Subgroups — C₁₀₁×#15189 #38797340

Order 25856.

1474 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#391×C₁₀₁	C₂	1	3	yes	maximal
2	C₂×#389×C₁₀₁	C₂	1	3	yes	maximal
3	C₁₀₁×#1443	C₂	1	2	yes	maximal
4	C₁₀₁×#1441	C₂	1	6	yes	maximal
5	C₄×#95×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	6	yes	—
8	C₂²×C₄⋊C₄×C₁₀₁	C₂²	1	3	yes	—
9	C₂²×C₄²×C₁₀₁	C₂²	1	3	yes	—
10	#391×C₁₀₁	—	2	12	no	—
11	#389×C₁₀₁	—	2	12	no	—
12	C₂×#95×C₁₀₁	C₂²	1	3	yes	—
13	C₂×#95×C₁₀₁	C₂²	1	6	yes	—
14	C₂×#95×C₁₀₁	C₂²	1	6	yes	—
15	C₂×#95×C₁₀₁	C₂²	1	6	yes	—
16	C₂×#95×C₁₀₁	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₁₀₁	—	2	6	no	—
18	C₂²×Q₈×C₁₀₁	—	2	6	no	—
19	C₂²×Q₈×C₁₀₁	Q₈	1	4	yes	—
20	C₂³×C₄×C₁₀₁	C₂³	1	6	yes	—
21	C₂³×C₄×C₁₀₁	C₂³	1	3	yes	—
22	C₂³×C₄×C₁₀₁	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₀₁	C₂³	1	3	yes	—
24	C₂³×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₁₀₁	—	2	6	no	—
26	C₂×C₄⋊C₄×C₁₀₁	—	2	6	no	—
27	C₂×C₄⋊C₄×C₁₀₁	—	2	12	no	—
28	C₂×C₄⋊C₄×C₁₀₁	—	2	12	no	—
29	C₂×C₄⋊C₄×C₁₀₁	—	4	6	no	—
30	C₂×C₄⋊C₄×C₁₀₁	D₄	1	12	yes	—
31	C₂×C₄²×C₁₀₁	—	2	6	no	—
32	C₂×C₄²×C₁₀₁	—	2	6	no	—
33	C₂×C₄²×C₁₀₁	—	2	6	no	—
34	#95×C₁₀₁	—	2	12	no	—
35	#95×C₁₀₁	—	2	12	no	—
36	#95×C₁₀₁	—	2	12	no	—
37	#95×C₁₀₁	—	2	6	no	—
38	#95×C₁₀₁	Q₈	1	4	yes	complemented
39	C₂⁴×C₁₀₁	C₂⁴	1	1	yes	characteristic
40	C₂×Q₈×C₁₀₁	—	2	24	no	—
41	C₂×Q₈×C₁₀₁	—	4	12	no	—
42	C₂×Q₈×C₁₀₁	—	4	4	no	—
43	C₂²×C₄×C₁₀₁	—	2	3	no	—
44	C₂²×C₄×C₁₀₁	—	2	3	no	—
45	C₂²×C₄×C₁₀₁	—	2	3	no	—
46	C₂²×C₄×C₁₀₁	—	2	6	no	—
47	C₂²×C₄×C₁₀₁	—	2	6	no	—
48	C₂²×C₄×C₁₀₁	—	2	3	no	—
49	C₂²×C₄×C₁₀₁	—	2	3	no	—
50	C₂²×C₄×C₁₀₁	—	2	3	no	—
51	C₂²×C₄×C₁₀₁	—	2	3	no	—
52	C₂²×C₄×C₁₀₁	—	2	3	no	—
53	C₂²×C₄×C₁₀₁	—	2	6	no	—
54	C₂²×C₄×C₁₀₁	—	2	6	no	—
55	C₂²×C₄×C₁₀₁	—	2	6	no	—
56	C₂²×C₄×C₁₀₁	—	2	6	no	—
57	C₂²×C₄×C₁₀₁	—	2	3	no	—
58	C₂²×C₄×C₁₀₁	—	2	6	no	—
59	C₂²×C₄×C₁₀₁	—	2	6	no	—
60	C₂²×C₄×C₁₀₁	—	2	3	no	—
61	C₂²×C₄×C₁₀₁	—	2	3	no	—
62	C₂²×C₄×C₁₀₁	—	2	3	no	—
63	C₂²×C₄×C₁₀₁	C₂×D₄	1	12	yes	—
64	C₂²×C₄×C₁₀₁	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₁₀₁	C₂×Q₈	1	6	yes	—
