Subgroups — #30761653

Order 17152.

950 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863262	C₂	1	2	yes	maximal
2	C₂×#16863159	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	—
5	#16863262	—	2	4	no	—
6	#16863262	C₄	1	8	yes	complemented
7	C₂×C₆₇⋊#95	—	2	4	no	—
8	#16863159	—	2	8	no	—
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₇	—	2	4	no	—
12	C₂×C₄×C₈×C₆₇	C₄	1	4	yes	complemented
13	C₂³×Dic₆₇	—	2	4	no	—
14	C₂×C₄⋊Dic₆₇	—	2	8	no	—
15	C₂×C₄⋊Dic₆₇	—	2	4	no	—
16	C₂×C₄⋊Dic₆₇	—	2	8	no	—
17	C₂×C₄⋊Dic₆₇	C₂×C₄	1	8	yes	—
18	C₆₇⋊#95	—	2	16	no	—
19	C₂³×C₄×C₆₇	D₄	1	2	yes	—
20	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₆₇	—	2	4	no	—
22	C₂²×C₈×C₆₇	—	2	4	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₆₇	D₄	1	4	yes	—
25	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
26	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₆₇	D₄	1	4	yes	—
29	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
30	C₄×C₈×C₆₇	—	2	8	no	—
31	C₂²×Dic₆₇	—	2	8	no	—
32	C₂²×Dic₆₇	—	4	4	no	—
33	C₂²×Dic₆₇	—	4	4	no	—
34	C₂²×Dic₆₇	—	4	16	no	—
35	C₄⋊Dic₆₇	—	2	16	no	—
36	C₄⋊Dic₆₇	—	4	8	no	—
37	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	—	2	8	no	—
39	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
47	C₂²×C₄×C₆₇	#41	1	4	yes	—
48	C₂×C₈×C₆₇	—	2	4	no	—
49	C₂×C₈×C₆₇	—	2	4	no	—
50	C₂×C₈×C₆₇	—	2	8	no	—
51	C₂×C₈×C₆₇	—	2	8	no	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
53	C₄²×C₆₇	C₄²	1	4	yes	—
54	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
56	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
57	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂³×C₆₇	#124	1	2	yes	—
59	C₂³×C₆₇	#127	1	1	yes	characteristic
60	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₂³×C₆₇	#135	1	8	yes	—
62	C₂×C₄×C₆₇	—	2	16	no	—
63	C₂×C₄×C₆₇	—	2	8	no	—
64	C₂×C₄×C₆₇	#95	1	8	yes	—
65	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₆₇	#117	1	2	yes	—
70	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
71	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
72	C₂×C₄×C₆₇	#121	1	4	yes	—
73	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
74	C₂×C₄×C₆₇	#125	1	2	yes	—
75	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
76	C₂×Dic₆₇	—	4	8	no	—
77	C₂×Dic₆₇	—	4	16	no	—
78	C₂×Dic₆₇	—	8	8	no	—
79	C₂×Dic₆₇	—	8	8	no	—
80	C₈×C₆₇	—	2	8	no	—
81	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
82	C₂²×C₆₇	#382	1	1	yes	characteristic
83	C₂²×C₆₇	#384	1	1	yes	characteristic
84	C₂²×C₆₇	#390	1	2	yes	—
85	C₂²×C₆₇	#391	1	2	yes	—
86	C₂²×C₆₇	#416	1	8	yes	—
87	C₂²×C₆₇	#427	1	4	yes	—
88	C₂²×C₆₇	#488	1	8	yes	—
89	C₂²×C₆₇	#495	1	4	yes	—
90	C₂²×C₆₇	#499	1	4	yes	—
91	C₄×C₆₇	—	2	8	no	—
92	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
93	C₄×C₆₇	#380	1	2	yes	—
94	C₄×C₆₇	#389	1	4	yes	—
95	Dic₆₇	—	8	8	no	—
96	#10842	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
98	C₂×C₆₇	#1735	1	2	yes	—
99	C₂×C₆₇	#1835	1	2	yes	—
100	C₂×C₆₇	#1846	1	2	yes	—
101	C₂×C₆₇	#1958	1	4	yes	—
102	C₂×C₆₇	#1991	1	4	yes	—
103	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
104	C₂×#382	—	67	2	no	—
105	C₂×#340	—	67	4	no	—
106	C₆₇	#10842	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₈	C₂×D₆₇	1	2	yes	—
108	C₂²×C₄⋊C₄	—	67	4	no	—
