Subgroups — C₁₀₁×#12890 #38795041

Order 25856.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346×C₁₀₁	C₂	1	1	yes	maximal, characteristic
2	C₁₀₁×#1846	C₂	1	1	yes	maximal, characteristic
3	C₁₀₁×#1845	C₂	1	1	yes	maximal, characteristic
4	C₁₀₁×#1380	C₂	1	4	yes	maximal
5	C₂²×C₁₆×C₁₀₁	C₂²	1	2	yes	—
6	C₂×#107×C₁₀₁	—	2	1	no	—
7	C₂×#107×C₁₀₁	C₂²	1	2	yes	—
8	C₂×#107×C₁₀₁	C₂²	1	2	yes	—
9	C₂×#106×C₁₀₁	—	2	1	no	—
10	C₂×C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
11	#384×C₁₀₁	—	2	1	no	—
12	#382×C₁₀₁	—	2	1	no	—
13	#346×C₁₀₁	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₀₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
17	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₀₁	—	4	2	no	—
19	C₂×C₄²×C₁₀₁	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₁₀₁	—	2	2	no	—
21	C₂×C₁₆×C₁₀₁	—	2	2	no	—
22	C₂×C₁₆×C₁₀₁	C₂×C₄	1	4	yes	—
23	#107×C₁₀₁	—	2	4	no	—
24	#107×C₁₀₁	—	2	4	no	—
25	#107×C₁₀₁	—	4	2	no	—
26	#106×C₁₀₁	—	4	2	no	—
27	C₄×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₀₁	Q₈	1	2	yes	—
29	#95×C₁₀₁	—	4	2	no	—
30	C₂²×C₄×C₁₀₁	—	4	2	no	—
31	C₂²×C₄×C₁₀₁	—	4	2	no	—
32	C₂²×C₄×C₁₀₁	D₈	1	2	yes	—
33	C₂²×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₀₁	SD₁₆	1	2	yes	—
36	C₂×C₈×C₁₀₁	SD₁₆	1	2	yes	—
37	C₂×C₈×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₀₁	#41	1	2	yes	—
40	C₂×C₈×C₁₀₁	#41	1	2	yes	—
41	C₄⋊C₄×C₁₀₁	—	4	2	no	—
42	C₄⋊C₄×C₁₀₁	—	4	2	no	—
43	C₄⋊C₄×C₁₀₁	—	4	2	no	—
44	C₄⋊C₄×C₁₀₁	—	4	2	no	—
45	C₄⋊C₄×C₁₀₁	—	8	4	no	—
46	C₄²×C₁₀₁	C₄⋊C₄	1	2	yes	—
47	C₄²×C₁₀₁	Q₁₆	1	2	yes	—
48	C₁₆×C₁₀₁	—	2	4	no	—
49	C₂³×C₁₀₁	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	—	2	2	no	—
51	C₂×C₄×C₁₀₁	—	2	2	no	—
52	C₂×C₄×C₁₀₁	—	8	2	no	—
53	C₂×C₄×C₁₀₁	—	8	2	no	—
54	C₂×C₄×C₁₀₁	—	8	4	no	—
55	C₂×C₄×C₁₀₁	—	8	2	no	—
56	C₂×C₄×C₁₀₁	—	8	2	no	—
57	C₂×C₄×C₁₀₁	#107	1	4	yes	—
58	C₂×C₄×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₁	#124	1	1	yes	characteristic
60	C₂×C₄×C₁₀₁	#124	1	1	yes	characteristic
61	C₂×C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₈×C₁₀₁	#106	1	4	yes	—
65	C₈×C₁₀₁	#117	1	2	yes	—
66	C₈×C₁₀₁	#125	1	2	yes	—
67	C₂²×C₁₀₁	C₂×#107	1	1	yes	characteristic
68	C₂²×C₁₀₁	#486	1	1	yes	characteristic
69	C₂²×C₁₀₁	#486	1	1	yes	characteristic
70	C₂²×C₁₀₁	#509	1	2	yes	—
71	C₂²×C₁₀₁	#510	1	2	yes	—
72	C₄×C₁₀₁	—	2	4	no	—
73	C₄×C₁₀₁	—	16	2	no	—
74	C₄×C₁₀₁	—	16	2	no	—
75	C₄×C₁₀₁	#382	1	1	yes	characteristic
76	C₄×C₁₀₁	C₂×#106	1	1	yes	characteristic
77	C₄×C₁₀₁	#487	1	1	yes	characteristic
78	C₄×C₁₀₁	#487	1	1	yes	characteristic
79	#12890	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₁₀₁	#1834	1	1	yes	characteristic
81	C₂×C₁₀₁	#2159	1	2	yes	—
82	C₂×C₁₀₁	#2243	1	2	yes	—
83	C₂×C₁₀₁	#2244	1	1	yes	characteristic
84	C₂×C₁₀₁	#2245	1	1	yes	characteristic
85	C₂×#346	C₂×C₁₀₁	1	1	yes	characteristic
