Subgroups — #31015856

Order 17152.

5126 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863874	C₂	1	2	yes	maximal, complemented
2	Q₈×C₂²⋊Dic₆₇	C₂	1	16	yes	maximal, complemented
3	#22562795	C₂	1	6	yes	maximal, complemented
4	#22562592	C₂	1	6	yes	maximal
5	C₂⁴×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863874	—	2	16	no	—
7	C₂×Q₈×Dic₆₇	C₂²	1	16	yes	complemented
8	C₂×#14524002	C₂²	1	12	yes	—
9	#16863303	C₂²	1	48	yes	complemented
10	#16863259	C₂²	1	12	yes	—
11	#16863256	C₂²	1	24	yes	—
12	#16863252	C₂²	1	4	yes	—
13	C₄×C₂²⋊Dic₆₇	C₂²	1	24	yes	—
14	C₂³×Q₈×C₆₇	—	2	16	no	—
15	C₂³×Q₈×C₆₇	C₄	1	16	yes	complemented
16	C₂³×Q₈×C₆₇	C₂²	1	8	yes	—
17	C₂³×Q₈×C₆₇	C₂²	1	3	yes	—
18	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₆₇	C₂²	1	3	yes	—
20	Q₈×Dic₆₇	—	2	64	no	—
21	#14524002	—	2	48	no	—
22	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	—
23	C₂×C₂²⋊Dic₆₇	C₂³	1	16	yes	—
24	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
25	C₂×C₄×Dic₆₇	C₂³	1	24	yes	—
26	C₆₇⋊#95	C₂³	1	48	yes	—
27	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
28	C₂²×Q₈×C₆₇	—	2	128	no	—
29	C₂²×Q₈×C₆₇	—	2	64	no	—
30	C₂²×Q₈×C₆₇	—	2	24	no	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	64	yes	—
32	C₂²×Q₈×C₆₇	C₂×C₄	1	24	yes	—
33	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
34	C₂²×Q₈×C₆₇	D₄	1	16	yes	—
35	C₂²×Q₈×C₆₇	C₂³	1	12	yes	—
36	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
37	C₂³×C₄×C₆₇	—	2	24	no	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	24	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
42	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
43	C₄⋊Dic₆₇	—	2	48	no	—
44	C₂²⋊Dic₆₇	—	2	16	no	—
45	C₂²⋊Dic₆₇	—	2	32	no	—
46	C₄×Dic₆₇	—	2	48	no	—
47	C₂⁴×C₆₇	—	2	4	no	—
48	C₂⁴×C₆₇	—	2	8	no	—
49	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
50	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
52	C₂×Q₈×C₆₇	—	2	256	no	—
53	C₂×Q₈×C₆₇	—	2	96	no	—
54	C₂×Q₈×C₆₇	—	2	48	no	—
55	C₂×Q₈×C₆₇	C₂²⋊C₄	1	64	yes	complemented
56	C₂×Q₈×C₆₇	C₂²×C₄	1	48	yes	—
57	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
58	C₂×Q₈×C₆₇	C₂×D₄	1	32	yes	—
59	C₂²×C₄×C₆₇	—	2	12	no	—
60	C₂²×C₄×C₆₇	—	2	96	no	—
61	C₂²×C₄×C₆₇	—	2	48	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	48	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
68	C₂³×C₆₇	—	2	32	no	—
69	C₂³×C₆₇	—	2	8	no	—
70	C₂³×C₆₇	—	2	16	no	—
71	C₂³×C₆₇	—	2	4	no	—
72	C₂³×C₆₇	—	2	4	no	—
73	C₂³×C₆₇	—	2	4	no	—
74	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
75	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
76	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
