Subgroups — #546×C₇₁ #17985677

Order 4544.

334 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#124×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#123×C₇₁	C₂	1	2	yes	maximal, complemented
5	C₂²⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#121×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#121×C₇₁	C₂	1	2	yes	maximal, complemented
8	#120×C₇₁	C₂	1	2	yes	maximal, complemented
9	C₄×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂²⋊C₄×C₇₁	C₂	1	2	yes	maximal, complemented
12	C₂⁴×C₇₁	C₂²	1	2	yes	—
13	#41×C₇₁	—	2	4	no	—
14	C₂×Q₈×C₇₁	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄×C₇₁	—	2	4	no	—
16	C₂×D₄×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₇₁	C₂²	1	2	yes	—
18	C₂×D₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₇₁	C₂²	1	2	yes	complemented
21	C₂×D₄×C₇₁	C₂²	1	2	yes	complemented
22	C₂²×C₄×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	C₂²	1	2	yes	complemented
24	C₂²×C₄×C₇₁	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄×C₇₁	C₂²	1	2	yes	complemented
26	C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic, complemented
27	C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic, complemented
28	C₄⋊C₄×C₇₁	C₂²	1	2	yes	complemented
29	C₂²⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
30	C₂²⋊C₄×C₇₁	C₂²	1	2	yes	—
31	C₂²⋊C₄×C₇₁	C₂²	1	2	yes	complemented
32	C₂²⋊C₄×C₇₁	C₂²	1	2	yes	complemented
33	C₂²⋊C₄×C₇₁	C₂²	1	2	yes	complemented
34	C₂²⋊C₄×C₇₁	C₂²	1	1	yes	characteristic, complemented
35	C₂²⋊C₄×C₇₁	C₂²	1	2	yes	complemented
36	C₄²×C₇₁	C₂²	1	1	yes	characteristic, complemented
37	C₂³×C₇₁	—	2	4	no	—
38	C₂³×C₇₁	—	2	4	no	—
39	C₂³×C₇₁	—	4	2	no	—
40	C₂³×C₇₁	C₂³	1	2	yes	—
41	C₂³×C₇₁	C₂³	1	1	yes	characteristic
42	C₂³×C₇₁	C₂³	1	2	yes	—
43	C₂³×C₇₁	C₂³	1	1	yes	characteristic
44	Q₈×C₇₁	—	2	2	no	—
45	D₄×C₇₁	—	2	4	no	—
46	D₄×C₇₁	—	2	4	no	—
47	D₄×C₇₁	—	2	2	no	—
48	D₄×C₇₁	—	2	2	no	—
49	D₄×C₇₁	—	4	2	no	—
50	D₄×C₇₁	D₄	1	4	yes	complemented
51	C₂×C₄×C₇₁	—	2	4	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	—	4	2	no	—
55	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂³	1	2	yes	—
58	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂³	1	2	yes	—
60	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
62	C₂²×C₇₁	—	2	1	no	—
63	C₂²×C₇₁	—	2	2	no	—
64	C₂²×C₇₁	—	2	2	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	2	1	no	—
67	C₂²×C₇₁	—	2	2	no	—
68	C₂²×C₇₁	—	2	1	no	—
69	C₂²×C₇₁	—	2	2	no	—
70	C₂²×C₇₁	—	2	1	no	—
71	C₂²×C₇₁	—	2	2	no	—
72	C₂²×C₇₁	—	4	8	no	—
73	C₂²×C₇₁	C₂×D₄	1	4	yes	—
74	C₂²×C₇₁	#41	1	2	yes	complemented
75	C₂²×C₇₁	C₂⁴	1	1	yes	characteristic