86	#1846	C₂×C₁₀₁	1	1	yes	characteristic
87	#1845	C₂×C₁₀₁	1	1	yes	characteristic
88	#1380	C₂×C₁₀₁	1	4	yes	—
89	C₁₀₁	#12890	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₁₆	C₂²×C₁₀₁	1	2	yes	—
91	C₂×#107	—	2	1	no	—
92	C₂×#107	C₂²×C₁₀₁	1	2	yes	—
93	C₂×#107	C₂²×C₁₀₁	1	2	yes	—
94	C₂×#106	—	2	1	no	—
95	C₂×C₄×C₈	C₂²×C₁₀₁	1	1	yes	characteristic
96	#384	—	2	1	no	—
97	#382	—	2	1	no	—
98	#346	C₄×C₁₀₁	1	4	yes	complemented
99	C₂²×C₈	D₄×C₁₀₁	1	1	yes	characteristic
100	C₂²×C₈	C₂³×C₁₀₁	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	4	2	no	—
104	C₂×C₄²	D₄×C₁₀₁	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×C₄×C₁₀₁	1	4	yes	—
108	#107	—	2	4	no	—
109	#107	—	2	4	no	—
110	#107	—	4	2	no	—
111	#106	—	4	2	no	—
112	C₄×C₈	C₂×C₄×C₁₀₁	1	2	yes	—
113	C₄×C₈	Q₈×C₁₀₁	1	2	yes	—
114	#95	—	4	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	D₈×C₁₀₁	1	2	yes	—
118	C₂²×C₄	C₂×D₄×C₁₀₁	1	1	yes	characteristic
119	C₂×C₈	C₄⋊C₄×C₁₀₁	1	2	yes	—
120	C₂×C₈	SD₁₆×C₁₀₁	1	2	yes	—
121	C₂×C₈	SD₁₆×C₁₀₁	1	2	yes	—
122	C₂×C₈	C₂²×C₄×C₁₀₁	1	1	yes	characteristic
123	C₂×C₈	C₂×Q₈×C₁₀₁	1	1	yes	characteristic
124	C₂×C₈	#41×C₁₀₁	1	2	yes	—
125	C₂×C₈	#41×C₁₀₁	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄²	C₄⋊C₄×C₁₀₁	1	2	yes	—
132	C₄²	Q₁₆×C₁₀₁	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₈×C₁₀₁	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	#107×C₁₀₁	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊C₄×C₁₀₁	1	1	yes	characteristic
144	C₂×C₄	#124×C₁₀₁	1	1	yes	characteristic
145	C₂×C₄	#124×C₁₀₁	1	1	yes	characteristic
146	C₂×C₄	C₂×SD₁₆×C₁₀₁	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₁₆×C₁₀₁	1	1	yes	characteristic
148	C₂×C₄	C₂×Q₁₆×C₁₀₁	1	1	yes	characteristic
149	C₈	#106×C₁₀₁	1	4	yes	—
150	C₈	#117×C₁₀₁	1	2	yes	—
151	C₈	#125×C₁₀₁	1	2	yes	—
152	C₂²	C₂×#107×C₁₀₁	1	1	yes	characteristic
153	C₂²	#486×C₁₀₁	1	1	yes	characteristic
154	C₂²	#486×C₁₀₁	1	1	yes	characteristic
155	C₂²	#509×C₁₀₁	1	2	yes	—
156	C₂²	#510×C₁₀₁	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	16	2	no	—
160	C₄	#382×C₁₀₁	1	1	yes	characteristic
161	C₄	C₂×#106×C₁₀₁	1	1	yes	characteristic
162	C₄	#487×C₁₀₁	1	1	yes	characteristic
163	C₄	#487×C₁₀₁	1	1	yes	characteristic
164	C₂	C₁₀₁×#1834	1	1	yes	characteristic
165	C₂	C₁₀₁×#2159	1	2	yes	—
166	C₂	C₁₀₁×#2243	1	2	yes	—
167	C₂	C₁₀₁×#2244	1	1	yes	characteristic
168	C₂	C₁₀₁×#2245	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ C₂×C₄×C₈×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. #346×C₁₀₁	C₄	semidirect
79. #12890	C₁₀₁	direct (N × H)
89. C₁₀₁	#12890	direct (N × H)
98. #346	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×#346×C₁₀₁	2
C₁₀₁×#1846	2
C₁₀₁×#1845	2
C₁₀₁×#1380	2
#12890	101

Subgroups — C101×#12890 #38795041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₁×#12890 #38795041