Subgroups — C₁₀₁×#21334 #38803485

Order 25856.

1142 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#2316	C₂	1	1	yes	maximal, characteristic
3	C₁₀₁×#2274	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#1996	C₂	1	8	yes	maximal
5	C₁₀₁×#1740	C₂	1	4	yes	maximal
6	#544×C₁₀₁	—	2	4	no	—
7	#544×C₁₀₁	—	2	4	no	—
8	C₂×C₄⋊Q₈×C₁₀₁	C₄	1	4	yes	complemented
9	C₂×#125×C₁₀₁	C₄	1	4	yes	complemented
10	C₂×C₂²⋊Q₈×C₁₀₁	C₂²	1	4	yes	—
11	C₂×#117×C₁₀₁	C₂²	1	2	yes	—
12	C₂×#117×C₁₀₁	C₂²	1	2	yes	—
13	C₂×#117×C₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
15	#389×C₁₀₁	—	2	4	no	—
16	#380×C₁₀₁	C₂²	1	4	yes	—
17	C₄×C₄⋊C₄×C₁₀₁	—	2	4	no	—
18	C₄×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	—
19	#342×C₁₀₁	C₂²	1	16	yes	—
20	C₂²×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
22	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₁₀₁	—	2	16	no	—
24	#125×C₁₀₁	—	2	16	no	—
25	C₂²⋊Q₈×C₁₀₁	—	2	8	no	—
26	C₂²⋊Q₈×C₁₀₁	—	2	8	no	—
27	#117×C₁₀₁	—	2	4	no	—
28	#117×C₁₀₁	—	2	4	no	—
29	#117×C₁₀₁	—	2	4	no	—
30	#117×C₁₀₁	—	2	8	no	—
31	#117×C₁₀₁	Q₈	1	8	yes	complemented
32	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₀₁	D₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₀₁	D₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₀₁	—	2	2	no	—
46	C₂×C₄²×C₁₀₁	—	2	2	no	—
47	C₂×C₄²×C₁₀₁	—	2	4	no	—
48	C₂×C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
49	#95×C₁₀₁	—	2	4	no	—
50	#95×C₁₀₁	—	2	4	no	—
51	C₂⁴×C₁₀₁	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₁₀₁	—	2	8	no	—
53	C₂×Q₈×C₁₀₁	—	8	4	no	—
54	C₂²×C₄×C₁₀₁	—	2	4	no	—
55	C₂²×C₄×C₁₀₁	—	2	4	no	—
56	C₂²×C₄×C₁₀₁	—	8	2	no	—
57	C₂²×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
62	C₂²×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₁₀₁	#41	1	4	yes	—
67	C₄⋊C₄×C₁₀₁	—	2	4	no	—
68	C₄⋊C₄×C₁₀₁	—	2	4	no	—
69	C₄⋊C₄×C₁₀₁	—	2	8	no	—
70	C₄⋊C₄×C₁₀₁	—	2	16	no	—
71	C₄⋊C₄×C₁₀₁	—	2	8	no	—
72	C₄⋊C₄×C₁₀₁	—	4	4	no	—
73	C₄⋊C₄×C₁₀₁	—	8	4	no	—
74	C₄⋊C₄×C₁₀₁	C₂²⋊C₄	1	16	yes	—
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	8	no	—
79	C₂²⋊C₄×C₁₀₁	C₂×Q₈	1	8	yes	—
80	C₄²×C₁₀₁	—	2	8	no	—
81	C₄²×C₁₀₁	—	2	8	no	—
82	C₄²×C₁₀₁	—	4	4	no	—
83	C₄²×C₁₀₁	—	8	4	no	—
84	C₂³×C₁₀₁	—	4	1	no	—
85	C₂³×C₁₀₁	—	4	1	no	—
86	C₂³×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
87	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
88	C₂³×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
89	C₂³×C₁₀₁	C₂²×Q₈	1	2	yes	—
90	C₂³×C₁₀₁	C₂×#41	1	2	yes	—
91	Q₈×C₁₀₁	—	8	8	no	—
92	C₂×C₄×C₁₀₁	—	2	4	no	—
93	C₂×C₄×C₁₀₁	—	2	4	no	—
94	C₂×C₄×C₁₀₁	—	2	8	no	—
95	C₂×C₄×C₁₀₁	—	4	4	no	—
96	C₂×C₄×C₁₀₁	—	4	4	no	—
97	C₂×C₄×C₁₀₁	—	4	4	no	—
98	C₂×C₄×C₁₀₁	—	4	4	no	—
99	C₂×C₄×C₁₀₁	—	4	4	no	—
100	C₂×C₄×C₁₀₁	—	4	4	no	—
