Subgroups — C₁₀₁×#8038 #38790189

Order 25856.

650 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₀₁	C₂	1	2	yes	maximal
2	C₁₀₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₁₀₁×#1296	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₀₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
6	#431×C₁₀₁	—	2	4	no	—
7	#431×C₁₀₁	C₄	1	8	yes	complemented
8	C₂×C₄⋊C₈×C₁₀₁	C₂²	1	4	yes	—
9	C₂×C₂²⋊C₈×C₁₀₁	—	2	2	no	—
10	C₂×C₄×C₈×C₁₀₁	—	2	2	no	—
11	#384×C₁₀₁	C₂²	1	2	yes	—
12	#382×C₁₀₁	C₄	1	2	yes	—
13	#380×C₁₀₁	—	2	2	no	—
14	C₄×C₄⋊C₄×C₁₀₁	—	2	2	no	—
15	C₂³×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₀₁	D₄	1	2	yes	—
17	C₂²×C₈×C₁₀₁	—	2	4	no	—
18	#117×C₁₀₁	—	4	4	no	—
19	C₂×C₄⋊C₄×C₁₀₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₁₀₁	C₂×C₄	1	4	yes	—
21	C₂×C₂²⋊C₄×C₁₀₁	—	2	2	no	—
22	C₂×C₄²×C₁₀₁	—	2	2	no	—
23	C₂×C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₀₁	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
26	C₂×C₄²×C₁₀₁	D₄	1	2	yes	—
27	C₂×C₄²×C₁₀₁	Q₈	1	2	yes	—
28	C₂×C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₁₀₁	—	2	4	no	—
30	C₄⋊C₈×C₁₀₁	C₂×C₄	1	8	yes	—
31	C₂²⋊C₈×C₁₀₁	—	2	8	no	—
32	C₄×C₈×C₁₀₁	—	2	8	no	—
33	#95×C₁₀₁	—	2	4	no	—
34	C₂⁴×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₁	—	2	2	no	—
36	C₂²×C₄×C₁₀₁	—	2	2	no	—
37	C₂²×C₄×C₁₀₁	—	2	8	no	—
38	C₂²×C₄×C₁₀₁	—	2	4	no	—
39	C₂²×C₄×C₁₀₁	C₄²	1	2	yes	—
40	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₀₁	—	2	8	no	—
48	C₂×C₈×C₁₀₁	—	4	8	no	—
49	C₄⋊C₄×C₁₀₁	—	4	4	no	—
50	C₄⋊C₄×C₁₀₁	—	4	8	no	—
51	C₂²⋊C₄×C₁₀₁	—	4	4	no	—
52	C₄²×C₁₀₁	—	4	4	no	—
53	C₄²×C₁₀₁	C₄²	1	2	yes	—
54	C₄²×C₁₀₁	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₁₀₁	C₄⋊C₄	1	2	yes	—
56	C₄²×C₁₀₁	C₄⋊C₄	1	4	yes	—
57	C₄²×C₁₀₁	C₂²×C₄	1	2	yes	—
58	C₄²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
60	C₂³×C₁₀₁	—	2	1	no	—
61	C₂³×C₁₀₁	—	2	1	no	—
62	C₂³×C₁₀₁	—	2	4	no	—
63	C₂³×C₁₀₁	#95	1	2	yes	—
64	C₂³×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₁	—	2	1	no	—
66	C₂×C₄×C₁₀₁	—	2	1	no	—
67	C₂×C₄×C₁₀₁	—	2	4	no	—
68	C₂×C₄×C₁₀₁	—	2	4	no	—
69	C₂×C₄×C₁₀₁	—	2	4	no	—
70	C₂×C₄×C₁₀₁	—	2	8	no	—
71	C₂×C₄×C₁₀₁	—	2	8	no	—
72	C₂×C₄×C₁₀₁	—	4	4	no	—
73	C₂×C₄×C₁₀₁	—	4	8	no	—
74	C₂×C₄×C₁₀₁	#95	1	4	yes	—
75	C₂×C₄×C₁₀₁	#95	1	2	yes	—
76	C₂×C₄×C₁₀₁	#100	1	2	yes	—
77	C₂×C₄×C₁₀₁	#101	1	2	yes	—
78	C₂×C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
80	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₁₀₁	#135	1	8	yes	—
83	C₈×C₁₀₁	—	4	8	no	—
84	C₂²×C₁₀₁	—	2	4	no	—
85	C₂²×C₁₀₁	—	2	2	no	—
86	C₂²×C₁₀₁	—	4	2	no	—
87	C₂²×C₁₀₁	—	4	2	no	—
88	C₂²×C₁₀₁	C₂×#95	1	1	yes	characteristic
89	C₂²×C₁₀₁	C₂×#100	1	1	yes	characteristic
90	C₂²×C₁₀₁	C₂×#101	1	1	yes	characteristic
91	C₂²×C₁₀₁	#416	1	4	yes	—
92	C₄×C₁₀₁	—	2	4	no	—
93	C₄×C₁₀₁	—	2	1	no	—
94	C₄×C₁₀₁	—	2	1	no	—
95	C₄×C₁₀₁	—	8	4	no	—
96	C₄×C₁₀₁	#343	1	4	yes	—
97	C₄×C₁₀₁	#416	1	4	yes	—
98	C₄×C₁₀₁	#427	1	4	yes	—
99	#8038	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₁₀₁	—	4	2	no	—
101	C₂×C₁₀₁	#1543	1	2	yes	—
102	C₂×C₁₀₁	#1544	1	2	yes	—
103	C₂×C₁₀₁	#1735	1	2	yes	—
104	C₂×C₁₀₁	C₂×#343	1	1	yes	characteristic
105	C₂×#431	C₂×C₁₀₁	1	2	yes	—
106	#2285	C₂×C₁₀₁	1	1	yes	characteristic
107	#1296	C₂×C₁₀₁	1	4	yes	complemented
108	C₁₀₁	#8038	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×#117	C₄×C₁₀₁	1	2	yes	—
