Subgroups — #31008875

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568809	C₂	1	2	yes	maximal
2	#22561997	C₂	1	6	yes	maximal, complemented
3	#22561993	C₂	1	3	yes	maximal, complemented
4	C₂×#16863222	C₂	1	3	yes	maximal, complemented
5	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863879	—	2	2	no	—
7	#16863438	—	2	6	no	—
8	#16863434	C₂²	1	6	yes	—
9	C₂×#14523977	C₂²	1	6	yes	—
10	#16863224	C₂²	1	6	yes	—
11	#16863222	C₂²	1	3	yes	—
12	#16863222	C₂²	1	6	yes	—
13	#16863222	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	#504×C₆₇	C₄	1	8	yes	complemented
16	C₂×M₃₂×C₆₇	C₂²	1	3	yes	—
17	C₂²×C₁₆×C₆₇	C₂²	1	3	yes	—
18	Q₈×Dic₆₇	—	4	2	no	—
19	D₄×Dic₆₇	—	4	6	no	—
20	#14524002	—	2	6	no	—
21	C₂×C₄⋊Dic₆₇	—	2	6	no	—
22	#14523977	—	2	6	no	—
23	#14523977	C₂³	1	6	yes	—
24	#14523977	C₂³	1	2	yes	—
25	#14523975	—	2	6	no	—
26	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
27	#131×C₆₇	C₂×C₄	1	4	yes	—
28	#131×C₆₇	Q₈	1	4	yes	—
29	C₂×M₁₆×C₆₇	D₄	1	3	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
31	M₃₂×C₆₇	C₂×C₄	1	12	yes	complemented
32	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	complemented
33	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
35	C₂²×Dic₆₇	—	4	6	no	—
36	C₄⋊Dic₆₇	—	2	2	no	—
37	C₄⋊Dic₆₇	—	2	6	no	—
38	C₄⋊Dic₆₇	—	4	6	no	—
39	C₂²⋊Dic₆₇	—	4	6	no	—
40	C₄×Dic₆₇	—	4	6	no	—
41	#41×C₆₇	C₄⋊C₄	1	4	yes	—
42	#41×C₆₇	Q₁₆	1	4	yes	—
43	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
44	C₂×D₄×C₆₇	D₈	1	3	yes	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
46	M₁₆×C₆₇	C₄⋊C₄	1	12	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
48	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
49	C₂×C₈×C₆₇	C₂×Q₈	1	6	yes	—
50	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
52	C₁₆×C₆₇	C₂²×C₄	1	6	yes	—
53	C₂³×C₆₇	C₂×D₈	1	3	yes	—
54	Q₈×C₆₇	#107	1	4	yes	—
55	D₄×C₆₇	#107	1	12	yes	—
56	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
57	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
59	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
60	C₂×Dic₆₇	—	4	2	no	—
61	C₂×Dic₆₇	—	4	6	no	—
62	C₂×Dic₆₇	—	16	6	no	—
63	C₈×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
64	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₆₇	—	2	3	no	—
67	C₂²×C₆₇	—	2	3	no	—
68	C₂²×C₆₇	C₂×#107	1	6	yes	—
69	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
70	C₄×C₆₇	C₂×#107	1	6	yes	—
71	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
73	Dic₆₇	—	8	6	no	—
74	Dic₆₇	—	8	2	no	—
75	#33410	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	—	2	6	no	—
77	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
78	C₂×C₆₇	#3415	1	1	yes	characteristic
79	C₂×C₆₇	#3417	1	1	yes	characteristic
80	C₂×#504	D₆₇	1	1	yes	characteristic
81	#2913	—	67	2	no	—
82	#2159	—	67	6	no	—
83	#2158	—	67	3	no	—
84	C₂×#366	—	67	3	no	—
85	C₆₇	#33410	1	1	yes	characteristic, complemented, Sylow
