Subgroups — C₂×#575×C₁₀₇ #28334263

Order 13696.

1484 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₀₇	C₂	1	2	yes	maximal, complemented
2	#575×C₁₀₇	C₂	1	16	yes	maximal, direct product factor
3	C₂×#137×C₁₀₇	C₂	1	4	yes	maximal, complemented
4	C₂×AGL(1,ℤ/8)×C₁₀₇	C₂	1	4	yes	maximal, complemented
5	C₂×#135×C₁₀₇	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₁₀₇	—	2	2	no	—
8	C₂×#41×C₁₀₇	—	2	8	no	—
9	C₂×#41×C₁₀₇	—	2	2	no	—
10	C₂×#41×C₁₀₇	C₂²	1	4	yes	—
11	C₂×#41×C₁₀₇	C₂²	1	4	yes	complemented
12	C₂×#41×C₁₀₇	C₂²	1	16	yes	complemented
13	C₂²×Q₈×C₁₀₇	C₂²	1	2	yes	complemented
14	C₂²×D₄×C₁₀₇	—	2	2	no	—
15	C₂²×D₄×C₁₀₇	C₂²	1	2	yes	complemented
16	C₂³×C₄×C₁₀₇	—	2	2	no	—
17	C₂³×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
18	#137×C₁₀₇	C₂²	1	32	yes	complemented
19	AGL(1,ℤ/8)×C₁₀₇	C₂²	1	32	yes	complemented
20	#135×C₁₀₇	C₂²	1	32	yes	complemented
21	C₂×Q₁₆×C₁₀₇	C₂²	1	4	yes	complemented
22	C₂×SD₁₆×C₁₀₇	C₂²	1	8	yes	complemented
23	C₂×D₈×C₁₀₇	C₂²	1	4	yes	complemented
24	C₂×M₁₆×C₁₀₇	C₂²	1	4	yes	complemented
25	C₂×M₁₆×C₁₀₇	C₂²	1	8	yes	complemented
26	C₂²×C₈×C₁₀₇	C₂²	1	2	yes	complemented
27	C₂⁴×C₁₀₇	—	2	2	no	—
28	C₂⁴×C₁₀₇	D₄	1	1	yes	characteristic
29	#41×C₁₀₇	—	2	8	no	—
30	#41×C₁₀₇	—	2	16	no	—
31	#41×C₁₀₇	—	2	16	no	—
32	#41×C₁₀₇	—	2	8	no	—
33	#41×C₁₀₇	C₂³	1	16	yes	—
34	#41×C₁₀₇	C₂³	1	16	yes	complemented
35	C₂×Q₈×C₁₀₇	—	2	2	no	—
36	C₂×Q₈×C₁₀₇	—	2	4	no	—
37	C₂×Q₈×C₁₀₇	C₂³	1	4	yes	—
38	C₂×Q₈×C₁₀₇	C₂³	1	8	yes	complemented
39	C₂×D₄×C₁₀₇	—	2	4	no	—
40	C₂×D₄×C₁₀₇	—	2	4	no	—
41	C₂×D₄×C₁₀₇	—	2	16	no	—
42	C₂×D₄×C₁₀₇	—	2	2	no	—
43	C₂×D₄×C₁₀₇	—	2	4	no	—
44	C₂×D₄×C₁₀₇	C₂³	1	4	yes	—
45	C₂×D₄×C₁₀₇	C₂³	1	8	yes	complemented
46	C₂²×C₄×C₁₀₇	—	2	2	no	—
47	C₂²×C₄×C₁₀₇	—	2	4	no	—
48	C₂²×C₄×C₁₀₇	—	2	16	no	—
49	C₂²×C₄×C₁₀₇	—	2	4	no	—
50	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₀₇	D₄	1	4	yes	—
52	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
53	C₂²×C₄×C₁₀₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₀₇	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
56	Q₁₆×C₁₀₇	C₂³	1	16	yes	complemented
57	SD₁₆×C₁₀₇	C₂³	1	32	yes	complemented
58	D₈×C₁₀₇	C₂³	1	16	yes	complemented
59	M₁₆×C₁₀₇	C₂³	1	16	yes	complemented
60	C₂×C₈×C₁₀₇	C₂³	1	4	yes	—
61	C₂×C₈×C₁₀₇	C₂³	1	8	yes	complemented
62	C₂³×C₁₀₇	—	2	4	no	—
63	C₂³×C₁₀₇	—	2	8	no	—
64	C₂³×C₁₀₇	—	4	2	no	—
65	C₂³×C₁₀₇	—	4	8	no	—
66	C₂³×C₁₀₇	C₂×D₄	1	2	yes	—
