Subgroups — C₂³×#14523979 #31719520

Order 17152.

2846 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761612	C₂	1	2	yes	maximal
2	C₂²×#14523979	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	—	2	14	no	—
5	C₂²×#11761612	C₂²	1	28	yes	—
6	C₂³×C₆₇⋊C₈	—	2	2	no	—
7	C₂×#14523979	C₂²	1	112	yes	direct product factor
8	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₄	1	14	yes	—
10	C₂³×C₈×C₆₇	C₄	1	2	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	14	yes	—
12	C₂×#11761612	—	2	84	no	—
13	C₂×#11761612	C₂³	1	56	yes	—
14	C₂²×C₆₇⋊C₈	—	2	28	no	—
15	#14523979	C₂³	1	64	yes	direct product factor
16	C₂⁵×C₆₇	Q₈	1	1	yes	characteristic
17	C₂³×C₄×C₆₇	C₂×C₄	1	7	yes	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₆₇	D₄	1	7	yes	—
20	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	Q₈	1	7	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	84	yes	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	28	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	28	yes	—
26	#11761612	—	2	56	no	—
27	#11761612	C₂⁴	1	16	yes	—
28	C₂×C₆₇⋊C₈	—	2	56	no	—
29	C₂⁴×C₆₇	—	2	8	no	—
30	C₂⁴×C₆₇	C₄⋊C₄	1	7	yes	—
31	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂⁴×C₆₇	C₂×Q₈	1	7	yes	—
33	C₂²×C₄×C₆₇	C₄⋊C₄	1	42	yes	—
34	C₂²×C₄×C₆₇	C₄⋊C₄	1	7	yes	—
35	C₂²×C₄×C₆₇	C₄⋊C₄	1	7	yes	—
36	C₂²×C₄×C₆₇	C₂²×C₄	1	21	yes	—
37	C₂²×C₄×C₆₇	C₂²×C₄	1	7	yes	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	7	yes	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
40	C₂²×C₄×C₆₇	C₂×Q₈	1	21	yes	—
41	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	56	yes	—
43	C₂×C₈×C₆₇	C₂²×C₄	1	56	yes	—
44	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
45	C₂³×C₆₇	—	2	28	no	—
46	C₂³×C₆₇	—	2	28	no	—
47	C₂³×C₆₇	#108	1	8	yes	complemented
48	C₂³×C₆₇	C₂×C₄⋊C₄	1	21	yes	—
49	C₂³×C₆₇	C₂×C₄⋊C₄	1	7	yes	—
50	C₂³×C₆₇	C₂²×Q₈	1	7	yes	—
51	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	42	yes	—
52	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	21	yes	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	21	yes	—
54	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
55	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
56	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
57	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
58	C₂×C₄×C₆₇	C₂²×Q₈	1	7	yes	—
59	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
60	C₈×C₆₇	C₂³×C₄	1	16	yes	—
61	C₆₇⋊C₈	—	2	16	no	—
62	C₂²×C₆₇	—	2	42	no	—
63	C₂²×C₆₇	—	2	14	no	—
64	C₂²×C₆₇	C₂×#108	1	28	yes	complemented
65	C₂²×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
66	C₂²×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
67	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
68	C₄×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
69	C₄×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
70	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
71	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
72	C₂³×#108	—	67	1	no	maximal, Sylow
73	C₂×C₆₇	—	2	7	no	—
74	C₂×C₆₇	—	2	1	no	—
75	C₂×C₆₇	C₂²×#108	1	14	yes	complemented
76	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
77	C₂³×M₁₆	—	67	2	no	—
