Subgroups — C₂×C₁₀₇×#2262 #41059942

Order 27392.

470 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₀₇	C₂	1	2	yes	maximal, complemented
2	C₁₀₇×#2262	C₂	1	8	yes	maximal, direct product factor
3	C₂×Q₆₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₆₄×C₁₀₇	C₂	1	2	yes	maximal, complemented
5	C₂×D₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃₂×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₁₀₇	—	2	2	no	—
8	#508×C₁₀₇	—	2	4	no	—
9	#508×C₁₀₇	C₂²	1	8	yes	complemented
10	C₂×Q₃₂×C₁₀₇	C₂²	1	2	yes	—
11	C₂×SD₃₂×C₁₀₇	—	2	2	no	—
12	C₂×D₁₆×C₁₀₇	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
14	Q₆₄×C₁₀₇	C₂²	1	4	yes	complemented
15	SD₆₄×C₁₀₇	C₂²	1	8	yes	complemented
16	D₃₂×C₁₀₇	C₂²	1	4	yes	complemented
17	C₂×C₃₂×C₁₀₇	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₁₀₇	C₂²	1	1	yes	characteristic
19	C₂×C₃₂×C₁₀₇	C₂²	1	4	yes	complemented
20	C₂×#41×C₁₀₇	—	4	2	no	—
21	#135×C₁₀₇	—	2	8	no	—
22	#135×C₁₀₇	—	4	4	no	—
23	C₂×Q₁₆×C₁₀₇	—	2	2	no	—
24	C₂×SD₁₆×C₁₀₇	—	4	2	no	—
25	C₂×D₈×C₁₀₇	—	2	2	no	—
26	C₂²×C₈×C₁₀₇	D₄	1	1	yes	characteristic
27	Q₃₂×C₁₀₇	—	2	2	no	—
28	Q₃₂×C₁₀₇	C₂³	1	4	yes	—
29	SD₃₂×C₁₀₇	—	2	8	no	—
30	D₁₆×C₁₀₇	—	2	2	no	—
31	D₁₆×C₁₀₇	C₂³	1	4	yes	—
32	C₂×C₁₆×C₁₀₇	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₁₀₇	D₄	1	2	yes	—
34	C₂×C₁₆×C₁₀₇	C₂³	1	1	yes	characteristic
35	C₂×C₁₆×C₁₀₇	C₂³	1	2	yes	—
36	C₃₂×C₁₀₇	C₂³	1	2	yes	—
37	C₃₂×C₁₀₇	C₂³	1	2	yes	—
38	#41×C₁₀₇	—	4	8	no	—
39	#41×C₁₀₇	—	8	4	no	—
40	C₂×Q₈×C₁₀₇	—	4	2	no	—
41	C₂×D₄×C₁₀₇	—	4	2	no	—
42	C₂×D₄×C₁₀₇	—	8	2	no	—
43	C₂²×C₄×C₁₀₇	—	8	2	no	—
44	C₂²×C₄×C₁₀₇	D₈	1	1	yes	characteristic
45	Q₁₆×C₁₀₇	—	2	4	no	—
46	Q₁₆×C₁₀₇	—	4	2	no	—
47	SD₁₆×C₁₀₇	—	4	8	no	—
48	D₈×C₁₀₇	—	2	4	no	—
49	D₈×C₁₀₇	—	4	2	no	—
50	C₂×C₈×C₁₀₇	D₈	1	1	yes	characteristic
51	C₂×C₈×C₁₀₇	D₈	1	2	yes	—
52	C₂×C₈×C₁₀₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₀₇	C₂×D₄	1	2	yes	—
54	C₁₆×C₁₀₇	C₂×D₄	1	1	yes	characteristic
55	C₁₆×C₁₀₇	C₂×D₄	1	2	yes	—
56	C₁₆×C₁₀₇	C₂⁴	1	1	yes	characteristic
57	C₂³×C₁₀₇	—	8	2	no	—
58	C₂³×C₁₀₇	D₁₆	1	1	yes	characteristic
59	Q₈×C₁₀₇	—	4	4	no	—
60	Q₈×C₁₀₇	—	8	2	no	—
61	D₄×C₁₀₇	—	4	4	no	—
62	D₄×C₁₀₇	—	8	8	no	—
63	D₄×C₁₀₇	—	8	2	no	—
64	C₂×C₄×C₁₀₇	—	8	2	no	—
65	C₂×C₄×C₁₀₇	—	8	8	no	—
66	C₂×C₄×C₁₀₇	D₁₆	1	2	yes	—
67	C₂×C₄×C₁₀₇	D₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₁₀₇	C₂×D₈	1	2	yes	—