67	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
68	C₂³×C₁₀₇	C₂×D₄	1	4	yes	—
69	Q₈×C₁₀₇	—	2	4	no	—
70	Q₈×C₁₀₇	—	2	8	no	—
71	Q₈×C₁₀₇	C₂⁴	1	8	yes	—
72	D₄×C₁₀₇	—	2	16	no	—
73	D₄×C₁₀₇	—	2	16	no	—
74	D₄×C₁₀₇	—	2	4	no	—
75	D₄×C₁₀₇	—	2	8	no	—
76	D₄×C₁₀₇	C₂⁴	1	8	yes	—
77	C₂×C₄×C₁₀₇	—	2	4	no	—
78	C₂×C₄×C₁₀₇	—	2	8	no	—
79	C₂×C₄×C₁₀₇	—	2	16	no	—
80	C₂×C₄×C₁₀₇	—	2	16	no	—
81	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₀₇	C₂×D₄	1	8	yes	—
83	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
84	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₀₇	C₂×D₄	1	4	yes	—
86	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₁₀₇	C₂⁴	1	2	yes	—
89	C₂×C₄×C₁₀₇	C₂⁴	1	4	yes	—
90	C₈×C₁₀₇	C₂⁴	1	8	yes	—
91	C₂²×C₁₀₇	—	2	4	no	—
92	C₂²×C₁₀₇	—	2	2	no	—
93	C₂²×C₁₀₇	—	2	8	no	—
94	C₂²×C₁₀₇	—	4	4	no	—
95	C₂²×C₁₀₇	—	4	16	no	—
96	C₂²×C₁₀₇	—	4	8	no	—
97	C₂²×C₁₀₇	C₂²×D₄	1	4	yes	—
98	C₂²×C₁₀₇	C₂²×D₄	1	2	yes	—
99	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₁₀₇	—	2	8	no	—
101	C₄×C₁₀₇	C₂²×D₄	1	2	yes	—
102	C₄×C₁₀₇	C₂²×D₄	1	4	yes	—
103	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₁₀₇	C₂⁵	1	1	yes	characteristic
105	C₂×C₁₀₇	—	2	4	no	—
106	C₂×C₁₀₇	—	2	2	no	—
107	C₂×C₁₀₇	—	4	8	no	—
108	C₂×C₁₀₇	#575	1	2	yes	direct product factor
109	C₂×C₁₀₇	C₂³×D₄	1	1	yes	characteristic
110	C₂×#575	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₀₇	C₂×#575	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×#41	C₂×C₁₀₇	1	2	yes	complemented
113	#575	C₂×C₁₀₇	1	16	yes	direct product factor
114	C₂×#137	C₂×C₁₀₇	1	4	yes	complemented
115	C₂×AGL(1,ℤ/8)	C₂×C₁₀₇	1	4	yes	complemented
116	C₂×#135	C₂×C₁₀₇	1	4	yes	complemented
117	C₂²×M₁₆	C₂×C₁₀₇	1	1	yes	characteristic, complemented
118	C₂×#41	—	2	2	no	—
119	C₂×#41	—	2	2	no	—
120	C₂×#41	—	2	8	no	—
121	C₂×#41	C₂²×C₁₀₇	1	4	yes	—
122	C₂×#41	C₂²×C₁₀₇	1	4	yes	complemented
123	C₂×#41	C₂²×C₁₀₇	1	16	yes	complemented
124	C₂²×Q₈	C₂²×C₁₀₇	1	2	yes	complemented
125	C₂²×D₄	—	2	2	no	—
126	C₂²×D₄	C₂²×C₁₀₇	1	2	yes	complemented
127	C₂³×C₄	—	2	2	no	—
128	C₂³×C₄	C₂²×C₁₀₇	1	1	yes	characteristic
129	#137	C₂²×C₁₀₇	1	32	yes	complemented
130	AGL(1,ℤ/8)	C₂²×C₁₀₇	1	32	yes	complemented
131	#135	C₂²×C₁₀₇	1	32	yes	complemented
132	C₂×Q₁₆	C₂²×C₁₀₇	1	4	yes	complemented
133	C₂×SD₁₆	C₂²×C₁₀₇	1	8	yes	complemented
134	C₂×D₈	C₂²×C₁₀₇	1	4	yes	complemented
135	C₂×M₁₆	C₂²×C₁₀₇	1	4	yes	complemented
136	C₂×M₁₆	C₂²×C₁₀₇	1	8	yes	complemented