66	C₂²×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₀₁	#41	1	6	yes	—
69	C₂²×C₄×C₁₀₁	#41	1	6	yes	—
70	C₄⋊C₄×C₁₀₁	—	2	24	no	—
71	C₄⋊C₄×C₁₀₁	—	4	24	no	—
72	C₄²×C₁₀₁	—	4	12	no	—
73	C₂³×C₁₀₁	C₂²×D₄	1	3	yes	—
74	C₂³×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
75	C₂³×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
76	C₂³×C₁₀₁	C₂×#41	1	3	yes	—
77	C₂³×C₁₀₁	C₂×#41	1	3	yes	—
78	C₂³×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
79	C₂³×C₁₀₁	2₋¹⁺⁴	1	3	yes	—
80	Q₈×C₁₀₁	—	4	16	no	—
81	C₂×C₄×C₁₀₁	—	2	12	no	—
82	C₂×C₄×C₁₀₁	—	2	12	no	—
83	C₂×C₄×C₁₀₁	—	2	6	no	—
84	C₂×C₄×C₁₀₁	—	2	6	no	—
85	C₂×C₄×C₁₀₁	—	2	6	no	—
86	C₂×C₄×C₁₀₁	—	2	12	no	—
87	C₂×C₄×C₁₀₁	—	2	12	no	—
88	C₂×C₄×C₁₀₁	—	2	6	no	—
89	C₂×C₄×C₁₀₁	—	2	6	no	—
90	C₂×C₄×C₁₀₁	—	2	6	no	—
91	C₂×C₄×C₁₀₁	—	4	3	no	—
92	C₂×C₄×C₁₀₁	—	4	3	no	—
93	C₂×C₄×C₁₀₁	—	4	6	no	—
94	C₂×C₄×C₁₀₁	—	4	2	no	—
95	C₂×C₄×C₁₀₁	—	4	6	no	—
96	C₂×C₄×C₁₀₁	—	4	3	no	—
97	C₂×C₄×C₁₀₁	—	4	3	no	—
98	C₂×C₄×C₁₀₁	—	4	6	no	—
99	C₂×C₄×C₁₀₁	—	4	6	no	—
100	C₂×C₄×C₁₀₁	—	4	3	no	—
101	C₂×C₄×C₁₀₁	—	4	3	no	—
102	C₂×C₄×C₁₀₁	—	4	3	no	—
103	C₂×C₄×C₁₀₁	—	4	6	no	—
104	C₂×C₄×C₁₀₁	—	4	3	no	—
105	C₂×C₄×C₁₀₁	#121	1	12	yes	—
106	C₂×C₄×C₁₀₁	C₂²⋊Q₈	1	12	yes	—
107	C₂×C₄×C₁₀₁	#126	1	4	yes	—
108	C₂²×C₁₀₁	C₂×#121	1	3	yes	—
109	C₂²×C₁₀₁	C₂×C₂²⋊Q₈	1	3	yes	—
110	C₂²×C₁₀₁	C₂×#126	1	1	yes	characteristic
111	C₂²×C₁₀₁	#529	1	3	yes	—
112	C₂²×C₁₀₁	#533	1	3	yes	—
113	C₂²×C₁₀₁	#541	1	3	yes	—
114	C₂²×C₁₀₁	#542	1	3	yes	—
115	C₂²×C₁₀₁	#546	1	3	yes	—
116	C₂²×C₁₀₁	#547	1	3	yes	—
117	C₂²×C₁₀₁	#548	1	3	yes	—
118	C₂²×C₁₀₁	D₄×Q₈	1	3	yes	—
119	C₂²×C₁₀₁	#557	1	3	yes	—
120	C₂²×C₁₀₁	Q₈²	1	1	yes	characteristic
121	C₄×C₁₀₁	—	4	4	no	—
122	C₄×C₁₀₁	—	4	12	no	—
123	C₄×C₁₀₁	—	4	6	no	—
124	C₄×C₁₀₁	—	4	6	no	—
125	C₄×C₁₀₁	#396	1	8	yes	—
126	#15189	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₁₀₁	C₂×#396	1	1	yes	characteristic
128	C₂×C₁₀₁	#2427	1	3	yes	—
129	C₂×C₁₀₁	#2429	1	3	yes	—
130	C₂×C₁₀₁	#2451	1	3	yes	—
131	C₂×C₁₀₁	#2453	1	3	yes	—
132	C₂×C₁₀₁	#2645	1	1	yes	characteristic
133	C₂×C₁₀₁	#2863	1	1	yes	characteristic
