Subgroups — #30772257

Order 17152.

334 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	C₂×#16863219	C₂	1	4	yes	maximal
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22560366	C₂	1	1	yes	maximal, characteristic
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523979	—	2	1	no	—
6	C₂×#14523979	C₂²	1	2	yes	—
7	C₂×#14523975	—	2	1	no	—
8	C₂×#14523975	C₂²	1	2	yes	—
9	#16863420	—	2	1	no	—
10	#16863266	—	2	1	no	—
11	#16863219	—	2	8	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#346×C₆₇	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₆₇	C₄	1	2	yes	complemented
16	C₂×#11761612	—	2	2	no	—
17	C₂×C₄⋊Dic₆₇	—	2	2	no	—
18	C₂×C₄⋊Dic₆₇	—	4	2	no	—
19	#14523979	—	2	4	no	—
20	#14523979	—	4	2	no	—
21	#14523975	—	2	4	no	—
22	#14523975	—	4	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
27	M₃₂×C₆₇	D₄	1	4	yes	—
28	M₃₂×C₆₇	Q₈	1	4	yes	—
29	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
34	C₂²×Dic₆₇	—	4	2	no	—
35	#11761612	—	4	2	no	—
36	#11761612	—	4	2	no	—
37	C₂×C₆₇⋊C₈	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	8	4	no	—
41	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
46	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₆₇	SD₁₆	1	2	yes	—
54	C₁₆×C₆₇	C₄⋊C₄	1	8	yes	complemented
55	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	4	no	—
57	C₂×C₄×C₆₇	#106	1	2	yes	—
58	C₂×C₄×C₆₇	#106	1	2	yes	—
59	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×Dic₆₇	—	8	4	no	—
66	C₂×Dic₆₇	—	8	2	no	—
67	C₈×C₆₇	#106	1	2	yes	—
68	C₈×C₆₇	#106	1	2	yes	—
69	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
71	C₆₇⋊C₈	—	8	2	no	—
72	C₂²×C₆₇	—	2	2	no	—
73	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
74	C₂²×C₆₇	#492	1	1	yes	characteristic
75	C₂²×C₆₇	#498	1	1	yes	characteristic
76	C₄×C₆₇	—	2	2	no	—
77	C₄×C₆₇	#365	1	4	yes	complemented
78	C₄×C₆₇	#384	1	1	yes	characteristic
79	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
80	C₄×C₆₇	#492	1	1	yes	characteristic
81	C₄×C₆₇	#498	1	1	yes	characteristic
82	Dic₆₇	—	16	2	no	—
83	#12903	—	67	1	no	maximal, Sylow
84	C₂×C₆₇	—	2	1	no	—
85	C₂×C₆₇	—	2	1	no	—
86	C₂×C₆₇	#1844	1	1	yes	characteristic
87	C₂×C₆₇	C₂×#365	1	2	yes	—
88	C₂×#365	—	67	4	no	—
89	C₄×M₃₂	D₆₇	1	1	yes	characteristic
90	#1848	—	67	1	no	—
91	#1844	—	67	1	no	—
92	C₆₇	#12903	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
94	C₂×#108	—	67	2	no	—
95	C₂×#108	—	134	1	no	—
96	C₂×#106	—	67	2	no	—
97	C₂×#106	—	134	1	no	—
98	#423	—	134	1	no	—
99	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
100	#384	—	134	1	no	—
101	#365	—	134	8	no	—
102	#346	Dic₆₇	1	2	yes	complemented
103	C₄×C₁₆	Dic₆₇	1	2	yes	complemented
104	C₂×M₁₆	—	134	2	no	—
105	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	268	2	no	—
108	C₂×C₄⋊C₄	—	134	2	no	—
109	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
110	M₃₂	Dic₁₃₄	1	4	yes	—
111	M₃₂	D₂₆₈	1	4	yes	—
112	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
113	#108	—	268	2	no	—
114	#108	—	134	4	no	—
115	#106	—	268	2	no	—
116	#106	—	134	4	no	—
117	C₄⋊C₈	—	268	2	no	—
118	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
119	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
121	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
122	C₂²×C₄	—	268	2	no	—
123	C₂²×C₄	#11761617	1	2	yes	—
124	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
125	M₁₆	—	268	2	no	—
126	M₁₆	—	268	2	no	—
127	C₂×C₈	—	268	2	no	—
128	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
129	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
130	C₂×C₈	#11761617	1	2	yes	—
131	C₂×C₈	#11761617	1	2	yes	—
132	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
133	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
134	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
135	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
136	C₄⋊C₄	—	268	2	no	—
137	C₄⋊C₄	—	268	2	no	—
138	C₄⋊C₄	—	536	4	no	—
139	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
140	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
141	C₄²	#11761617	1	2	yes	—
142	C₁₆	C₄⋊Dic₆₇	1	8	yes	complemented
143	C₂³	C₂×#11761617	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	536	4	no	—
146	C₂×C₄	—	536	2	no	—
147	C₂×C₄	#14523975	1	2	yes	—
148	C₂×C₄	#14523975	1	2	yes	—
149	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
150	C₂×C₄	#14524040	1	1	yes	characteristic
151	C₂×C₄	#14524043	1	1	yes	characteristic
152	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
153	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
154	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
155	C₈	—	536	2	no	—
156	C₈	#14523975	1	2	yes	—
157	C₈	#14523975	1	2	yes	—
158	C₈	C₂×C₄⋊Dic₆₇	1	2	yes	—
159	C₈	#14524043	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#14523975	1	1	yes	characteristic
162	C₂²	#16863769	1	1	yes	characteristic
163	C₂²	#16863794	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	1072	2	no	—
166	C₄	#16863219	1	4	yes	complemented
167	C₄	#16863266	1	1	yes	characteristic
168	C₄	C₂×#14523975	1	1	yes	characteristic
169	C₄	#16863769	1	1	yes	characteristic
170	C₄	#16863794	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#22560366	1	1	yes	characteristic
174	C₂	C₂×#16863219	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523975
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#12903
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
14. #346×C₆₇	C₄	semidirect
15. C₄×C₁₆×C₆₇	C₄	semidirect
54. C₁₆×C₆₇	C₄⋊C₄	semidirect
77. C₄×C₆₇	#365	semidirect
92. C₆₇	#12903	semidirect
102. #346	Dic₆₇	semidirect
103. C₄×C₁₆	Dic₆₇	semidirect
142. C₁₆	C₄⋊Dic₆₇	semidirect
166. C₄	#16863219	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863219	2
#22560381	2
#22560366	2
C₄×M₃₂×C₆₇	2
#12903	67