Subgroups — C₆₇×#20507 #30699758

Order 17152.

1054 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₆₇	C₂	1	3	yes	maximal
2	C₆₇×#2276	C₂	1	2	yes	maximal
3	C₄×#107×C₆₇	C₂	1	3	yes	maximal
4	C₆₇×#1763	C₂	1	6	yes	maximal
5	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₆₇	C₂²	1	6	yes	—
8	C₄×Q₁₆×C₆₇	C₄	1	24	yes	complemented
9	C₂×#107×C₆₇	C₂²	1	3	yes	—
10	C₂×#103×C₆₇	C₂²	1	6	yes	—
11	C₂×#97×C₆₇	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
13	C₄×C₄⋊C₄×C₆₇	—	2	6	no	—
14	C₄×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
15	#376×C₆₇	—	2	2	no	—
16	#376×C₆₇	C₂²	1	1	yes	characteristic
17	#340×C₆₇	C₂²	1	6	yes	—
18	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
19	C₂×Q₁₆×C₆₇	C₂×C₄	1	12	yes	—
20	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
21	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₆₇	—	2	12	no	—
23	C₄×Q₈×C₆₇	C₂×C₄	1	24	yes	—
24	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
25	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
26	C₂×C₄²×C₆₇	—	2	6	no	—
27	C₂×C₄²×C₆₇	—	2	6	no	—
28	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
29	#107×C₆₇	C₂×C₄	1	12	yes	—
30	#103×C₆₇	C₂×C₄	1	24	yes	—
31	#97×C₆₇	—	2	6	no	—
32	C₄×C₈×C₆₇	C₂×C₄	1	12	yes	—
33	#95×C₆₇	—	2	6	no	—
34	#95×C₆₇	D₄	1	1	yes	characteristic
35	#95×C₆₇	D₄	1	3	yes	—
36	C₂×Q₈×C₆₇	—	2	6	no	—
37	C₂×Q₈×C₆₇	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₆₇	—	2	2	no	—
39	C₂²×C₄×C₆₇	—	2	6	no	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
43	Q₁₆×C₆₇	C₄²	1	16	yes	complemented
44	C₂×C₈×C₆₇	—	2	6	no	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
46	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
47	C₄⋊C₄×C₆₇	—	2	24	no	—
48	C₄⋊C₄×C₆₇	—	2	6	no	—
49	C₄⋊C₄×C₆₇	C₂²×C₄	1	12	yes	—
50	C₄²×C₆₇	—	2	24	no	—
51	C₄²×C₆₇	—	8	6	no	—
52	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
53	C₄²×C₆₇	#41	1	6	yes	—
54	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
55	Q₈×C₆₇	—	2	12	no	—
56	Q₈×C₆₇	C₂×C₄²	1	8	yes	—
57	C₂×C₄×C₆₇	—	2	12	no	—
58	C₂×C₄×C₆₇	—	2	3	no	—
59	C₂×C₄×C₆₇	—	2	6	no	—
60	C₂×C₄×C₆₇	—	2	3	no	—
61	C₂×C₄×C₆₇	—	2	12	no	—
62	C₂×C₄×C₆₇	—	4	6	no	—
63	C₂×C₄×C₆₇	—	4	6	no	—
64	C₂×C₄×C₆₇	C₄×D₄	1	6	yes	—
65	C₂×C₄×C₆₇	C₄×D₄	1	6	yes	—
66	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
67	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
68	C₈×C₆₇	—	2	6	no	—
69	C₈×C₆₇	C₂×C₄²	1	4	yes	—
70	C₂²×C₆₇	C₂×C₄×D₄	1	3	yes	—
71	C₂²×C₆₇	#576	1	1	yes	characteristic
72	C₂²×C₆₇	#578	1	3	yes	—
73	C₄×C₆₇	—	2	6	no	—
74	C₄×C₆₇	—	2	6	no	—
75	C₄×C₆₇	—	2	8	no	—
76	C₄×C₆₇	—	4	12	no	—
77	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
78	C₄×C₆₇	C₄×#41	1	3	yes	—
79	#20507	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₆₇	C₄²×D₄	1	1	yes	characteristic
81	C₂×C₆₇	#2950	1	3	yes	—
82	C₂×C₆₇	#2951	1	3	yes	—
83	C₂×C₄×Q₁₆	C₂×C₆₇	1	3	yes	—
84	#2276	C₂×C₆₇	1	2	yes	—
85	C₄×#107	C₂×C₆₇	1	3	yes	—
86	#1763	C₂×C₆₇	1	6	yes	—
87	#1726	C₂×C₆₇	1	1	yes	characteristic