137	C₂²×C₈	C₂²×C₁₀₇	1	2	yes	complemented
138	C₂⁴	—	2	2	no	—
139	C₂⁴	D₄×C₁₀₇	1	1	yes	characteristic
140	#41	—	2	8	no	—
141	#41	—	2	16	no	—
142	#41	—	2	16	no	—
143	#41	—	2	8	no	—
144	#41	C₂³×C₁₀₇	1	16	yes	—
145	#41	C₂³×C₁₀₇	1	16	yes	complemented
146	C₂×Q₈	—	2	2	no	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	C₂³×C₁₀₇	1	4	yes	—
149	C₂×Q₈	C₂³×C₁₀₇	1	8	yes	complemented
150	C₂×D₄	—	2	4	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	2	16	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	C₂³×C₁₀₇	1	4	yes	—
156	C₂×D₄	C₂³×C₁₀₇	1	8	yes	complemented
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	16	no	—
161	C₂²×C₄	D₄×C₁₀₇	1	2	yes	—
162	C₂²×C₄	D₄×C₁₀₇	1	1	yes	characteristic
163	C₂²×C₄	D₄×C₁₀₇	1	4	yes	—
164	C₂²×C₄	C₂³×C₁₀₇	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₁₀₇	1	2	yes	—
166	C₂²×C₄	C₂³×C₁₀₇	1	4	yes	—
167	Q₁₆	C₂³×C₁₀₇	1	16	yes	complemented
168	SD₁₆	C₂³×C₁₀₇	1	32	yes	complemented
169	D₈	C₂³×C₁₀₇	1	16	yes	complemented
170	M₁₆	C₂³×C₁₀₇	1	16	yes	complemented
171	C₂×C₈	C₂³×C₁₀₇	1	4	yes	—
172	C₂×C₈	C₂³×C₁₀₇	1	8	yes	complemented
173	C₂³	—	2	4	no	—
174	C₂³	—	2	8	no	—
175	C₂³	—	4	2	no	—
176	C₂³	—	4	8	no	—
177	C₂³	C₂×D₄×C₁₀₇	1	2	yes	—
178	C₂³	C₂×D₄×C₁₀₇	1	1	yes	characteristic
179	C₂³	C₂×D₄×C₁₀₇	1	4	yes	—
180	Q₈	—	2	4	no	—
181	Q₈	—	2	8	no	—
182	Q₈	C₂⁴×C₁₀₇	1	8	yes	—
183	D₄	—	2	16	no	—
184	D₄	—	2	16	no	—
185	D₄	—	2	4	no	—
186	D₄	—	2	8	no	—
187	D₄	C₂⁴×C₁₀₇	1	8	yes	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	16	no	—
191	C₂×C₄	—	2	16	no	—
192	C₂×C₄	C₂×D₄×C₁₀₇	1	1	yes	characteristic
193	C₂×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
194	C₂×C₄	C₂×D₄×C₁₀₇	1	8	yes	—
195	C₂×C₄	C₂×D₄×C₁₀₇	1	4	yes	—
196	C₂×C₄	C₂×D₄×C₁₀₇	1	2	yes	—
197	C₂×C₄	C₂×D₄×C₁₀₇	1	4	yes	—
198	C₂×C₄	C₂⁴×C₁₀₇	1	1	yes	characteristic
199	C₂×C₄	C₂⁴×C₁₀₇	1	2	yes	—
200	C₂×C₄	C₂⁴×C₁₀₇	1	4	yes	—
201	C₈	C₂⁴×C₁₀₇	1	8	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	8	no	—
205	C₂²	—	4	4	no	—
206	C₂²	—	4	16	no	—
207	C₂²	—	4	8	no	—
208	C₂²	C₂²×D₄×C₁₀₇	1	4	yes	—
209	C₂²	C₂²×D₄×C₁₀₇	1	2	yes	—
210	C₂²	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
211	C₄	—	2	8	no	—
212	C₄	C₂²×D₄×C₁₀₇	1	2	yes	—
213	C₄	C₂²×D₄×C₁₀₇	1	4	yes	—
214	C₄	C₂²×D₄×C₁₀₇	1	1	yes	characteristic
215	C₄	C₂⁵×C₁₀₇	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	—	2	2	no	—
218	C₂	—	4	8	no	—
