Subgroups — C₆₇×#2581 #22557012

Order 8576.

468 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×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#400×C₆₇	C₂	1	2	yes	maximal
4	#395×C₆₇	C₂	1	2	yes	maximal
5	#391×C₆₇	C₂	1	2	yes	maximal
6	#389×C₆₇	C₂	1	2	yes	maximal
7	#387×C₆₇	C₂	1	4	yes	maximal
8	#382×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₆₇	—	2	4	no	—
12	C₄×Q₈×C₆₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#95×C₆₇	C₂²	1	2	yes	—
23	#95×C₆₇	C₂²	1	2	yes	—
24	#95×C₆₇	C₂²	1	4	yes	—
25	#95×C₆₇	C₂²	1	4	yes	—
26	#95×C₆₇	C₂²	1	4	yes	—
27	C₂×Q₈×C₆₇	—	2	2	no	—
28	C₂×Q₈×C₆₇	—	2	2	no	—
29	C₂×Q₈×C₆₇	—	2	4	no	—
30	C₂×Q₈×C₆₇	—	2	2	no	—
31	C₂×Q₈×C₆₇	Q₈	1	4	yes	complemented
32	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
35	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₄⋊C₄×C₆₇	—	2	8	no	—
45	C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₄²×C₆₇	—	2	4	no	—
47	C₄²×C₆₇	—	2	4	no	—
48	C₄²×C₆₇	D₄	1	4	yes	—
49	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
50	Q₈×C₆₇	—	2	8	no	—
51	Q₈×C₆₇	—	2	8	no	—
52	C₂×C₄×C₆₇	—	2	2	no	—
53	C₂×C₄×C₆₇	—	2	2	no	—
54	C₂×C₄×C₆₇	—	2	2	no	—
55	C₂×C₄×C₆₇	—	2	2	no	—
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	4	no	—
61	C₂×C₄×C₆₇	—	2	4	no	—
62	C₂×C₄×C₆₇	—	2	4	no	—
63	C₂×C₄×C₆₇	—	2	4	no	—
64	C₂×C₄×C₆₇	—	2	4	no	—
65	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
69	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₆₇	#41	1	4	yes	—
71	C₂×C₄×C₆₇	#41	1	2	yes	—
72	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
75	C₂²×C₆₇	C₂×#41	1	2	yes	—
76	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
77	C₄×C₆₇	—	2	4	no	—
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	—	2	4	no	—
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	—	4	2	no	—
82	C₄×C₆₇	—	4	2	no	—
83	C₄×C₆₇	—	4	4	no	—
84	C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
85	C₂×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₆₇	#525	1	1	yes	characteristic
87	C₂×C₆₇	#536	1	1	yes	characteristic
88	C₂×C₆₇	#556	1	2	yes	—
89	C₂×C₆₇	#557	1	2	yes	—
90	#2581	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₆₇	#2581	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
93	C₂×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
94	#400	C₂×C₆₇	1	2	yes	—
95	#395	C₂×C₆₇	1	2	yes	—
96	#391	C₂×C₆₇	1	2	yes	—
97	#389	C₂×C₆₇	1	2	yes	—
98	#387	C₂×C₆₇	1	4	yes	—
99	#382	C₂×C₆₇	1	1	yes	characteristic
100	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
101	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
102	C₄×Q₈	—	2	4	no	—
103	C₄×Q₈	—	2	4	no	—
104	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
106	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
109	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
110	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
111	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
112	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
113	#95	C₂²×C₆₇	1	2	yes	—
114	#95	C₂²×C₆₇	1	2	yes	—
115	#95	C₂²×C₆₇	1	4	yes	—
116	#95	C₂²×C₆₇	1	4	yes	—
117	#95	C₂²×C₆₇	1	4	yes	—
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×Q₈	—	2	4	no	—
121	C₂×Q₈	—	2	2	no	—
122	C₂×Q₈	Q₈×C₆₇	1	4	yes	complemented
123	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
124	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
125	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
126	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
129	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
130	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	4	no	—
139	C₄²	D₄×C₆₇	1	4	yes	—
140	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
141	Q₈	—	2	8	no	—
142	Q₈	—	2	8	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
157	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
158	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
159	C₂×C₄	C₂×Q₈×C₆₇	1	4	yes	—
160	C₂×C₄	C₂×Q₈×C₆₇	1	2	yes	—
161	C₂×C₄	#41×C₆₇	1	4	yes	—
162	C₂×C₄	#41×C₆₇	1	2	yes	—
163	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
164	C₂²	C₂²×Q₈×C₆₇	1	1	yes	characteristic
165	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
166	C₂²	C₂×#41×C₆₇	1	2	yes	—
167	C₂²	2₋¹⁺⁴×C₆₇	1	2	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	4	no	—
171	C₄	—	2	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	4	no	—
175	C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
176	C₂	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
177	C₂	#525×C₆₇	1	1	yes	characteristic
178	C₂	#536×C₆₇	1	1	yes	characteristic
179	C₂	#556×C₆₇	1	2	yes	—
180	C₂	#557×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
31. C₂×Q₈×C₆₇	Q₈	semidirect
90. #2581	C₆₇	direct (N × H)
91. C₆₇	#2581	direct (N × H)
122. C₂×Q₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₆₇	2
#400×C₆₇	2
#395×C₆₇	2
#391×C₆₇	2
#389×C₆₇	2
#387×C₆₇	2
#382×C₆₇	2
#2581	67

Subgroups — C67×#2581 #22557012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2581 #22557012