70	C₈×C₁₀₇	C₂×D₈	1	1	yes	characteristic
71	C₈×C₁₀₇	C₂×D₈	1	2	yes	—
72	C₈×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₁₀₇	—	2	2	no	—
74	C₂²×C₁₀₇	—	8	4	no	—
75	C₂²×C₁₀₇	—	16	4	no	—
76	C₂²×C₁₀₇	C₂×D₁₆	1	2	yes	—
77	C₂²×C₁₀₇	C₂×D₁₆	1	1	yes	characteristic
78	C₄×C₁₀₇	—	8	4	no	—
79	C₄×C₁₀₇	C₂×D₁₆	1	2	yes	—
80	C₄×C₁₀₇	C₂×D₁₆	1	1	yes	characteristic
81	C₄×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
82	C₂×#2262	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×C₁₀₇	—	2	2	no	—
84	C₂×C₁₀₇	—	16	4	no	—
85	C₂×C₁₀₇	#2262	1	2	yes	direct product factor
86	C₂×C₁₀₇	C₂²×D₁₆	1	1	yes	characteristic
87	C₂×#508	C₂×C₁₀₇	1	2	yes	complemented
88	#2262	C₂×C₁₀₇	1	8	yes	direct product factor
89	C₂×Q₆₄	C₂×C₁₀₇	1	1	yes	characteristic, complemented
90	C₂×SD₆₄	C₂×C₁₀₇	1	2	yes	complemented
91	C₂×D₃₂	C₂×C₁₀₇	1	1	yes	characteristic, complemented
92	C₂²×C₃₂	C₂×C₁₀₇	1	1	yes	characteristic, complemented
93	C₁₀₇	C₂×#2262	1	1	yes	characteristic, direct product factor, Sylow
94	C₂×#135	—	2	2	no	—
95	#508	—	2	4	no	—
96	#508	C₂²×C₁₀₇	1	8	yes	complemented
97	C₂×Q₃₂	C₂²×C₁₀₇	1	2	yes	—
98	C₂×SD₃₂	—	2	2	no	—
99	C₂×D₁₆	C₂²×C₁₀₇	1	2	yes	—
100	C₂²×C₁₆	C₂²×C₁₀₇	1	1	yes	characteristic
101	Q₆₄	C₂²×C₁₀₇	1	4	yes	complemented
102	SD₆₄	C₂²×C₁₀₇	1	8	yes	complemented
103	D₃₂	C₂²×C₁₀₇	1	4	yes	complemented
104	C₂×C₃₂	C₂²×C₁₀₇	1	1	yes	characteristic
105	C₂×C₃₂	C₂²×C₁₀₇	1	1	yes	characteristic
106	C₂×C₃₂	C₂²×C₁₀₇	1	4	yes	complemented
107	C₂×#41	—	4	2	no	—
108	#135	—	2	8	no	—
109	#135	—	4	4	no	—
110	C₂×Q₁₆	—	2	2	no	—
111	C₂×SD₁₆	—	4	2	no	—
112	C₂×D₈	—	2	2	no	—
113	C₂²×C₈	D₄×C₁₀₇	1	1	yes	characteristic
114	Q₃₂	—	2	2	no	—
115	Q₃₂	C₂³×C₁₀₇	1	4	yes	—
116	SD₃₂	—	2	8	no	—
117	D₁₆	—	2	2	no	—
118	D₁₆	C₂³×C₁₀₇	1	4	yes	—
119	C₂×C₁₆	D₄×C₁₀₇	1	1	yes	characteristic
120	C₂×C₁₆	D₄×C₁₀₇	1	2	yes	—
121	C₂×C₁₆	C₂³×C₁₀₇	1	1	yes	characteristic
122	C₂×C₁₆	C₂³×C₁₀₇	1	2	yes	—
123	C₃₂	C₂³×C₁₀₇	1	2	yes	—
124	C₃₂	C₂³×C₁₀₇	1	2	yes	—
125	#41	—	4	8	no	—
126	#41	—	8	4	no	—
127	C₂×Q₈	—	4	2	no	—
128	C₂×D₄	—	4	2	no	—
129	C₂×D₄	—	8	2	no	—
130	C₂²×C₄	—	8	2	no	—
131	C₂²×C₄	D₈×C₁₀₇	1	1	yes	characteristic
132	Q₁₆	—	2	4	no	—
133	Q₁₆	—	4	2	no	—
134	SD₁₆	—	4	8	no	—
135	D₈	—	2	4	no	—
136	D₈	—	4	2	no	—
137	C₂×C₈	D₈×C₁₀₇	1	1	yes	characteristic
138	C₂×C₈	D₈×C₁₀₇	1	2	yes	—
139	C₂×C₈	C₂×D₄×C₁₀₇	1	1	yes	characteristic