219	C₂	#575×C₁₀₇	1	2	yes	direct product factor
220	C₂	C₂³×D₄×C₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₀₇	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₁₀₇
socle	C₂²×C₁₀₇	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₇ ◁ C₂³×C₄×C₁₀₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₁₀₇	C₂	semidirect
2. #575×C₁₀₇	C₂	direct (N × H)
3. C₂×#137×C₁₀₇	C₂	semidirect
4. C₂×AGL(1,ℤ/8)×C₁₀₇	C₂	semidirect
5. C₂×#135×C₁₀₇	C₂	semidirect
6. C₂²×M₁₆×C₁₀₇	C₂	semidirect
11. C₂×#41×C₁₀₇	C₂²	semidirect
12. C₂×#41×C₁₀₇	C₂²	semidirect
13. C₂²×Q₈×C₁₀₇	C₂²	semidirect
15. C₂²×D₄×C₁₀₇	C₂²	semidirect
18. #137×C₁₀₇	C₂²	semidirect
19. AGL(1,ℤ/8)×C₁₀₇	C₂²	semidirect
20. #135×C₁₀₇	C₂²	semidirect
21. C₂×Q₁₆×C₁₀₇	C₂²	semidirect
22. C₂×SD₁₆×C₁₀₇	C₂²	semidirect
23. C₂×D₈×C₁₀₇	C₂²	semidirect
24. C₂×M₁₆×C₁₀₇	C₂²	semidirect
25. C₂×M₁₆×C₁₀₇	C₂²	semidirect
26. C₂²×C₈×C₁₀₇	C₂²	semidirect
34. #41×C₁₀₇	C₂³	semidirect
38. C₂×Q₈×C₁₀₇	C₂³	semidirect
45. C₂×D₄×C₁₀₇	C₂³	semidirect
56. Q₁₆×C₁₀₇	C₂³	semidirect
57. SD₁₆×C₁₀₇	C₂³	semidirect
58. D₈×C₁₀₇	C₂³	semidirect
59. M₁₆×C₁₀₇	C₂³	semidirect
61. C₂×C₈×C₁₀₇	C₂³	semidirect
108. C₂×C₁₀₇	#575	direct (N × H)
110. C₂×#575	C₁₀₇	direct (N × H)
111. C₁₀₇	C₂×#575	direct (N × H)
112. C₂²×#41	C₂×C₁₀₇	semidirect
113. #575	C₂×C₁₀₇	direct (N × H)
114. C₂×#137	C₂×C₁₀₇	semidirect
115. C₂×AGL(1,ℤ/8)	C₂×C₁₀₇	semidirect
116. C₂×#135	C₂×C₁₀₇	semidirect
117. C₂²×M₁₆	C₂×C₁₀₇	semidirect
122. C₂×#41	C₂²×C₁₀₇	semidirect
123. C₂×#41	C₂²×C₁₀₇	semidirect
124. C₂²×Q₈	C₂²×C₁₀₇	semidirect
126. C₂²×D₄	C₂²×C₁₀₇	semidirect
129. #137	C₂²×C₁₀₇	semidirect
130. AGL(1,ℤ/8)	C₂²×C₁₀₇	semidirect
131. #135	C₂²×C₁₀₇	semidirect
132. C₂×Q₁₆	C₂²×C₁₀₇	semidirect
133. C₂×SD₁₆	C₂²×C₁₀₇	semidirect
134. C₂×D₈	C₂²×C₁₀₇	semidirect
135. C₂×M₁₆	C₂²×C₁₀₇	semidirect
136. C₂×M₁₆	C₂²×C₁₀₇	semidirect
137. C₂²×C₈	C₂²×C₁₀₇	semidirect
145. #41	C₂³×C₁₀₇	semidirect
149. C₂×Q₈	C₂³×C₁₀₇	semidirect
156. C₂×D₄	C₂³×C₁₀₇	semidirect
167. Q₁₆	C₂³×C₁₀₇	semidirect
168. SD₁₆	C₂³×C₁₀₇	semidirect
169. D₈	C₂³×C₁₀₇	semidirect
170. M₁₆	C₂³×C₁₀₇	semidirect
172. C₂×C₈	C₂³×C₁₀₇	semidirect
219. C₂	#575×C₁₀₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41×C₁₀₇	2
#575×C₁₀₇	2
C₂×#137×C₁₀₇	2
C₂×AGL(1,ℤ/8)×C₁₀₇	2
C₂×#135×C₁₀₇	2
C₂²×M₁₆×C₁₀₇	2
C₂×#575	107

Subgroups — C2×#575×C107 #28334263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#575×C₁₀₇ #28334263