110	C₂²×C₄²	C₂²×C₁₀₁	1	1	yes	characteristic
111	#431	—	2	4	no	—
112	#431	C₄×C₁₀₁	1	8	yes	complemented
113	C₂×C₄⋊C₈	C₂²×C₁₀₁	1	4	yes	—
114	C₂×C₂²⋊C₈	—	2	2	no	—
115	C₂×C₄×C₈	—	2	2	no	—
116	#384	C₂²×C₁₀₁	1	2	yes	—
117	#382	C₄×C₁₀₁	1	2	yes	—
118	#380	—	2	2	no	—
119	C₄×C₄⋊C₄	—	2	2	no	—
120	C₂³×C₄	C₂×C₄×C₁₀₁	1	1	yes	characteristic
121	C₂³×C₄	D₄×C₁₀₁	1	2	yes	—
122	C₂²×C₈	—	2	4	no	—
123	#117	—	4	4	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	C₂×C₄×C₁₀₁	1	4	yes	—
126	C₂×C₂²⋊C₄	—	2	2	no	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	C₂×C₄×C₁₀₁	1	1	yes	characteristic
129	C₂×C₄²	C₂×C₄×C₁₀₁	1	4	yes	—
130	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
131	C₂×C₄²	D₄×C₁₀₁	1	2	yes	—
132	C₂×C₄²	Q₈×C₁₀₁	1	2	yes	—
133	C₂×C₄²	C₂³×C₁₀₁	1	1	yes	characteristic
134	C₄⋊C₈	—	2	4	no	—
135	C₄⋊C₈	C₂×C₄×C₁₀₁	1	8	yes	—
136	C₂²⋊C₈	—	2	8	no	—
137	C₄×C₈	—	2	8	no	—
138	#95	—	2	4	no	—
139	C₂⁴	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₄²×C₁₀₁	1	2	yes	—
145	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	2	yes	—
146	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	4	yes	—
147	C₂²×C₄	C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
148	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	4	yes	—
149	C₂²×C₄	C₄⋊C₄×C₁₀₁	1	2	yes	—
150	C₂²×C₄	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₁₀₁	1	2	yes	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	4	8	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₂²⋊C₄	—	4	4	no	—
157	C₄²	—	4	4	no	—
158	C₄²	C₄²×C₁₀₁	1	2	yes	—
159	C₄²	C₂²⋊C₄×C₁₀₁	1	4	yes	—
160	C₄²	C₄⋊C₄×C₁₀₁	1	2	yes	—
161	C₄²	C₄⋊C₄×C₁₀₁	1	4	yes	—
162	C₄²	C₂²×C₄×C₁₀₁	1	2	yes	—
163	C₄²	C₂×D₄×C₁₀₁	1	1	yes	characteristic
164	C₄²	C₂×Q₈×C₁₀₁	1	1	yes	characteristic
165	C₂³	—	2	1	no	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	4	no	—
168	C₂³	#95×C₁₀₁	1	2	yes	—
169	C₂³	C₂×C₂²⋊C₄×C₁₀₁	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	#95×C₁₀₁	1	4	yes	—
180	C₂×C₄	#95×C₁₀₁	1	2	yes	—
181	C₂×C₄	#100×C₁₀₁	1	2	yes	—
182	C₂×C₄	#101×C₁₀₁	1	2	yes	—
183	C₂×C₄	C₂×C₄²×C₁₀₁	1	1	yes	characteristic
184	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₁	1	2	yes	—
185	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	2	yes	—
187	C₂×C₄	#135×C₁₀₁	1	8	yes	—
188	C₈	—	4	8	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	C₂×#95×C₁₀₁	1	1	yes	characteristic
194	C₂²	C₂×#100×C₁₀₁	1	1	yes	characteristic
195	C₂²	C₂×#101×C₁₀₁	1	1	yes	characteristic
196	C₂²	#416×C₁₀₁	1	4	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	8	4	no	—
201	C₄	#343×C₁₀₁	1	4	yes	—
202	C₄	#416×C₁₀₁	1	4	yes	—
203	C₄	#427×C₁₀₁	1	4	yes	—
204	C₂	—	4	2	no	—
205	C₂	C₁₀₁×#1543	1	2	yes	—
206	C₂	C₁₀₁×#1544	1	2	yes	—
207	C₂	C₁₀₁×#1735	1	2	yes	—
208	C₂	C₂×#343×C₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
socle	C₂³×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₁	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂²×C₄²×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₀₁×#1296	C₂	semidirect
7. #431×C₁₀₁	C₄	semidirect
99. #8038	C₁₀₁	direct (N × H)
107. #1296	C₂×C₁₀₁	semidirect
108. C₁₀₁	#8038	direct (N × H)
112. #431	C₄×C₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₀₁	2
C₁₀₁×#2285	2
C₁₀₁×#1296	2
#8038	101

Subgroups — C101×#8038 #38790189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#8038 #38790189