101	C₂×C₄×C₁₀₁	—	4	4	no	—
102	C₂×C₄×C₁₀₁	—	4	4	no	—
103	C₂×C₄×C₁₀₁	—	4	4	no	—
104	C₂×C₄×C₁₀₁	—	4	4	no	—
105	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	8	yes	—
106	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
107	C₂×C₄×C₁₀₁	C₄×Q₈	1	8	yes	—
108	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	4	yes	—
110	C₂×C₄×C₁₀₁	#124	1	4	yes	—
111	C₂×C₄×C₁₀₁	C₄⋊Q₈	1	4	yes	—
112	C₂²×C₁₀₁	—	2	1	no	—
113	C₂²×C₁₀₁	—	2	1	no	—
114	C₂²×C₁₀₁	—	2	1	no	—
115	C₂²×C₁₀₁	—	2	1	no	—
116	C₂²×C₁₀₁	—	2	1	no	—
117	C₂²×C₁₀₁	—	2	1	no	—
118	C₂²×C₁₀₁	—	4	2	no	—
119	C₂²×C₁₀₁	—	4	2	no	—
120	C₂²×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
122	C₂²×C₁₀₁	C₂×C₂²⋊Q₈	1	2	yes	—
123	C₂²×C₁₀₁	C₂×#124	1	1	yes	characteristic
124	C₂²×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₄×C₁₀₁	—	2	8	no	—
126	C₄×C₁₀₁	—	4	8	no	—
127	C₄×C₁₀₁	—	4	4	no	—
128	C₄×C₁₀₁	—	4	4	no	—
129	C₄×C₁₀₁	—	8	4	no	—
130	C₄×C₁₀₁	#391	1	8	yes	—
131	#21334	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₁₀₁	—	2	2	no	—
133	C₂×C₁₀₁	—	2	1	no	—
134	C₂×C₁₀₁	—	2	2	no	—
135	C₂×C₁₀₁	—	2	1	no	—
136	C₂×C₁₀₁	C₂×#391	1	1	yes	characteristic
137	C₂×C₁₀₁	#2883	1	1	yes	characteristic
138	C₂×C₁₀₁	#2884	1	1	yes	characteristic
139	C₂×#544	C₂×C₁₀₁	1	1	yes	characteristic
140	#2316	C₂×C₁₀₁	1	1	yes	characteristic
141	#2274	C₂×C₁₀₁	1	1	yes	characteristic
142	#1996	C₂×C₁₀₁	1	8	yes	—
143	#1740	C₂×C₁₀₁	1	4	yes	—
144	C₁₀₁	#21334	1	1	yes	characteristic, direct product factor, Sylow
145	#544	—	2	4	no	—
146	#544	—	2	4	no	—
147	C₂×C₄⋊Q₈	C₄×C₁₀₁	1	4	yes	complemented
148	C₂×#125	C₄×C₁₀₁	1	4	yes	complemented
149	C₂×C₂²⋊Q₈	C₂²×C₁₀₁	1	4	yes	—
150	C₂×#117	C₂²×C₁₀₁	1	2	yes	—
151	C₂×#117	C₂²×C₁₀₁	1	2	yes	—
152	C₂×#117	C₂²×C₁₀₁	1	2	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	1	yes	characteristic
154	#389	—	2	4	no	—
155	#380	C₂²×C₁₀₁	1	4	yes	—
156	C₄×C₄⋊C₄	—	2	4	no	—
157	C₄×C₂²⋊C₄	C₂²×C₁₀₁	1	4	yes	—
158	#342	C₂²×C₁₀₁	1	16	yes	—
159	C₂²×Q₈	C₂×C₄×C₁₀₁	1	4	yes	—
160	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
161	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	16	no	—
163	#125	—	2	16	no	—
164	C₂²⋊Q₈	—	2	8	no	—
165	C₂²⋊Q₈	—	2	8	no	—
166	#117	—	2	4	no	—
167	#117	—	2	4	no	—
168	#117	—	2	4	no	—
169	#117	—	2	8	no	—
170	#117	Q₈×C₁₀₁	1	8	yes	complemented
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	2	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	8	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
179	C₂×C₄⋊C₄	D₄×C₁₀₁	1	4	yes	—
180	C₂×C₄⋊C₄	D₄×C₁₀₁	1	4	yes	—
181	C₂×C₂²⋊C₄	C₂³×C₁₀₁	1	4	yes	—
182	C₂×C₂²⋊C₄	C₂³×C₁₀₁	1	4	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₁₀₁	1	4	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	C₂×C₄×C₁₀₁	1	4	yes	—