76	C₄×C₇₁	—	2	1	no	—
77	C₄×C₇₁	—	2	1	no	—
78	C₄×C₇₁	—	2	2	no	—
79	C₄×C₇₁	—	2	1	no	—
80	C₄×C₇₁	—	2	1	no	—
81	C₄×C₇₁	—	2	2	no	—
82	C₄×C₇₁	C₂×D₄	1	2	yes	—
83	C₂×C₇₁	—	2	2	no	—
84	C₂×C₇₁	—	2	4	no	—
85	C₂×C₇₁	—	4	1	no	—
86	C₂×C₇₁	—	4	2	no	—
87	C₂×C₇₁	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₇₁	C₂×#41	1	1	yes	characteristic
89	C₂×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
90	C₇₁	#546	1	1	yes	characteristic, direct product factor, Sylow
91	#546	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂×#41	C₂×C₇₁	1	1	yes	characteristic, complemented
93	C₂²×D₄	C₂×C₇₁	1	1	yes	characteristic, complemented
94	#124	C₂×C₇₁	1	1	yes	characteristic, complemented
95	#123	C₂×C₇₁	1	2	yes	complemented
96	C₂²⋊Q₈	C₂×C₇₁	1	1	yes	characteristic, complemented
97	#121	C₂×C₇₁	1	1	yes	characteristic, complemented
98	#121	C₂×C₇₁	1	2	yes	complemented
99	#120	C₂×C₇₁	1	2	yes	complemented
100	C₄×D₄	C₂×C₇₁	1	1	yes	characteristic, complemented
101	C₄×D₄	C₂×C₇₁	1	1	yes	characteristic, complemented
102	C₂×C₂²⋊C₄	C₂×C₇₁	1	2	yes	complemented
103	C₂⁴	C₂²×C₇₁	1	2	yes	—
104	#41	—	2	4	no	—
105	C₂×Q₈	C₂²×C₇₁	1	1	yes	characteristic, complemented
106	C₂×D₄	—	2	4	no	—
107	C₂×D₄	C₂²×C₇₁	1	1	yes	characteristic
108	C₂×D₄	C₂²×C₇₁	1	2	yes	—
109	C₂×D₄	C₂²×C₇₁	1	1	yes	characteristic
110	C₂×D₄	C₂²×C₇₁	1	1	yes	characteristic
111	C₂×D₄	C₂²×C₇₁	1	2	yes	complemented
112	C₂×D₄	C₂²×C₇₁	1	2	yes	complemented
113	C₂²×C₄	C₂²×C₇₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²×C₇₁	1	2	yes	complemented
115	C₂²×C₄	C₂²×C₇₁	1	1	yes	characteristic, complemented
116	C₂²×C₄	C₂²×C₇₁	1	2	yes	complemented
117	C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic, complemented
118	C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic, complemented
119	C₄⋊C₄	C₂²×C₇₁	1	2	yes	complemented
120	C₂²⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
121	C₂²⋊C₄	C₂²×C₇₁	1	2	yes	—
122	C₂²⋊C₄	C₂²×C₇₁	1	2	yes	complemented
123	C₂²⋊C₄	C₂²×C₇₁	1	2	yes	complemented
124	C₂²⋊C₄	C₂²×C₇₁	1	2	yes	complemented
125	C₂²⋊C₄	C₂²×C₇₁	1	1	yes	characteristic, complemented
126	C₂²⋊C₄	C₂²×C₇₁	1	2	yes	complemented
127	C₄²	C₂²×C₇₁	1	1	yes	characteristic, complemented
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	—	4	2	no	—
131	C₂³	C₂³×C₇₁	1	2	yes	—
132	C₂³	C₂³×C₇₁	1	1	yes	characteristic
133	C₂³	C₂³×C₇₁	1	2	yes	—
134	C₂³	C₂³×C₇₁	1	1	yes	characteristic
135	Q₈	—	2	2	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	2	2	no	—
140	D₄	—	4	2	no	—
141	D₄	D₄×C₇₁	1	4	yes	complemented
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	C₂³×C₇₁	1	1	yes	characteristic
147	C₂×C₄	C₂³×C₇₁	1	1	yes	characteristic
