Subgroups — #30743015

Order 17152.

208 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863217	C₂	1	1	yes	maximal, characteristic
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	1	no	—
5	C₂×#14523979	—	2	1	no	—
6	C₂×#14523979	—	2	1	no	—
7	#16863420	C₄	1	2	yes	—
8	#16863217	—	2	1	no	—
9	#16863217	C₄	1	2	yes	—
10	C₂×M₃₂×C₆₇	—	2	1	no	—
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#364×C₆₇	—	2	1	no	—
13	#364×C₆₇	C₄	1	2	yes	—
14	C₂×#11761612	—	2	2	no	—
15	#14523983	—	2	2	no	—
16	#14523983	—	4	1	no	—
17	C₂×C₆₇⋊C₁₆	—	4	1	no	—
18	#14523979	—	4	1	no	—
19	#14523979	—	4	1	no	—
20	#14523979	—	4	1	no	—
21	#14523979	—	4	1	no	—
22	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
23	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₆₇	—	2	2	no	—
27	M₃₂×C₆₇	—	4	1	no	—
28	C₂×C₁₆×C₆₇	—	4	1	no	—
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
33	#11761612	—	4	2	no	—
34	#11761612	—	4	2	no	—
35	C₂×C₆₇⋊C₈	—	4	2	no	—
36	C₆₇⋊C₁₆	—	4	2	no	—
37	C₂²×C₄×C₆₇	—	2	1	no	—
38	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂²⋊C₄	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	D₈	1	1	yes	characteristic
44	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
45	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
50	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
51	C₄²×C₆₇	—	2	1	no	—
52	C₄²×C₆₇	C₄²	1	1	yes	characteristic
53	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₆₇	—	4	2	no	—
55	C₂³×C₆₇	#101	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	1	no	—
57	C₂×C₄×C₆₇	—	2	1	no	—
58	C₂×C₄×C₆₇	—	4	2	no	—
59	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#100	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
65	C₈×C₆₇	#102	1	1	yes	characteristic
66	C₈×C₆₇	#102	1	1	yes	characteristic
67	C₈×C₆₇	#103	1	1	yes	characteristic
68	C₈×C₆₇	#103	1	1	yes	characteristic
69	C₈×C₆₇	#106	1	1	yes	characteristic
70	C₈×C₆₇	#106	1	1	yes	characteristic
71	C₈×C₆₇	#107	1	1	yes	characteristic
72	C₈×C₆₇	#107	1	1	yes	characteristic
73	C₆₇⋊C₈	—	8	2	no	—
74	C₂²×C₆₇	—	4	1	no	—
75	C₂²×C₆₇	#332	1	1	yes	characteristic
76	C₂²×C₆₇	#332	1	1	yes	characteristic
77	C₂²×C₆₇	#343	1	1	yes	characteristic
78	C₄×C₆₇	—	2	2	no	—
79	C₄×C₆₇	—	4	1	no	—
80	C₄×C₆₇	#331	1	1	yes	characteristic
81	C₄×C₆₇	#333	1	1	yes	characteristic
82	C₄×C₆₇	#340	1	1	yes	characteristic
83	C₄×C₆₇	#340	1	1	yes	characteristic
84	#7273	—	67	1	no	maximal, Sylow
85	C₂×C₆₇	—	2	1	no	—
86	C₂×C₆₇	—	2	1	no	—
87	C₂×C₆₇	—	2	1	no	—
88	C₂×C₆₇	#1296	1	1	yes	characteristic
89	C₂×#364	—	67	1	no	—
90	C₂×#364	D₆₇	1	1	yes	characteristic
91	#1848	—	67	1	no	—
92	C₆₇	#7273	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₃₂	—	2	1	no	—
94	C₂×M₃₂	—	134	1	no	—
95	C₂×#108	—	134	1	no	—
96	C₂×#108	—	134	1	no	—
97	#423	—	67	2	no	—
98	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
99	#364	—	2	1	no	—
100	#364	—	67	2	no	—
101	#364	—	134	1	no	—
102	#364	Dic₆₇	1	2	yes	—
103	C₂×M₁₆	—	134	2	no	—
104	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
106	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	4	1	no	—
109	M₃₂	—	268	1	no	—
110	M₃₂	—	134	2	no	—
111	C₂×C₁₆	—	4	1	no	—
112	C₂×C₁₆	—	268	1	no	—
113	#108	—	268	1	no	—
114	#108	—	268	1	no	—
115	#108	—	268	1	no	—
116	#108	—	268	1	no	—
117	C₄⋊C₈	—	268	2	no	—
118	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
119	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
124	M₁₆	—	268	2	no	—
125	M₁₆	—	268	2	no	—
126	C₂×C₈	—	268	2	no	—
127	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
129	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
130	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
131	C₂×C₈	#11761613	1	1	yes	characteristic
132	C₂×C₈	D₅₃₆	1	1	yes	characteristic
133	C₂×C₈	#11761615	1	1	yes	characteristic
134	C₂×C₈	#11761616	1	1	yes	characteristic
135	C₂×C₈	#11761617	1	1	yes	characteristic
136	C₂×C₈	#11761617	1	1	yes	characteristic
137	C₂×C₈	#11761618	1	1	yes	characteristic
138	C₂×C₈	Dic₂₆₈	1	1	yes	characteristic
139	C₄²	—	2	1	no	—
140	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
141	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
142	C₁₆	—	4	2	no	—
143	C₁₆	—	268	2	no	—
144	C₂³	#14523961	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
149	C₂×C₄	#14523959	1	1	yes	characteristic
150	C₂×C₄	#14523963	1	1	yes	characteristic
151	C₂×C₄	#14523965	1	1	yes	characteristic
152	C₂×C₄	#14523966	1	1	yes	characteristic
153	C₂×C₄	#14523968	1	1	yes	characteristic
154	C₈	—	536	2	no	—
155	C₈	#14523964	1	1	yes	characteristic
156	C₈	#14523965	1	1	yes	characteristic
157	C₈	#14523967	1	1	yes	characteristic
158	C₈	#14523968	1	1	yes	characteristic
159	C₈	#14523974	1	1	yes	characteristic
160	C₈	#14523975	1	1	yes	characteristic
161	C₈	#14523976	1	1	yes	characteristic
162	C₈	#14523977	1	1	yes	characteristic
163	C₂²	—	4	1	no	—
164	C₂²	#16863141	1	1	yes	characteristic
165	C₂²	#16863143	1	1	yes	characteristic
166	C₂²	#16863164	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	#16863139	1	1	yes	characteristic
170	C₄	#16863144	1	1	yes	characteristic
171	C₄	#16863158	1	1	yes	characteristic
172	C₄	#16863159	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#22558093	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863144
commutator	C₄²×C₆₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂×C₆₇	#1296
prime radical	C₂³×C₆₇	#101
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#7273
LCS	G ▷ C₄²×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
92. C₆₇	#7273	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
#22560381	2
C₂×#364×C₆₇	2
#7273	67