₀₁	1	1	yes	characteristic
301	C₂²	#563×C₁₀₁	1	2	yes	—
302	C₂²	#563×C₁₀₁	1	2	yes	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	4	no	—
307	C₄	—	8	4	no	—
308	C₄	—	8	4	no	—
309	C₂	C₁₀₁×#2721	1	4	yes	—
310	C₂	C₁₀₁×#2724	1	1	yes	characteristic
311	C₂	C₁₀₁×#2733	1	1	yes	characteristic
312	C₂	C₁₀₁×#2769	1	2	yes	—
313	C₂	C₁₀₁×#2771	1	2	yes	—
314	C₂	C₁₀₁×#2817	1	1	yes	characteristic
315	C₂	C₁₀₁×#2833	1	2	yes	—
316	C₂	C₁₀₁×#2835	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₁	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₀₁
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
150. #17228	C₁₀₁	direct (N × H)
165. C₁₀₁	#17228	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×#400×C₁₀₁	2
C₁₀₁×#1445	2
C₁₀₁×#1444	2
C₁₀₁×#1443	2
C₁₀₁×#1442	2
#17228	101

Subgroups — C101×#17228 #38799379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#17228 #38799379