Subgroups — #30772256

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×#16863218	C₂	1	4	yes	maximal
2	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
3	#22560381	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523979	—	2	1	no	—
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863420	—	2	1	no	—
9	#16863218	—	2	8	no	—
10	#16863172	C₄	1	2	yes	complemented
11	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
12	C₂×#106×C₆₇	—	2	1	no	—
13	C₂×#106×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#384×C₆₇	—	2	1	no	—
16	C₂×#11761612	—	2	2	no	—
17	#14523983	D₄	1	4	yes	—
18	#14523983	Q₈	1	4	yes	—
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
20	#14523979	—	2	4	no	—
21	#14523979	—	4	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
23	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
27	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
28	#106×C₆₇	—	2	4	no	—
29	#106×C₆₇	—	4	2	no	—
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	#11761612	—	4	2	no	—
35	#11761612	—	4	2	no	—
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₆₇⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
38	C₂²×C₄×C₆₇	—	4	2	no	—
39	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
44	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄×C₆₇	—	4	2	no	—
50	C₄⋊C₄×C₆₇	—	4	2	no	—
51	C₄⋊C₄×C₆₇	—	8	4	no	—
52	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₆₇	SD₁₆	1	2	yes	—
55	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	4	no	—
57	C₂×C₄×C₆₇	—	8	4	no	—
58	C₂×C₄×C₆₇	—	8	2	no	—
59	C₂×C₄×C₆₇	#106	1	2	yes	—
60	C₂×C₄×C₆₇	#106	1	2	yes	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
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	—	16	2	no	—
78	C₄×C₆₇	#365	1	4	yes	complemented
79	C₄×C₆₇	#384	1	1	yes	characteristic
80	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
81	C₄×C₆₇	#492	1	1	yes	characteristic
82	C₄×C₆₇	#498	1	1	yes	characteristic
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	—	67	1	no	—
90	#1848	—	67	1	no	—
91	#1844	D₆₇	1	1	yes	characteristic
92	C₆₇	#12903	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₃₂	—	67	2	no	—
94	C₂×#108	—	67	2	no	—
95	C₂×#108	—	134	1	no	—
96	C₂×#106	—	2	1	no	—
97	C₂×#106	C₂×D₆₇	1	2	yes	—
98	#423	—	134	1	no	—
99	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
100	#384	—	2	1	no	—
101	#365	—	134	8	no	—
102	#346	—	67	2	no	—
103	C₄×C₁₆	—	67	2	no	—
104	C₂×M₁₆	—	134	2	no	—
105	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	4	2	no	—
109	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
110	M₃₂	—	67	4	no	—
111	M₃₂	—	67	4	no	—
112	C₂×C₁₆	—	67	4	no	—
113	#108	—	268	2	no	—
114	#108	—	134	4	no	—
115	#106	—	2	4	no	—
116	#106	—	4	2	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	C₄×D₆₇	1	1	yes	characteristic
121	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	#11761616	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
125	M₁₆	—	268	2	no	—
126	M₁₆	—	268	2	no	—
127	C₂×C₈	—	268	2	no	—
128	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
129	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
130	C₂×C₈	#11761615	1	2	yes	—
131	C₂×C₈	#11761616	1	2	yes	—
132	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
133	C₂×C₈	D₄×D₆₇	1	2	yes	—
134	C₂×C₈	Q₈×D₆₇	1	2	yes	—
135	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	8	4	no	—
139	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
140	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
141	C₄²	#11761615	1	2	yes	—
142	C₁₆	—	67	8	no	—
143	C₂³	C₂×#11761616	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	#14523974	1	2	yes	—
148	C₂×C₄	#14523974	1	2	yes	—
149	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
150	C₂×C₄	#14524039	1	1	yes	characteristic
151	C₂×C₄	#14524041	1	1	yes	characteristic
152	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
153	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
154	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
155	C₈	—	536	2	no	—
156	C₈	#14523974	1	2	yes	—
157	C₈	#14523974	1	2	yes	—
158	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
159	C₈	#14524042	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	C₂×#14523974	1	1	yes	characteristic
162	C₂²	#16863768	1	1	yes	characteristic
163	C₂²	#16863792	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	16	2	no	—
166	C₄	#16863218	1	4	yes	complemented
167	C₄	#16863265	1	1	yes	characteristic
168	C₄	C₂×#14523974	1	1	yes	characteristic
169	C₄	#16863770	1	1	yes	characteristic
170	C₄	#16863792	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#22560365	1	1	yes	characteristic
174	C₂	C₂×#16863218	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523974
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1844	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1844	D₆₇
hypocenter	C₆₇	#12903
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
10. #16863172	C₄	semidirect
11. C₄×C₆₇⋊C₁₆	C₄	semidirect
37. C₆₇⋊C₁₆	C₄⋊C₄	semidirect
78. C₄×C₆₇	#365	semidirect
92. C₆₇	#12903	semidirect
166. C₄	#16863218	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863218	2
C₄×#14523983	2
#22560381	2
C₆₇×#1844	2
#12903	67