Subgroups — #30743003

Order 17152.

238 conjugacy classes of subgroups.

The subgroup lattice has 184 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	#22560366	C₂	1	1	yes	maximal, characteristic
3	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523983	—	2	1	no	—
5	C₂×#14523975	—	2	1	no	—
6	C₂×#14523975	—	2	1	no	—
7	#16863266	C₄	1	2	yes	—
8	#16863217	—	2	1	no	—
9	#16863217	C₄	1	2	yes	complemented
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	complemented
14	C₂×C₄⋊Dic₆₇	—	2	2	no	—
15	C₂×C₄⋊Dic₆₇	—	4	2	no	—
16	#14523983	—	2	2	no	—
17	#14523983	—	4	1	no	—
18	C₂×C₆₇⋊C₁₆	—	4	1	no	—
19	#14523975	—	4	1	no	—
20	#14523975	—	4	1	no	—
21	#14523975	—	4	1	no	—
22	#14523975	—	4	1	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	C₂²×Dic₆₇	—	4	2	no	—
34	C₄⋊Dic₆₇	—	4	2	no	—
35	C₄⋊Dic₆₇	—	4	2	no	—
36	C₄⋊Dic₆₇	—	8	4	no	—
37	C₆₇⋊C₁₆	—	4	2	no	—
38	C₂²×C₄×C₆₇	—	2	1	no	—
39	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
45	C₂×C₈×C₆₇	D₈	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	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
51	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
52	C₄²×C₆₇	—	2	1	no	—
53	C₄²×C₆₇	C₄²	1	1	yes	characteristic
54	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₁₆×C₆₇	—	4	2	no	—
56	C₂³×C₆₇	#100	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	4	no	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	1	no	—
60	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
61	C₂×C₄×C₆₇	#101	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
65	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
66	C₂×Dic₆₇	—	8	4	no	—
67	C₂×Dic₆₇	—	8	2	no	—
68	C₈×C₆₇	#102	1	1	yes	characteristic
69	C₈×C₆₇	#102	1	1	yes	characteristic
70	C₈×C₆₇	#103	1	1	yes	characteristic
71	C₈×C₆₇	#103	1	1	yes	characteristic
72	C₈×C₆₇	#106	1	1	yes	characteristic
73	C₈×C₆₇	#106	1	1	yes	characteristic
74	C₈×C₆₇	#107	1	1	yes	characteristic
75	C₈×C₆₇	#107	1	1	yes	characteristic
76	C₂²×C₆₇	—	2	2	no	—
77	C₂²×C₆₇	#331	1	1	yes	characteristic
78	C₂²×C₆₇	#333	1	1	yes	characteristic
79	C₂²×C₆₇	#343	1	1	yes	characteristic
80	C₄×C₆₇	—	2	2	no	—
81	C₄×C₆₇	—	2	2	no	—
82	C₄×C₆₇	#332	1	1	yes	characteristic
83	C₄×C₆₇	#332	1	1	yes	characteristic
84	C₄×C₆₇	#340	1	1	yes	characteristic
85	C₄×C₆₇	#340	1	1	yes	characteristic
86	Dic₆₇	—	16	2	no	—
87	#7268	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	1	no	—
89	C₂×C₆₇	—	2	1	no	—
90	C₂×C₆₇	#1296	1	1	yes	characteristic
91	C₂×C₆₇	#1359	1	2	yes	—
92	C₂×#364	—	67	1	no	—
93	C₂×#364	D₆₇	1	1	yes	characteristic
94	#1844	—	67	1	no	—
95	C₆₇	#7268	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₃₂	—	2	1	no	—
97	C₂×M₃₂	—	134	1	no	—
98	C₂×#106	—	134	1	no	—
99	C₂×#106	—	134	1	no	—
100	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
101	#384	—	67	2	no	—
102	#364	—	2	1	no	—
103	#364	—	67	2	no	—
104	#364	—	134	1	no	—
105	#364	Dic₆₇	1	2	yes	complemented
106	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
107	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	268	2	no	—
109	C₂×C₄⋊C₄	—	134	2	no	—
110	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
111	M₃₂	—	2	2	no	—
112	M₃₂	—	4	1	no	—
113	M₃₂	—	268	1	no	—
114	M₃₂	—	134	2	no	—
115	C₂×C₁₆	—	4	1	no	—
116	C₂×C₁₆	—	268	1	no	—
117	#106	—	268	1	no	—
118	#106	—	268	1	no	—
119	#106	—	268	1	no	—
120	#106	—	268	1	no	—
121	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
122	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
123	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
124	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	268	2	no	—
127	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
128	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
129	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
130	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
131	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
132	C₂×C₈	#11761613	1	1	yes	characteristic
133	C₂×C₈	D₅₃₆	1	1	yes	characteristic
134	C₂×C₈	#11761615	1	1	yes	characteristic
135	C₂×C₈	#11761616	1	1	yes	characteristic
136	C₂×C₈	#11761617	1	1	yes	characteristic
137	C₂×C₈	#11761617	1	1	yes	characteristic
138	C₂×C₈	#11761618	1	1	yes	characteristic
139	C₂×C₈	Dic₂₆₈	1	1	yes	characteristic
140	C₄⋊C₄	—	268	2	no	—
141	C₄⋊C₄	—	268	2	no	—
142	C₄⋊C₄	—	536	4	no	—
143	C₄²	—	2	1	no	—
144	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
145	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
146	C₁₆	—	4	2	no	—
147	C₁₆	—	268	2	no	—
148	C₂³	#14523959	1	1	yes	characteristic
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	536	4	no	—
153	C₂×C₄	—	536	2	no	—
154	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
155	C₂×C₄	#14523961	1	1	yes	characteristic
156	C₂×C₄	#14523963	1	1	yes	characteristic
157	C₂×C₄	#14523965	1	1	yes	characteristic
158	C₂×C₄	#14523966	1	1	yes	characteristic
159	C₂×C₄	#14523968	1	1	yes	characteristic
160	C₈	#14523964	1	1	yes	characteristic
161	C₈	#14523965	1	1	yes	characteristic
162	C₈	#14523967	1	1	yes	characteristic
163	C₈	#14523968	1	1	yes	characteristic
164	C₈	#14523974	1	1	yes	characteristic
165	C₈	#14523975	1	1	yes	characteristic
166	C₈	#14523976	1	1	yes	characteristic
167	C₈	#14523977	1	1	yes	characteristic
168	C₂²	—	2	2	no	—
169	C₂²	#16863139	1	1	yes	characteristic
170	C₂²	#16863144	1	1	yes	characteristic
171	C₂²	#16863164	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	1072	2	no	—
175	C₄	#16863141	1	1	yes	characteristic
176	C₄	#16863143	1	1	yes	characteristic
177	C₄	#16863158	1	1	yes	characteristic
178	C₄	#16863159	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#22558093	1	1	yes	characteristic
182	C₂	#22558249	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863139
commutator	C₄²×C₆₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	#331
prime radical	C₂³×C₆₇	#100
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#7268
LCS	G ▷ C₄²×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
9. #16863217	C₄	semidirect
13. #364×C₆₇	C₄	semidirect
95. C₆₇	#7268	semidirect
105. #364	Dic₆₇	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
#22560366	2
C₂×#364×C₆₇	2
#7268	67