134	C₂×#391	C₂×C₁₀₁	1	3	yes	—
135	C₂×#389	C₂×C₁₀₁	1	3	yes	—
136	#1443	C₂×C₁₀₁	1	2	yes	—
137	#1441	C₂×C₁₀₁	1	6	yes	—
138	C₄×#95	C₂×C₁₀₁	1	1	yes	characteristic
139	C₁₀₁	#15189	1	1	yes	characteristic, direct product factor, Sylow
140	C₂³×Q₈	C₂²×C₁₀₁	1	1	yes	characteristic
141	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	6	yes	—
142	C₂²×C₄⋊C₄	C₂²×C₁₀₁	1	3	yes	—
143	C₂²×C₄²	C₂²×C₁₀₁	1	3	yes	—
144	#391	—	2	12	no	—
145	#389	—	2	12	no	—
146	C₂×#95	C₂²×C₁₀₁	1	3	yes	—
147	C₂×#95	C₂²×C₁₀₁	1	6	yes	—
148	C₂×#95	C₂²×C₁₀₁	1	6	yes	—
149	C₂×#95	C₂²×C₁₀₁	1	6	yes	—
150	C₂×#95	C₂²×C₁₀₁	1	1	yes	characteristic
151	C₂²×Q₈	—	2	6	no	—
152	C₂²×Q₈	—	2	6	no	—
153	C₂²×Q₈	Q₈×C₁₀₁	1	4	yes	—
154	C₂³×C₄	C₂³×C₁₀₁	1	6	yes	—
155	C₂³×C₄	C₂³×C₁₀₁	1	2	yes	—
156	C₂³×C₄	C₂³×C₁₀₁	1	3	yes	—
157	C₂³×C₄	C₂³×C₁₀₁	1	3	yes	—
158	C₂³×C₄	C₂³×C₁₀₁	1	1	yes	characteristic
159	C₂×C₄⋊C₄	—	2	6	no	—
160	C₂×C₄⋊C₄	—	2	6	no	—
161	C₂×C₄⋊C₄	—	2	12	no	—
162	C₂×C₄⋊C₄	—	2	12	no	—
163	C₂×C₄⋊C₄	—	4	6	no	—
164	C₂×C₄⋊C₄	D₄×C₁₀₁	1	12	yes	—
165	C₂×C₄²	—	2	6	no	—
166	C₂×C₄²	—	2	6	no	—
167	C₂×C₄²	—	2	6	no	—
168	#95	—	2	12	no	—
169	#95	—	2	12	no	—
170	#95	—	2	12	no	—
171	#95	—	2	6	no	—
172	#95	Q₈×C₁₀₁	1	4	yes	complemented
173	C₂⁴	C₂⁴×C₁₀₁	1	1	yes	characteristic
174	C₂×Q₈	—	2	24	no	—
175	C₂×Q₈	—	4	12	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂²×C₄	—	2	3	no	—
178	C₂²×C₄	—	2	3	no	—
179	C₂²×C₄	—	2	3	no	—
180	C₂²×C₄	—	2	6	no	—
181	C₂²×C₄	—	2	6	no	—
182	C₂²×C₄	—	2	3	no	—
183	C₂²×C₄	—	2	3	no	—
184	C₂²×C₄	—	2	3	no	—
185	C₂²×C₄	—	2	3	no	—
186	C₂²×C₄	—	2	3	no	—
187	C₂²×C₄	—	2	6	no	—
188	C₂²×C₄	—	2	6	no	—
189	C₂²×C₄	—	2	6	no	—
190	C₂²×C₄	—	2	6	no	—
191	C₂²×C₄	—	2	3	no	—
192	C₂²×C₄	—	2	6	no	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	2	3	no	—
195	C₂²×C₄	—	2	3	no	—
196	C₂²×C₄	—	2	3	no	—
197	C₂²×C₄	C₂×D₄×C₁₀₁	1	12	yes	—
198	C₂²×C₄	C₂×D₄×C₁₀₁	1	6	yes	—
199	C₂²×C₄	C₂×Q₈×C₁₀₁	1	6	yes	—
200	C₂²×C₄	C₂×Q₈×C₁₀₁	1	4	yes	—
201	C₂²×C₄	C₂×Q₈×C₁₀₁	1	2	yes	—
202	C₂²×C₄	#41×C₁₀₁	1	6	yes	—
203	C₂²×C₄	#41×C₁₀₁	1	6	yes	—
204	C₄⋊C₄	—	2	24	no	—
205	C₄⋊C₄	—	4	24	no	—
