Subgroups — C₆₇×#59875 #30739126

Order 17152.

1524 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2810	C₂	1	4	yes	maximal
2	C₆₇×#2568	C₂	1	6	yes	maximal
3	C₆₇×#2514	C₂	1	12	yes	maximal
4	C₆₇×#2355	C₂	1	6	yes	maximal
5	C₄×C₄⋊Q₈×C₆₇	C₂	1	3	yes	maximal
6	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
7	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
8	#401×C₆₇	C₂²	1	8	yes	—
9	#400×C₆₇	C₂²	1	24	yes	—
10	#391×C₆₇	C₂²	1	6	yes	—
11	#391×C₆₇	C₂²	1	12	yes	—
12	#391×C₆₇	C₂²	1	12	yes	—
13	#389×C₆₇	C₂²	1	6	yes	—
14	#389×C₆₇	C₂²	1	6	yes	—
15	#387×C₆₇	C₂²	1	24	yes	—
16	#384×C₆₇	C₂²	1	3	yes	—
17	#383×C₆₇	C₂²	1	8	yes	—
18	#382×C₆₇	C₂²	1	12	yes	—
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	12	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
24	C₄⋊Q₈×C₆₇	—	2	12	no	—
25	C₄×Q₈×C₆₇	—	2	24	no	—
26	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
27	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
28	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
29	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
30	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
31	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
32	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
36	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
37	#95×C₆₇	C₂³	1	24	yes	—
38	#95×C₆₇	C₂³	1	12	yes	—
39	#95×C₆₇	C₂³	1	12	yes	—
40	#95×C₆₇	C₂³	1	24	yes	—
41	#95×C₆₇	C₂³	1	8	yes	—
42	C₂×Q₈×C₆₇	—	2	12	no	—
43	C₂×Q₈×C₆₇	—	2	12	no	—
44	C₂×Q₈×C₆₇	—	2	12	no	—
45	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
46	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
48	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
49	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
50	C₄⋊C₄×C₆₇	—	2	12	no	—
51	C₄⋊C₄×C₆₇	—	2	24	no	—
52	C₄⋊C₄×C₆₇	—	2	12	no	—
53	C₄⋊C₄×C₆₇	—	2	12	no	—
54	C₄⋊C₄×C₆₇	—	2	12	no	—
55	C₄⋊C₄×C₆₇	—	4	6	no	—
56	C₄²×C₆₇	—	2	24	no	—
57	C₄²×C₆₇	—	2	6	no	—
58	C₄²×C₆₇	—	4	1	no	—
59	C₄²×C₆₇	—	4	8	no	—
60	C₄²×C₆₇	#41	1	12	yes	—
61	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
62	Q₈×C₆₇	—	4	24	no	—
63	C₂×C₄×C₆₇	—	2	3	no	—
64	C₂×C₄×C₆₇	—	2	3	no	—
65	C₂×C₄×C₆₇	—	2	12	no	—
66	C₂×C₄×C₆₇	—	2	12	no	—
67	C₂×C₄×C₆₇	—	2	12	no	—
68	C₂×C₄×C₆₇	—	2	3	no	—
69	C₂×C₄×C₆₇	—	2	12	no	—
70	C₂×C₄×C₆₇	—	2	12	no	—
71	C₂×C₄×C₆₇	—	2	12	no	—
72	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
73	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
74	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
75	C₂×C₄×C₆₇	2₋¹⁺⁴	1	6	yes	—
76	C₂²×C₆₇	C₂²×#41	1	3	yes	—
77	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	3	yes	—
78	C₂²×C₆₇	#585	1	1	yes	characteristic
79	C₄×C₆₇	—	2	6	no	—
80	C₄×C₆₇	—	4	12	no	—
81	C₄×C₆₇	—	4	12	no	—
82	C₄×C₆₇	—	4	1	no	—
83	C₄×C₆₇	#556	1	12	yes	—
84	#59875	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₆₇	C₂×#556	1	3	yes	—
86	C₂×C₆₇	#3507	1	3	yes	—
87	C₂×C₆₇	#3556	1	1	yes	characteristic
88	#2810	C₂×C₆₇	1	4	yes	—
89	#2568	C₂×C₆₇	1	6	yes	—
90	#2514	C₂×C₆₇	1	12	yes	—
91	#2355	C₂×C₆₇	1	6	yes	—
92	C₄×C₄⋊Q₈	C₂×C₆₇	1	3	yes	—
93	C₆₇	#59875	1	1	yes	characteristic, direct product factor, Sylow
