Subgroups — #30770721

Order 17152.

554 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	2	yes	maximal
2	#22558303	C₂	1	4	yes	maximal, complemented
3	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	—	2	2	no	—
5	C₂×#14523979	—	2	2	no	—
6	C₂×#14523979	—	2	4	no	—
7	C₂×#14523979	C₄	1	8	yes	—
8	C₂×#14523979	C₂²	1	4	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₆₇	—	2	4	no	—
11	C₂×M₃₂×C₆₇	C₄	1	4	yes	—
12	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×#11761612	—	2	8	no	—
14	C₂×#11761612	—	2	4	no	—
15	C₂×#11761612	—	4	2	no	—
16	C₂×#11761612	—	4	4	no	—
17	C₂²×C₆₇⋊C₈	—	4	2	no	—
18	#14523979	—	2	8	no	—
19	#14523979	—	2	4	no	—
20	#14523979	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	D₄	1	4	yes	—
26	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	M₃₂×C₆₇	—	2	8	no	—
29	C₂×C₁₆×C₆₇	—	2	4	no	—
30	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
31	#11761612	—	2	8	no	—
32	#11761612	—	4	8	no	—
33	#11761612	—	4	4	no	—
34	C₂×C₆₇⋊C₈	—	4	8	no	—
35	C₂×C₆₇⋊C₈	—	4	4	no	—
36	C₂⁴×C₆₇	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₆₇	D₈	1	2	yes	—
40	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
41	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₄²	1	4	yes	—
44	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
46	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
47	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₆₇	—	2	4	no	—
53	C₂³×C₆₇	—	4	2	no	—
54	C₂³×C₆₇	#103	1	4	yes	—
55	C₂³×C₆₇	#107	1	2	yes	—
56	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₇	#95	1	4	yes	—
58	C₂×C₄×C₆₇	#102	1	4	yes	—
59	C₂×C₄×C₆₇	#102	1	4	yes	—
60	C₂×C₄×C₆₇	#103	1	4	yes	—
61	C₂×C₄×C₆₇	#106	1	4	yes	—
62	C₂×C₄×C₆₇	#107	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
67	C₈×C₆₇	#95	1	4	yes	—
68	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₆₇⋊C₈	—	4	8	no	—
72	C₂²×C₆₇	—	2	4	no	—
73	C₂²×C₆₇	—	2	2	no	—
74	C₂²×C₆₇	—	4	4	no	—
75	C₂²×C₆₇	#340	1	4	yes	—
76	C₂²×C₆₇	C₂×#103	1	2	yes	—
77	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
78	C₄×C₆₇	#340	1	4	yes	—
79	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
80	C₄×C₆₇	C₂×#102	1	2	yes	—
81	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
82	#12559	—	67	1	no	maximal, Sylow
83	C₂×C₆₇	—	2	4	no	—
84	C₂×C₆₇	—	2	2	no	—
85	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
86	C₂×C₆₇	#2161	1	2	yes	—
87	C₂²×M₃₂	D₆₇	1	1	yes	characteristic
88	C₂²×#108	—	67	2	no	—
89	#1381	—	67	4	no	—
90	C₆₇	#12559	1	1	yes	characteristic, complemented, Sylow
91	C₂²×M₁₆	—	134	2	no	—
92	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
93	C₂×M₃₂	—	2	4	no	—
94	C₂×M₃₂	Dic₆₇	1	4	yes	—
95	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
96	C₂×#108	—	67	8	no	—
97	C₂×#108	—	67	4	no	—
98	C₂×#108	—	134	2	no	—
99	C₂×#108	—	134	4	no	—
100	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
101	C₂×M₁₆	—	268	2	no	—
102	C₂×M₁₆	—	268	4	no	—
103	C₂×M₁₆	—	134	8	no	—
104	C₂×M₁₆	—	134	4	no	—
105	C₂²×C₈	—	268	2	no	—
106	C₂²×C₈	Dic₁₃₄	1	2	yes	—
107	C₂²×C₈	C₄×D₆₇	1	4	yes	—
108	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
110	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
111	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
112	M₃₂	—	2	8	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
115	#108	—	67	8	no	—
116	#108	—	134	8	no	—
117	#108	—	134	4	no	—
118	C₂⁴	Dic₂₆₈	1	1	yes	characteristic
119	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	4	yes	—
120	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
121	C₂²×C₄	D₅₃₆	1	2	yes	—
122	C₂²×C₄	#11761617	1	4	yes	—
123	C₂²×C₄	Dic₂₆₈	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
125	M₁₆	—	268	8	no	—
126	M₁₆	—	268	4	no	—
127	M₁₆	—	134	8	no	—
128	C₂×C₈	—	268	8	no	—
129	C₂×C₈	—	268	4	no	—
130	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
131	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
132	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
133	C₂×C₈	Dic₆₇⋊C₄	1	8	yes	—
134	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
135	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
136	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	#14523968	1	4	yes	—
142	C₂³	#14523977	1	2	yes	—
143	C₂³	C₂×Dic₂₆₈	1	1	yes	characteristic
144	C₂×C₄	C₆₇⋊#95	1	4	yes	—
145	C₂×C₄	#14523965	1	4	yes	—
146	C₂×C₄	#14523965	1	4	yes	—
147	C₂×C₄	#14523968	1	4	yes	—
148	C₂×C₄	#14523975	1	4	yes	—
149	C₂×C₄	#14523977	1	2	yes	—
150	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
152	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
153	C₂×C₄	C₂×#11761617	1	2	yes	—
154	C₈	—	268	8	no	—
155	C₈	C₆₇⋊#95	1	4	yes	—
156	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
157	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
158	C₈	C₂×Dic₆₇⋊C₄	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	4	no	—
162	C₂²	#16863159	1	4	yes	—
163	C₂²	C₂×#14523968	1	2	yes	—
164	C₂²	C₂×#14523977	1	1	yes	characteristic
165	C₄	#16863159	1	4	yes	—
166	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
167	C₄	C₂×#14523965	1	2	yes	—
168	C₄	C₂×#14523975	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	2	4	no	—
171	C₂	C₂×#16863159	1	1	yes	characteristic
172	C₂	#22562004	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₃₆
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂²×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂⁴×C₆₇	Q₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂²×M₃₂	D₆₇
hypocenter	C₆₇	#12559
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #22558303	C₂	semidirect
90. C₆₇	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523979	2
#22558303	2
C₂²×M₃₂×C₆₇	2
#12559	67