206	C₄²	—	4	12	no	—
207	C₂³	C₂²×D₄×C₁₀₁	1	3	yes	—
208	C₂³	C₂²×Q₈×C₁₀₁	1	1	yes	characteristic
209	C₂³	C₂²×Q₈×C₁₀₁	1	1	yes	characteristic
210	C₂³	C₂×#41×C₁₀₁	1	3	yes	—
211	C₂³	C₂×#41×C₁₀₁	1	3	yes	—
212	C₂³	H(5,𝔽₂)×C₁₀₁	1	1	yes	characteristic
213	C₂³	2₋¹⁺⁴×C₁₀₁	1	3	yes	—
214	Q₈	—	4	16	no	—
215	C₂×C₄	—	2	12	no	—
216	C₂×C₄	—	2	12	no	—
217	C₂×C₄	—	2	6	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	12	no	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	6	no	—
225	C₂×C₄	—	4	3	no	—
226	C₂×C₄	—	4	3	no	—
227	C₂×C₄	—	4	6	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	6	no	—
230	C₂×C₄	—	4	3	no	—
231	C₂×C₄	—	4	3	no	—
232	C₂×C₄	—	4	6	no	—
233	C₂×C₄	—	4	6	no	—
234	C₂×C₄	—	4	3	no	—
235	C₂×C₄	—	4	3	no	—
236	C₂×C₄	—	4	3	no	—
237	C₂×C₄	—	4	6	no	—
238	C₂×C₄	—	4	3	no	—
239	C₂×C₄	#121×C₁₀₁	1	12	yes	—
240	C₂×C₄	C₂²⋊Q₈×C₁₀₁	1	12	yes	—
241	C₂×C₄	#126×C₁₀₁	1	4	yes	—
242	C₂²	C₂×#121×C₁₀₁	1	3	yes	—
243	C₂²	C₂×C₂²⋊Q₈×C₁₀₁	1	3	yes	—
244	C₂²	C₂×#126×C₁₀₁	1	1	yes	characteristic
245	C₂²	#529×C₁₀₁	1	3	yes	—
246	C₂²	#533×C₁₀₁	1	3	yes	—
247	C₂²	#541×C₁₀₁	1	3	yes	—
248	C₂²	#542×C₁₀₁	1	3	yes	—
249	C₂²	#546×C₁₀₁	1	3	yes	—
250	C₂²	#547×C₁₀₁	1	3	yes	—
251	C₂²	#548×C₁₀₁	1	3	yes	—
252	C₂²	D₄×Q₈×C₁₀₁	1	3	yes	—
253	C₂²	#557×C₁₀₁	1	3	yes	—
254	C₂²	Q₈²×C₁₀₁	1	1	yes	characteristic
255	C₄	—	4	12	no	—
256	C₄	—	4	6	no	—
257	C₄	—	4	6	no	—
258	C₄	—	4	4	no	—
259	C₄	#396×C₁₀₁	1	8	yes	—
260	C₂	C₂×#396×C₁₀₁	1	1	yes	characteristic
261	C₂	C₁₀₁×#2427	1	3	yes	—
262	C₂	C₁₀₁×#2429	1	3	yes	—
263	C₂	C₁₀₁×#2451	1	3	yes	—
264	C₂	C₁₀₁×#2453	1	3	yes	—
265	C₂	C₁₀₁×#2645	1	1	yes	characteristic
266	C₂	C₁₀₁×#2863	1	1	yes	characteristic

	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
38. #95×C₁₀₁	Q₈	semidirect
126. #15189	C₁₀₁	direct (N × H)
139. C₁₀₁	#15189	direct (N × H)
172. #95	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×#391×C₁₀₁	2
C₂×#389×C₁₀₁	2
C₁₀₁×#1443	2
C₁₀₁×#1441	2
C₄×#95×C₁₀₁	2
#15189	101

Subgroups — C101×#15189 #38797340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#15189 #38797340