188	#95	—	2	4	no	—
189	#95	—	2	4	no	—
190	C₂⁴	C₂⁴×C₁₀₁	1	1	yes	characteristic
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	—	8	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	8	2	no	—
196	C₂²×C₄	C₂²×C₄×C₁₀₁	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₀₁	1	4	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₀₁	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₀₁	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
201	C₂²×C₄	C₂×D₄×C₁₀₁	1	4	yes	—
202	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
204	C₂²×C₄	C₂×Q₈×C₁₀₁	1	4	yes	—
205	C₂²×C₄	#41×C₁₀₁	1	4	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	2	16	no	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	8	4	no	—
213	C₄⋊C₄	C₂²⋊C₄×C₁₀₁	1	16	yes	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₂²⋊C₄	C₂×Q₈×C₁₀₁	1	8	yes	—
219	C₄²	—	2	8	no	—
220	C₄²	—	2	8	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	8	4	no	—
223	C₂³	—	4	1	no	—
224	C₂³	—	4	1	no	—
225	C₂³	C₂³×C₄×C₁₀₁	1	1	yes	characteristic
226	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
227	C₂³	C₂²×D₄×C₁₀₁	1	1	yes	characteristic
228	C₂³	C₂²×Q₈×C₁₀₁	1	2	yes	—
229	C₂³	C₂×#41×C₁₀₁	1	2	yes	—
230	Q₈	—	8	8	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	8	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	4	yes	—
246	C₂×C₄	C₄×Q₈×C₁₀₁	1	8	yes	—
247	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	4	yes	—
248	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	4	yes	—
249	C₂×C₄	#124×C₁₀₁	1	4	yes	—
250	C₂×C₄	C₄⋊Q₈×C₁₀₁	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	C₂²×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
260	C₂²	C₂×C₄×Q₈×C₁₀₁	1	2	yes	—
261	C₂²	C₂×C₂²⋊Q₈×C₁₀₁	1	2	yes	—
262	C₂²	C₂×#124×C₁₀₁	1	1	yes	characteristic
263	C₂²	C₂×C₄⋊Q₈×C₁₀₁	1	1	yes	characteristic
264	C₄	—	2	8	no	—
265	C₄	—	4	8	no	—
266	C₄	—	4	4	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	4	no	—
269	C₄	#391×C₁₀₁	1	8	yes	—
270	C₂	—	2	2	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	2	no	—
273	C₂	—	2	1	no	—
274	C₂	C₂×#391×C₁₀₁	1	1	yes	characteristic
275	C₂	C₁₀₁×#2883	1	1	yes	characteristic
276	C₂	C₁₀₁×#2884	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
8. C₂×C₄⋊Q₈×C₁₀₁	C₄	semidirect
9. C₂×#125×C₁₀₁	C₄	semidirect
31. #117×C₁₀₁	Q₈	semidirect
131. #21334	C₁₀₁	direct (N × H)
144. C₁₀₁	#21334	direct (N × H)
147. C₂×C₄⋊Q₈	C₄×C₁₀₁	semidirect
148. C₂×#125	C₄×C₁₀₁	semidirect
170. #117	Q₈×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#544×C₁₀₁	2
C₁₀₁×#2316	2
C₁₀₁×#2274	2
C₁₀₁×#1996	2
C₁₀₁×#1740	2
#21334	101

Subgroups — C101×#21334 #38803485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#21334 #38803485