148	C₂×C₄	C₂³×C₇₁	1	2	yes	—
149	C₂×C₄	C₂³×C₇₁	1	1	yes	characteristic
150	C₂×C₄	C₂³×C₇₁	1	2	yes	—
151	C₂×C₄	C₂³×C₇₁	1	1	yes	characteristic
152	C₂×C₄	C₂³×C₇₁	1	1	yes	characteristic
153	C₂²	—	2	1	no	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	8	no	—
164	C₂²	C₂×D₄×C₇₁	1	4	yes	—
165	C₂²	#41×C₇₁	1	2	yes	complemented
166	C₂²	C₂⁴×C₇₁	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	2	no	—
173	C₄	C₂×D₄×C₇₁	1	2	yes	—
174	C₂	—	2	2	no	—
175	C₂	—	2	4	no	—
176	C₂	—	4	2	no	—
177	C₂	—	4	1	no	—
178	C₂	C₂²×D₄×C₇₁	1	1	yes	characteristic
179	C₂	C₂×#41×C₇₁	1	1	yes	characteristic
180	C₂	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#41×C₇₁	C₂	semidirect
2. C₂²×D₄×C₇₁	C₂	semidirect
3. #124×C₇₁	C₂	semidirect
4. #123×C₇₁	C₂	semidirect
5. C₂²⋊Q₈×C₇₁	C₂	semidirect
6. #121×C₇₁	C₂	semidirect
7. #121×C₇₁	C₂	semidirect
8. #120×C₇₁	C₂	semidirect
9. C₄×D₄×C₇₁	C₂	semidirect
10. C₄×D₄×C₇₁	C₂	semidirect
11. C₂×C₂²⋊C₄×C₇₁	C₂	semidirect
14. C₂×Q₈×C₇₁	C₂²	semidirect
20. C₂×D₄×C₇₁	C₂²	semidirect
21. C₂×D₄×C₇₁	C₂²	semidirect
23. C₂²×C₄×C₇₁	C₂²	semidirect
24. C₂²×C₄×C₇₁	C₂²	semidirect
25. C₂²×C₄×C₇₁	C₂²	semidirect
26. C₄⋊C₄×C₇₁	C₂²	semidirect
27. C₄⋊C₄×C₇₁	C₂²	semidirect
28. C₄⋊C₄×C₇₁	C₂²	semidirect
31. C₂²⋊C₄×C₇₁	C₂²	semidirect
32. C₂²⋊C₄×C₇₁	C₂²	semidirect
33. C₂²⋊C₄×C₇₁	C₂²	semidirect
34. C₂²⋊C₄×C₇₁	C₂²	semidirect
35. C₂²⋊C₄×C₇₁	C₂²	semidirect
36. C₄²×C₇₁	C₂²	semidirect
50. D₄×C₇₁	D₄	semidirect
74. C₂²×C₇₁	#41	semidirect
90. C₇₁	#546	direct (N × H)
91. #546	C₇₁	direct (N × H)
92. C₂×#41	C₂×C₇₁	semidirect
93. C₂²×D₄	C₂×C₇₁	semidirect
94. #124	C₂×C₇₁	semidirect
95. #123	C₂×C₇₁	semidirect
96. C₂²⋊Q₈	C₂×C₇₁	semidirect
97. #121	C₂×C₇₁	semidirect
98. #121	C₂×C₇₁	semidirect
99. #120	C₂×C₇₁	semidirect
100. C₄×D₄	C₂×C₇₁	semidirect
101. C₄×D₄	C₂×C₇₁	semidirect
102. C₂×C₂²⋊C₄	C₂×C₇₁	semidirect
105. C₂×Q₈	C₂²×C₇₁	semidirect
111. C₂×D₄	C₂²×C₇₁	semidirect
112. C₂×D₄	C₂²×C₇₁	semidirect
114. C₂²×C₄	C₂²×C₇₁	semidirect
115. C₂²×C₄	C₂²×C₇₁	semidirect
116. C₂²×C₄	C₂²×C₇₁	semidirect
117. C₄⋊C₄	C₂²×C₇₁	semidirect
118. C₄⋊C₄	C₂²×C₇₁	semidirect
119. C₄⋊C₄	C₂²×C₇₁	semidirect
122. C₂²⋊C₄	C₂²×C₇₁	semidirect
123. C₂²⋊C₄	C₂²×C₇₁	semidirect
124. C₂²⋊C₄	C₂²×C₇₁	semidirect
125. C₂²⋊C₄	C₂²×C₇₁	semidirect
126. C₂²⋊C₄	C₂²×C₇₁	semidirect
127. C₄²	C₂²×C₇₁	semidirect
141. D₄	D₄×C₇₁	semidirect
165. C₂²	#41×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₇₁	2
C₂²×D₄×C₇₁	2
#124×C₇₁	2
#123×C₇₁	2
C₂²⋊Q₈×C₇₁	2
#121×C₇₁	2
#120×C₇₁	2
C₄×D₄×C₇₁	2
C₂×C₂²⋊C₄×C₇₁	2
#546	71

Subgroups — #546×C71 #17985677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #546×C₇₁ #17985677