140	C₂×C₈	C₂×D₄×C₁₀₇	1	2	yes	—
141	C₁₆	C₂×D₄×C₁₀₇	1	1	yes	characteristic
142	C₁₆	C₂×D₄×C₁₀₇	1	2	yes	—
143	C₁₆	C₂⁴×C₁₀₇	1	1	yes	characteristic
144	C₂³	—	8	2	no	—
145	C₂³	D₁₆×C₁₀₇	1	1	yes	characteristic
146	Q₈	—	4	4	no	—
147	Q₈	—	8	2	no	—
148	D₄	—	4	4	no	—
149	D₄	—	8	8	no	—
150	D₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	8	no	—
153	C₂×C₄	D₁₆×C₁₀₇	1	1	yes	characteristic
154	C₂×C₄	D₁₆×C₁₀₇	1	2	yes	—
155	C₂×C₄	C₂×D₈×C₁₀₇	1	1	yes	characteristic
156	C₂×C₄	C₂×D₈×C₁₀₇	1	2	yes	—
157	C₈	C₂×D₈×C₁₀₇	1	1	yes	characteristic
158	C₈	C₂×D₈×C₁₀₇	1	2	yes	—
159	C₈	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	—	8	4	no	—
162	C₂²	—	16	4	no	—
163	C₂²	C₂×D₁₆×C₁₀₇	1	2	yes	—
164	C₂²	C₂×D₁₆×C₁₀₇	1	1	yes	characteristic
165	C₄	—	8	4	no	—
166	C₄	C₂×D₁₆×C₁₀₇	1	2	yes	—
167	C₄	C₂×D₁₆×C₁₀₇	1	1	yes	characteristic
168	C₄	C₂²×D₈×C₁₀₇	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	16	4	no	—
171	C₂	C₁₀₇×#2262	1	2	yes	direct product factor
172	C₂	C₂²×D₁₆×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₇	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₀₇
socle	C₂²×C₁₀₇	C₂×D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂²×C₄×C₁₀₇ ◁ C₂²×C₈×C₁₀₇ ◁ C₂²×C₁₆×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₁₀₇	C₂	semidirect
2. C₁₀₇×#2262	C₂	direct (N × H)
3. C₂×Q₆₄×C₁₀₇	C₂	semidirect
4. C₂×SD₆₄×C₁₀₇	C₂	semidirect
5. C₂×D₃₂×C₁₀₇	C₂	semidirect
6. C₂²×C₃₂×C₁₀₇	C₂	semidirect
9. #508×C₁₀₇	C₂²	semidirect
14. Q₆₄×C₁₀₇	C₂²	semidirect
15. SD₆₄×C₁₀₇	C₂²	semidirect
16. D₃₂×C₁₀₇	C₂²	semidirect
19. C₂×C₃₂×C₁₀₇	C₂²	semidirect
82. C₂×#2262	C₁₀₇	direct (N × H)
85. C₂×C₁₀₇	#2262	direct (N × H)
87. C₂×#508	C₂×C₁₀₇	semidirect
88. #2262	C₂×C₁₀₇	direct (N × H)
89. C₂×Q₆₄	C₂×C₁₀₇	semidirect
90. C₂×SD₆₄	C₂×C₁₀₇	semidirect
91. C₂×D₃₂	C₂×C₁₀₇	semidirect
92. C₂²×C₃₂	C₂×C₁₀₇	semidirect
93. C₁₀₇	C₂×#2262	direct (N × H)
96. #508	C₂²×C₁₀₇	semidirect
101. Q₆₄	C₂²×C₁₀₇	semidirect
102. SD₆₄	C₂²×C₁₀₇	semidirect
103. D₃₂	C₂²×C₁₀₇	semidirect
106. C₂×C₃₂	C₂²×C₁₀₇	semidirect
171. C₂	C₁₀₇×#2262	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×#508×C₁₀₇	2
C₁₀₇×#2262	2
C₂×Q₆₄×C₁₀₇	2
C₂×SD₆₄×C₁₀₇	2
C₂×D₃₂×C₁₀₇	2
C₂²×C₃₂×C₁₀₇	2
C₂×#2262	107

Subgroups — C2×C107×#2262 #41059942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₇×#2262 #41059942