94	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
95	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
96	#401	C₂²×C₆₇	1	8	yes	—
97	#400	C₂²×C₆₇	1	24	yes	—
98	#391	C₂²×C₆₇	1	6	yes	—
99	#391	C₂²×C₆₇	1	12	yes	—
100	#391	C₂²×C₆₇	1	12	yes	—
101	#389	C₂²×C₆₇	1	6	yes	—
102	#389	C₂²×C₆₇	1	6	yes	—
103	#387	C₂²×C₆₇	1	24	yes	—
104	#384	C₂²×C₆₇	1	3	yes	—
105	#383	C₂²×C₆₇	1	8	yes	—
106	#382	C₂²×C₆₇	1	12	yes	—
107	C₄×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
108	C₄×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
109	C₄×C₄⋊C₄	C₂²×C₆₇	1	12	yes	—
110	C₄³	C₂²×C₆₇	1	1	yes	characteristic
111	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
112	C₄⋊Q₈	—	2	12	no	—
113	C₄×Q₈	—	2	24	no	—
114	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
115	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
116	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
117	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
118	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
119	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
120	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
121	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
122	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
124	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
125	#95	C₂³×C₆₇	1	24	yes	—
126	#95	C₂³×C₆₇	1	12	yes	—
127	#95	C₂³×C₆₇	1	12	yes	—
128	#95	C₂³×C₆₇	1	24	yes	—
129	#95	C₂³×C₆₇	1	8	yes	—
130	C₂×Q₈	—	2	12	no	—
131	C₂×Q₈	—	2	12	no	—
132	C₂×Q₈	—	2	12	no	—
133	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
134	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
135	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
136	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
137	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
138	C₄⋊C₄	—	2	12	no	—
139	C₄⋊C₄	—	2	24	no	—
140	C₄⋊C₄	—	2	12	no	—
141	C₄⋊C₄	—	2	12	no	—
142	C₄⋊C₄	—	2	12	no	—
143	C₄⋊C₄	—	4	6	no	—
144	C₄²	—	2	24	no	—
145	C₄²	—	2	6	no	—
146	C₄²	—	4	1	no	—
147	C₄²	—	4	8	no	—
148	C₄²	#41×C₆₇	1	12	yes	—
149	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
150	Q₈	—	4	24	no	—
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	2	12	no	—
155	C₂×C₄	—	2	12	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	—	2	12	no	—
158	C₂×C₄	—	2	12	no	—
159	C₂×C₄	—	2	12	no	—
160	C₂×C₄	C₂×#41×C₆₇	1	6	yes	—
161	C₂×C₄	C₂×#41×C₆₇	1	6	yes	—
162	C₂×C₄	C₂×#41×C₆₇	1	6	yes	—
163	C₂×C₄	2₋¹⁺⁴×C₆₇	1	6	yes	—
164	C₂²	C₂²×#41×C₆₇	1	3	yes	—
165	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	3	yes	—
166	C₂²	#585×C₆₇	1	1	yes	characteristic
167	C₄	—	2	6	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	12	no	—
170	C₄	—	4	12	no	—
171	C₄	#556×C₆₇	1	12	yes	—
172	C₂	C₂×#556×C₆₇	1	3	yes	—
173	C₂	C₆₇×#3507	1	3	yes	—
174	C₂	C₆₇×#3556	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×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
84. #59875	C₆₇	direct (N × H)
93. C₆₇	#59875	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2810	2
C₆₇×#2568	2
C₆₇×#2514	2
C₆₇×#2355	2
C₄×C₄⋊Q₈×C₆₇	2
#59875	67

Subgroups — C67×#59875 #30739126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59875 #30739126