78	C₂⁴×C₈	D₆₇	1	1	yes	characteristic
79	C₂²×#108	—	67	28	no	—
80	C₆₇	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
82	C₂²×M₁₆	—	67	28	no	—
83	C₂²×M₁₆	—	134	14	no	—
84	C₂³×C₈	—	134	2	no	—
85	C₂³×C₈	Dic₆₇	1	2	yes	—
86	C₂³×C₈	Dic₆₇	1	14	yes	—
87	C₂³×C₈	C₂×D₆₇	1	14	yes	—
88	C₂×#108	—	67	112	no	—
89	C₂⁵	Dic₁₃₄	1	1	yes	characteristic
90	C₂³×C₄	Dic₁₃₄	1	7	yes	—
91	C₂³×C₄	D₂₆₈	1	7	yes	—
92	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
93	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
94	C₂³×C₄	C₂×Dic₆₇	1	7	yes	—
95	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
96	C₂×M₁₆	—	67	56	no	—
97	C₂×M₁₆	—	134	84	no	—
98	C₂²×C₈	—	134	28	no	—
99	C₂²×C₈	C₂×Dic₆₇	1	84	yes	—
100	C₂²×C₈	C₂×Dic₆₇	1	28	yes	—
101	C₂²×C₈	C₂²×D₆₇	1	28	yes	—
102	#108	—	67	64	no	—
103	C₂⁴	—	2	8	no	—
104	C₂⁴	C₄⋊Dic₆₇	1	7	yes	—
105	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
106	C₂⁴	C₂×Dic₁₃₄	1	7	yes	—
107	C₂²×C₄	C₄⋊Dic₆₇	1	42	yes	—
108	C₂²×C₄	C₄⋊Dic₆₇	1	7	yes	—
109	C₂²×C₄	C₄⋊Dic₆₇	1	7	yes	—
110	C₂²×C₄	C₂²×Dic₆₇	1	21	yes	—
111	C₂²×C₄	C₂²×Dic₆₇	1	7	yes	—
112	C₂²×C₄	C₂×D₂₆₈	1	7	yes	—
113	C₂²×C₄	C₂×D₂₆₈	1	21	yes	—
114	C₂²×C₄	C₂×Dic₁₃₄	1	21	yes	—
115	C₂²×C₄	C₂³×D₆₇	1	7	yes	—
116	M₁₆	—	67	16	no	—
117	M₁₆	—	134	56	no	—
118	C₂×C₈	—	134	56	no	—
119	C₂×C₈	C₂²×Dic₆₇	1	56	yes	—
120	C₂×C₈	C₂²×Dic₆₇	1	56	yes	—
121	C₂×C₈	C₂³×D₆₇	1	8	yes	—
122	C₂³	—	2	28	no	—
123	C₂³	—	2	28	no	—
124	C₂³	#14523979	1	8	yes	direct product factor
125	C₂³	C₂×C₄⋊Dic₆₇	1	21	yes	—
126	C₂³	C₂×C₄⋊Dic₆₇	1	7	yes	—
127	C₂³	C₂²×Dic₁₃₄	1	7	yes	—
128	C₂×C₄	C₂×C₄⋊Dic₆₇	1	42	yes	—
129	C₂×C₄	C₂×C₄⋊Dic₆₇	1	21	yes	—
130	C₂×C₄	C₂×C₄⋊Dic₆₇	1	21	yes	—
131	C₂×C₄	C₂³×Dic₆₇	1	7	yes	—
132	C₂×C₄	C₂³×Dic₆₇	1	7	yes	—
133	C₂×C₄	C₂²×D₂₆₈	1	7	yes	—
134	C₂×C₄	C₂²×D₂₆₈	1	7	yes	—
135	C₂×C₄	C₂²×Dic₁₃₄	1	7	yes	—
136	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
137	C₈	—	134	16	no	—
138	C₈	C₂³×Dic₆₇	1	16	yes	—
139	C₂²	—	2	42	no	—
140	C₂²	—	2	14	no	—
141	C₂²	C₂×#14523979	1	28	yes	direct product factor
142	C₂²	C₂²×C₄⋊Dic₆₇	1	7	yes	—
143	C₂²	C₂²×C₄⋊Dic₆₇	1	7	yes	—
144	C₂²	C₂³×Dic₁₃₄	1	1	yes	characteristic
145	C₄	C₂²×C₄⋊Dic₆₇	1	7	yes	—
146	C₄	C₂²×C₄⋊Dic₆₇	1	7	yes	—
147	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
148	C₄	C₂³×D₂₆₈	1	1	yes	characteristic
149	C₂	—	2	7	no	—
150	C₂	—	2	1	no	—
151	C₂	C₂²×#14523979	1	14	yes	direct product factor
152	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₆₈
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle	C₂⁴×C₆₇	C₄⋊C₄
prime radical	C₂⁵×C₆₇	Q₈
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂³×#108
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#14523979	C₂	direct (N × H)
7. C₂×#14523979	C₂²	direct (N × H)
15. #14523979	C₂³	direct (N × H)
47. C₂³×C₆₇	#108	semidirect
64. C₂²×C₆₇	C₂×#108	semidirect
75. C₂×C₆₇	C₂²×#108	semidirect
80. C₆₇	C₂³×#108	semidirect
124. C₂³	#14523979	direct (N × H)
141. C₂²	C₂×#14523979	direct (N × H)
151. C₂	C₂²×#14523979	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11761612	2
C₂²×#14523979	2
C₂⁴×C₈×C₆₇	2
C₂³×#108	67

Subgroups — C23×#14523979 #31719520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523979 #31719520