109	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
110	C₂×C₄×C₈	—	2	4	no	—
111	C₂×C₄×C₈	Dic₆₇	1	4	yes	complemented
112	#382	—	67	8	no	—
113	#382	—	134	4	no	—
114	C₂×#95	—	134	4	no	—
115	#340	—	134	8	no	—
116	C₂³×C₄	—	134	4	no	—
117	C₂³×C₄	D₂₆₈	1	2	yes	—
118	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	Dic₁₃₄	1	4	yes	—
122	C₂²×C₈	D₂₆₈	1	4	yes	—
123	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
124	C₂×C₄⋊C₄	—	67	8	no	—
125	C₂×C₄⋊C₄	—	134	4	no	—
126	C₂×C₄⋊C₄	—	134	8	no	—
127	C₂×C₄⋊C₄	—	134	8	no	—
128	C₂×C₄²	Dic₁₃₄	1	2	yes	—
129	C₂×C₄²	C₄×D₆₇	1	4	yes	—
130	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₆₇	1	4	yes	—
132	C₄×C₈	—	2	8	no	—
133	#95	—	134	16	no	—
134	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	268	4	no	—
137	C₂²×C₄	—	268	4	no	—
138	C₂²×C₄	—	268	16	no	—
139	C₂²×C₄	—	134	8	no	—
140	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	8	yes	—
141	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
142	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
143	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
145	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
146	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
148	C₂²×C₄	#11761628	1	4	yes	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
154	C₄⋊C₄	—	268	8	no	—
155	C₄⋊C₄	—	134	16	no	—
156	C₄²	C₄×Dic₆₇	1	4	yes	—
157	C₄²	C₂²⋊Dic₆₇	1	4	yes	—
158	C₄²	Dic₆₇⋊C₄	1	8	yes	—
159	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
160	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
161	C₂³	#14524031	1	2	yes	—
162	C₂³	#14524040	1	1	yes	characteristic
163	C₂³	#14524043	1	1	yes	characteristic
164	C₂³	#14524065	1	8	yes	—
165	C₂×C₄	—	2	16	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	268	8	no	—
168	C₂×C₄	—	268	16	no	—
169	C₂×C₄	—	536	8	no	—
170	C₂×C₄	—	536	8	no	—
171	C₂×C₄	C₆₇⋊#95	1	8	yes	—
172	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
176	C₂×C₄	#14524002	1	2	yes	—
177	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
178	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
179	C₂×C₄	#14524017	1	4	yes	—
180	C₂×C₄	#14524022	1	4	yes	—
181	C₂×C₄	#14524035	1	2	yes	—
182	C₂×C₄	#14524043	1	2	yes	—
183	C₈	—	2	8	no	—
184	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
185	C₂²	#16863262	1	1	yes	characteristic
186	C₂²	#16863266	1	1	yes	characteristic
187	C₂²	#16863298	1	2	yes	—
188	C₂²	#16863302	1	2	yes	—
189	C₂²	#16863393	1	8	yes	—
190	C₂²	#16863434	1	4	yes	—
191	C₂²	#16863751	1	8	yes	—
192	C₂²	#16863780	1	4	yes	—
193	C₂²	#16863798	1	4	yes	—
194	C₄	—	2	8	no	—
195	C₄	—	536	8	no	—
196	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
197	C₄	#16863256	1	2	yes	—
198	C₄	#16863294	1	4	yes	—
199	C₂	#22558439	1	1	yes	characteristic
200	C₂	#22559926	1	2	yes	—
201	C₂	#22560340	1	2	yes	—
202	C₂	#22560373	1	2	yes	—
203	C₂	#22561020	1	4	yes	—
204	C₂	#22561174	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	#10842
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #16863262	C₄	semidirect
12. C₂×C₄×C₈×C₆₇	C₄	semidirect
106. C₆₇	#10842	semidirect
111. C₂×C₄×C₈	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863262	2
C₂×#16863159	2
C₂²×C₄×C₈×C₆₇	2
#10842	67