77	C₂³×C₆₇	2₋¹⁺⁴	1	8	yes	—
78	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
79	Q₈×C₆₇	—	2	96	no	—
80	Q₈×C₆₇	C₂×C₂²⋊C₄	1	64	yes	—
81	C₂×C₄×C₆₇	—	2	24	no	—
82	C₂×C₄×C₆₇	—	2	12	no	—
83	C₂×C₄×C₆₇	—	2	96	no	—
84	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	48	yes	—
85	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
86	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	24	yes	—
88	C₂×Dic₆₇	—	2	16	no	—
89	C₂×Dic₆₇	—	2	32	no	—
90	C₂²×C₆₇	—	2	32	no	—
91	C₂²×C₆₇	—	2	4	no	—
92	C₂²×C₆₇	—	2	4	no	—
93	C₂²×C₆₇	—	2	4	no	—
94	C₂²×C₆₇	—	2	8	no	—
95	C₂²×C₆₇	—	2	16	no	—
96	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	4	yes	—
97	C₂²×C₆₇	#519	1	8	yes	complemented
98	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
99	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
100	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	4	yes	—
101	C₄×C₆₇	—	2	12	no	—
102	C₄×C₆₇	C₂²×C₂²⋊C₄	1	24	yes	—
103	Dic₆₇	—	4	16	no	—
104	#34173	—	67	1	no	maximal, Sylow
105	C₂×C₆₇	—	2	4	no	—
106	C₂×C₆₇	—	2	8	no	—
107	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#519	1	2	yes	—
109	C₂×C₆₇	#3486	1	4	yes	—
110	C₂⁴×Q₈	D₆₇	1	1	yes	characteristic
111	C₂×#519	—	67	2	no	—
112	Q₈×C₂²⋊C₄	—	67	16	no	—
113	#2310	—	67	6	no	—
114	#2280	—	67	6	no	—
115	C₆₇	#34173	1	1	yes	characteristic, complemented, Sylow
116	C₂³×Q₈	—	2	16	no	—
117	C₂³×Q₈	Dic₆₇	1	16	yes	complemented
118	C₂³×Q₈	C₂×D₆₇	1	3	yes	—
119	C₂³×Q₈	C₂×D₆₇	1	8	yes	—
120	C₂³×Q₈	C₂×D₆₇	1	1	yes	characteristic
121	C₂⁴×C₄	C₂×D₆₇	1	3	yes	—
122	#519	—	134	16	no	—
123	C₂×C₄×Q₈	—	67	16	no	—
124	C₂×#117	—	67	12	no	—
125	#391	—	67	48	no	—
126	#381	—	67	12	no	—
127	#380	—	67	24	no	—
128	#379	—	67	4	no	—
129	C₄×C₂²⋊C₄	—	67	24	no	—
130	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
131	C₂²×Q₈	—	2	128	no	—
132	C₂²×Q₈	—	2	64	no	—
133	C₂²×Q₈	—	2	24	no	—
134	C₂²×Q₈	C₂×Dic₆₇	1	64	yes	—
135	C₂²×Q₈	C₂×Dic₆₇	1	24	yes	—
136	C₂²×Q₈	C₂×Dic₆₇	1	8	yes	—
137	C₂²×Q₈	C₂²⋊D₆₇	1	16	yes	—
138	C₂²×Q₈	C₂²×D₆₇	1	12	yes	—
139	C₂²×Q₈	C₂²×D₆₇	1	4	yes	—
140	C₂³×C₄	—	2	24	no	—
141	C₂³×C₄	C₂×Dic₆₇	1	24	yes	—
142	C₂³×C₄	C₂²×D₆₇	1	12	yes	—
143	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
144	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
145	C₄×Q₈	—	134	64	no	—
146	#117	—	134	48	no	—
147	C₂×C₄⋊C₄	—	67	24	no	—
148	C₂×C₂²⋊C₄	—	67	8	no	—
149	C₂×C₂²⋊C₄	—	67	16	no	—
150	C₂×C₄²	—	67	24	no	—
151	#95	—	67	48	no	—
152	C₂⁴	—	2	4	no	—
153	C₂⁴	—	2	8	no	—
154	C₂⁴	C₂²×Dic₆₇	1	4	yes	—
155	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
156	C₂⁴	C₂³×D₆₇	1	2	yes	—
157	C₂×Q₈	—	2	256	no	—