88	C₆₇	#20507	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
90	C₂×C₄×Q₈	C₂²×C₆₇	1	6	yes	—
91	C₄×Q₁₆	C₄×C₆₇	1	24	yes	complemented
92	C₂×#107	C₂²×C₆₇	1	3	yes	—
93	C₂×#103	C₂²×C₆₇	1	6	yes	—
94	C₂×#97	C₂²×C₆₇	1	3	yes	—
95	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
96	C₄×C₄⋊C₄	—	2	6	no	—
97	C₄×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
98	#376	—	2	2	no	—
99	#376	C₂²×C₆₇	1	1	yes	characteristic
100	#340	C₂²×C₆₇	1	6	yes	—
101	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
102	C₂×Q₁₆	C₂×C₄×C₆₇	1	12	yes	—
103	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
104	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
105	C₄×Q₈	—	2	12	no	—
106	C₄×Q₈	C₂×C₄×C₆₇	1	24	yes	—
107	C₂×C₄⋊C₄	—	2	6	no	—
108	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
109	C₂×C₄²	—	2	6	no	—
110	C₂×C₄²	—	2	6	no	—
111	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
112	#107	C₂×C₄×C₆₇	1	12	yes	—
113	#103	C₂×C₄×C₆₇	1	24	yes	—
114	#97	—	2	6	no	—
115	C₄×C₈	C₂×C₄×C₆₇	1	12	yes	—
116	#95	—	2	6	no	—
117	#95	D₄×C₆₇	1	1	yes	characteristic
118	#95	D₄×C₆₇	1	3	yes	—
119	C₂×Q₈	—	2	6	no	—
120	C₂×Q₈	C₂²×C₄×C₆₇	1	12	yes	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	6	no	—
123	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
124	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
125	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
126	Q₁₆	C₄²×C₆₇	1	16	yes	complemented
127	C₂×C₈	—	2	6	no	—
128	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
129	C₂×C₈	C₂²×C₄×C₆₇	1	6	yes	—
130	C₄⋊C₄	—	2	24	no	—
131	C₄⋊C₄	—	2	6	no	—
132	C₄⋊C₄	C₂²×C₄×C₆₇	1	12	yes	—
133	C₄²	—	2	24	no	—
134	C₄²	—	8	6	no	—
135	C₄²	C₂²×C₄×C₆₇	1	6	yes	—
136	C₄²	#41×C₆₇	1	6	yes	—
137	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
138	Q₈	—	2	12	no	—
139	Q₈	C₂×C₄²×C₆₇	1	8	yes	—
140	C₂×C₄	—	2	12	no	—
141	C₂×C₄	—	2	3	no	—
142	C₂×C₄	—	2	6	no	—
143	C₂×C₄	—	2	3	no	—
144	C₂×C₄	—	2	12	no	—
145	C₂×C₄	—	4	6	no	—
146	C₂×C₄	—	4	6	no	—
147	C₂×C₄	C₄×D₄×C₆₇	1	6	yes	—
148	C₂×C₄	C₄×D₄×C₆₇	1	6	yes	—
149	C₂×C₄	C₂³×C₄×C₆₇	1	3	yes	—
150	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
151	C₈	—	2	6	no	—
152	C₈	C₂×C₄²×C₆₇	1	4	yes	—
153	C₂²	C₂×C₄×D₄×C₆₇	1	3	yes	—
154	C₂²	#576×C₆₇	1	1	yes	characteristic
155	C₂²	#578×C₆₇	1	3	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	2	6	no	—
158	C₄	—	2	6	no	—
159	C₄	—	4	12	no	—
160	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
161	C₄	C₄×#41×C₆₇	1	3	yes	—
162	C₂	C₄²×D₄×C₆₇	1	1	yes	characteristic
163	C₂	C₆₇×#2950	1	3	yes	—
164	C₂	C₆₇×#2951	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₄×Q₁₆×C₆₇	C₄	semidirect
43. Q₁₆×C₆₇	C₄²	semidirect
79. #20507	C₆₇	direct (N × H)
88. C₆₇	#20507	direct (N × H)
91. C₄×Q₁₆	C₄×C₆₇	semidirect
126. Q₁₆	C₄²×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×Q₁₆×C₆₇	2
C₆₇×#2276	2
C₄×#107×C₆₇	2
C₆₇×#1763	2
C₆₇×#1726	2
#20507	67

Subgroups — C67×#20507 #30699758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20507 #30699758