Subgroups — #30774461

Order 17152.

342 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22558317	C₂	1	4	yes	maximal
4	C₂×#499×C₆₇	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	#16863172	—	2	1	no	—
10	C₄×C₆₇⋊C₁₆	—	2	1	no	—
11	C₂×#125×C₆₇	—	2	1	no	—
12	#499×C₆₇	—	2	1	no	—
13	#499×C₆₇	—	2	1	no	—
14	#499×C₆₇	C₄	1	2	yes	—
15	#499×C₆₇	C₄	1	2	yes	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#106×C₆₇	—	2	1	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	—	2	2	no	—
20	#14523983	—	2	2	no	—
21	#14523983	Q₈	1	4	yes	—
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	#14523979	—	2	4	no	—
24	#14523979	—	4	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	#125×C₆₇	—	2	2	no	—
29	#125×C₆₇	—	2	2	no	—
30	#125×C₆₇	—	4	2	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	#107×C₆₇	—	2	2	no	—
35	#107×C₆₇	C₂×C₄	1	4	yes	—
36	#106×C₆₇	—	2	2	no	—
37	#106×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	#11761612	—	4	2	no	—
43	#11761612	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₆₇⋊C₁₆	—	2	4	no	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
52	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₆₇	#41	1	2	yes	—
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	4	4	no	—
59	C₄⋊C₄×C₆₇	—	4	4	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₆₇	D₈	1	2	yes	—
64	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	4	2	no	—
66	C₂×C₄×C₆₇	—	4	4	no	—
67	C₂×C₄×C₆₇	—	8	4	no	—
68	C₂×C₄×C₆₇	#102	1	2	yes	—
69	C₂×C₄×C₆₇	#102	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₈×C₆₇	#103	1	2	yes	—
77	C₈×C₆₇	#103	1	2	yes	—
78	C₈×C₆₇	C₄×Q₈	1	2	yes	—
79	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
80	C₆₇⋊C₈	—	8	2	no	—
81	C₂²×C₆₇	—	2	2	no	—
82	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
83	C₂²×C₆₇	#487	1	1	yes	characteristic
84	C₂²×C₆₇	#498	1	1	yes	characteristic
85	C₄×C₆₇	—	2	2	no	—
86	C₄×C₆₇	—	2	2	no	—
87	C₄×C₆₇	—	8	4	no	—
88	C₄×C₆₇	#391	1	1	yes	characteristic
89	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
90	C₄×C₆₇	#486	1	1	yes	characteristic
91	C₄×C₆₇	#500	1	1	yes	characteristic
92	#13289	—	67	1	no	maximal, Sylow
93	C₂×C₆₇	—	2	1	no	—
94	C₂×C₆₇	—	2	1	no	—
95	C₂×C₆₇	#1990	1	1	yes	characteristic
96	C₂×C₆₇	#2148	1	2	yes	—
97	C₂×#499	D₆₇	1	1	yes	characteristic
98	C₄×M₃₂	—	67	1	no	—
99	#1848	—	67	1	no	—
100	#1387	—	67	4	no	—
101	C₆₇	#13289	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	—	2	1	no	—
103	C₂×M₃₂	—	67	2	no	—
104	#499	—	2	1	no	—
105	#499	—	2	1	no	—
106	#499	Dic₆₇	1	2	yes	—
107	#499	Dic₆₇	1	2	yes	—
108	C₂×#108	—	67	2	no	—
109	C₂×#108	—	134	1	no	—
110	C₂×#107	C₂×D₆₇	1	2	yes	—
111	C₂×#106	—	2	1	no	—
112	#423	—	134	1	no	—
113	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
114	#346	—	134	1	no	—
115	C₄×C₁₆	—	134	1	no	—
116	C₂×M₁₆	—	134	2	no	—
117	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
119	#125	—	2	2	no	—
120	#125	—	2	2	no	—
121	#125	—	4	2	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	4	2	no	—
124	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
125	M₃₂	—	67	4	no	—
126	M₃₂	—	134	2	no	—
127	C₂×C₁₆	—	134	2	no	—
128	#108	—	268	2	no	—
129	#108	—	134	4	no	—
130	#107	—	2	2	no	—
131	#107	C₂×Dic₆₇	1	4	yes	—
132	#106	—	2	2	no	—
133	#106	—	2	2	no	—
134	C₄⋊C₈	—	268	2	no	—
135	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	#11761616	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
142	M₁₆	—	268	2	no	—
143	M₁₆	—	268	2	no	—
144	C₂×C₈	—	268	2	no	—
145	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
146	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
147	C₂×C₈	#11761615	1	2	yes	—
148	C₂×C₈	#11761618	1	2	yes	—
149	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
150	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₆₇	1	2	yes	—
152	C₂×C₈	#11761629	1	2	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
158	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
159	C₄²	#11761613	1	2	yes	—
160	C₁₆	—	134	4	no	—
161	C₂³	C₂×#11761616	1	1	yes	characteristic
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	#14523964	1	2	yes	—
166	C₂×C₄	#14523964	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
168	C₂×C₄	#14524032	1	1	yes	characteristic
169	C₂×C₄	#14524041	1	1	yes	characteristic
170	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
171	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
172	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
173	C₈	—	536	2	no	—
174	C₈	#14523967	1	2	yes	—
175	C₈	#14523967	1	2	yes	—
176	C₈	Q₈×Dic₆₇	1	2	yes	—
177	C₈	#14524019	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₂×#14523964	1	1	yes	characteristic
180	C₂²	#16863748	1	1	yes	characteristic
181	C₂²	#16863792	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	#16863303	1	1	yes	characteristic
186	C₄	C₂×#14523967	1	1	yes	characteristic
187	C₄	#16863744	1	1	yes	characteristic
188	C₄	#16863799	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	#22561171	1	1	yes	characteristic
192	C₂	#22561958	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	#13289
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
101. C₆₇	#13289	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523983	2
#22560381	2
#22558317	2
C₂×#499×C₆₇	2
#13289	67