158	C₂×Q₈	—	2	96	no	—
159	C₂×Q₈	—	2	48	no	—
160	C₂×Q₈	C₂²⋊Dic₆₇	1	64	yes	complemented
161	C₂×Q₈	C₂²×Dic₆₇	1	48	yes	—
162	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
163	C₂×Q₈	C₂×C₂²⋊D₆₇	1	32	yes	—
164	C₂²×C₄	—	2	12	no	—
165	C₂²×C₄	—	2	96	no	—
166	C₂²×C₄	—	2	48	no	—
167	C₂²×C₄	—	67	16	no	—
168	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
169	C₂²×C₄	C₂²×Dic₆₇	1	48	yes	—
170	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
171	C₂²×C₄	C₂×C₂²⋊D₆₇	1	24	yes	—
172	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
173	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
174	C₄⋊C₄	—	134	48	no	—
175	C₂²⋊C₄	—	134	16	no	—
176	C₂²⋊C₄	—	134	32	no	—
177	C₄²	—	134	48	no	—
178	C₂³	—	2	32	no	—
179	C₂³	—	2	8	no	—
180	C₂³	—	2	16	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	4	no	—
184	C₂³	C₂³×Dic₆₇	1	2	yes	—
185	C₂³	C₂³×Dic₆₇	1	4	yes	—
186	C₂³	C₂²×C₂²⋊D₆₇	1	4	yes	—
187	C₂³	#14524091	1	8	yes	—
188	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
189	Q₈	—	2	96	no	—
190	Q₈	C₂×C₂²⋊Dic₆₇	1	64	yes	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	96	no	—
194	C₂×C₄	—	134	16	no	—
195	C₂×C₄	—	134	32	no	—
196	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	48	yes	—
197	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
198	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
199	C₂×C₄	C₂²×C₂²⋊D₆₇	1	24	yes	—
200	C₂²	—	2	32	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	2	16	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	C₂²×C₂²⋊Dic₆₇	1	4	yes	—
207	C₂²	#16863874	1	8	yes	complemented
208	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
209	C₂²	C₂³×C₂²⋊D₆₇	1	2	yes	—
210	C₂²	C₂×#14524091	1	4	yes	—
211	C₄	—	2	12	no	—
212	C₄	—	268	16	no	—
213	C₄	C₂²×C₂²⋊Dic₆₇	1	24	yes	—
214	C₂	—	2	4	no	—
215	C₂	—	2	8	no	—
216	C₂	C₂³×C₂²⋊Dic₆₇	1	1	yes	characteristic
217	C₂	C₂×#16863874	1	2	yes	—
218	C₂	#22573860	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Fitting	C₂⁴×Q₈×C₆₇	C₂
socle, cosocle	C₂³×C₆₇	C₂⁵
prime radical	C₂⁵×C₆₇	C₂³
hypercenter	C₂⁴×Q₈	D₆₇
hypocenter	C₆₇	#34173
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16863874	C₂	semidirect
2. Q₈×C₂²⋊Dic₆₇	C₂	semidirect
3. #22562795	C₂	semidirect
7. C₂×Q₈×Dic₆₇	C₂²	semidirect
9. #16863303	C₂²	semidirect
15. C₂³×Q₈×C₆₇	C₄	semidirect
55. C₂×Q₈×C₆₇	C₂²⋊C₄	semidirect
97. C₂²×C₆₇	#519	semidirect
115. C₆₇	#34173	semidirect
117. C₂³×Q₈	Dic₆₇	semidirect
160. C₂×Q₈	C₂²⋊Dic₆₇	semidirect
207. C₂²	#16863874	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863874	2
Q₈×C₂²⋊Dic₆₇	2
#22562795	2
#22562592	2
C₂⁴×Q₈×C₆₇	2
#34173	67