86	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
87	#520	—	134	2	no	—
88	#504	Dic₆₇	1	8	yes	complemented
89	C₂×M₃₂	C₂×D₆₇	1	3	yes	—
90	C₂²×C₁₆	C₂×D₆₇	1	3	yes	—
91	#428	—	134	6	no	—
92	#427	—	67	6	no	—
93	C₂×#107	—	67	6	no	—
94	#367	—	67	6	no	—
95	#366	—	67	3	no	—
96	#366	—	67	6	no	—
97	#366	—	67	1	no	—
98	C₂×#41	D₂₆₈	1	1	yes	characteristic
99	#131	Dic₁₃₄	1	4	yes	—
100	#131	C₂×Dic₆₇	1	4	yes	—
101	C₂×M₁₆	D₂₆₈	1	3	yes	—
102	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
103	C₄×Q₈	—	268	2	no	—
104	C₄×D₄	—	268	6	no	—
105	#117	—	134	6	no	—
106	C₂×C₄⋊C₄	—	134	6	no	—
107	M₃₂	C₂×Dic₆₇	1	12	yes	complemented
108	C₂×C₁₆	C₂×Dic₆₇	1	12	yes	complemented
109	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
110	C₂×C₁₆	C₂²×D₆₇	1	3	yes	—
111	#107	—	67	6	no	—
112	#107	—	67	2	no	—
113	#107	—	134	6	no	—
114	#106	—	134	6	no	—
115	#41	C₄⋊Dic₆₇	1	4	yes	—
116	#41	Dic₂₆₈	1	4	yes	—
117	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
118	C₂×D₄	D₅₃₆	1	3	yes	—
119	C₂²×C₄	—	268	6	no	—
120	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
121	M₁₆	C₄⋊Dic₆₇	1	12	yes	—
122	C₂×C₈	C₂²×Dic₆₇	1	6	yes	—
123	C₂×C₈	C₂×D₂₆₈	1	3	yes	—
124	C₂×C₈	C₂×Dic₁₃₄	1	6	yes	—
125	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
126	C₄⋊C₄	—	268	6	no	—
127	C₄⋊C₄	—	134	2	no	—
128	C₄⋊C₄	—	134	6	no	—
129	C₂²⋊C₄	—	268	6	no	—
130	C₄²	—	268	6	no	—
131	C₁₆	C₂²×Dic₆₇	1	2	yes	—
132	C₁₆	C₂²×Dic₆₇	1	6	yes	—
133	C₂³	C₂×D₅₃₆	1	3	yes	—
134	Q₈	#14523977	1	4	yes	—
135	D₄	#14523977	1	12	yes	—
136	C₂×C₄	—	268	2	no	—
137	C₂×C₄	—	268	6	no	—
138	C₂×C₄	—	1072	6	no	—
139	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
140	C₂×C₄	C₂×D₅₃₆	1	3	yes	—
141	C₂×C₄	C₂×Dic₂₆₈	1	6	yes	—
142	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
143	C₈	C₂×C₄⋊Dic₆₇	1	6	yes	—
144	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
145	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×#14523977	1	6	yes	—
149	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
150	C₄	—	536	6	no	—
151	C₄	—	536	2	no	—
152	C₄	C₂×#14523977	1	6	yes	—
153	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
154	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
155	C₂	—	2	6	no	—
156	C₂	C₂²×#14523977	1	1	yes	characteristic
157	C₂	#22573350	1	1	yes	characteristic
158	C₂	#22573362	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₃₆
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#33410
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #22561997	C₂	semidirect
3. #22561993	C₂	semidirect
4. C₂×#16863222	C₂	semidirect
15. #504×C₆₇	C₄	semidirect
31. M₃₂×C₆₇	C₂×C₄	semidirect
32. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
85. C₆₇	#33410	semidirect
88. #504	Dic₆₇	semidirect
107. M₃₂	C₂×Dic₆₇	semidirect
108. C₂×C₁₆	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568809	2
#22561997	2
#22561993	2
C₂×#16863222	2
C₂×#504×C₆₇	2
#33410	67