Subgroups — #31009193

Order 17152.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569061	C₂	1	2	yes	maximal, complemented
2	#22562085	C₂	1	4	yes	maximal, complemented
3	D₁₆×Dic₆₇	C₂	1	4	yes	maximal, complemented
4	#22561993	C₂	1	1	yes	maximal, characteristic, complemented
5	#22561925	C₂	1	2	yes	maximal, complemented
6	#22561805	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863871	—	2	2	no	—
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	#16863508	—	2	4	no	—
11	D₈×Dic₆₇	C₂²	1	8	yes	complemented
12	#16863434	C₂²	1	2	yes	—
13	#16863401	—	2	2	no	—
14	#16863351	C₂²	1	1	yes	characteristic
15	#16863224	C₂²	1	2	yes	complemented
16	#16863222	C₂²	1	2	yes	complemented
17	#16863197	C₂²	1	8	yes	complemented
18	#16863171	C₂²	1	2	yes	complemented
19	C₁₆×Dic₆₇	C₂²	1	2	yes	complemented
20	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
21	C₂×D₁₆×C₆₇	C₄	1	8	yes	complemented
22	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
23	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	D₄	1	1	yes	characteristic
26	D₄×Dic₆₇	—	2	8	no	—
27	D₄×Dic₆₇	—	4	4	no	—
28	#14524002	—	2	2	no	—
29	#14524002	D₄	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₆₇	—	4	4	no	—
31	#14523983	—	2	2	no	—
32	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
33	#14523977	C₂³	1	4	yes	—
34	#14523975	—	2	2	no	—
35	#14523964	—	2	8	no	—
36	#14523953	D₄	1	2	yes	—
37	C₈×Dic₆₇	C₂³	1	2	yes	—
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂×D₈×C₆₇	—	2	2	no	—
40	C₂×D₈×C₆₇	—	2	4	no	—
41	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
42	C₂×D₈×C₆₇	C₂³	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	D₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
45	D₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
46	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×Dic₆₇	—	4	8	no	—
50	#11761612	—	2	2	no	—
51	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
52	C₄⋊Dic₆₇	—	2	4	no	—
53	C₄⋊Dic₆₇	D₈	1	2	yes	—
54	C₂²⋊Dic₆₇	—	4	2	no	—
55	C₂²⋊Dic₆₇	—	4	8	no	—
56	C₂²⋊Dic₆₇	D₈	1	2	yes	—
57	C₄×Dic₆₇	—	4	2	no	—
58	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
59	C₆₇⋊C₁₆	—	2	2	no	—
60	C₂⁴×C₆₇	—	4	2	no	—
61	C₂×D₄×C₆₇	—	2	8	no	—
62	C₂×D₄×C₆₇	—	2	4	no	—
63	C₂×D₄×C₆₇	—	4	2	no	—
64	C₂×D₄×C₆₇	—	4	4	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	D₈×C₆₇	—	2	8	no	—
67	D₈×C₆₇	—	2	4	no	—
68	D₈×C₆₇	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	#41	1	2	yes	—
72	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
74	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₆₇	—	4	4	no	—
76	C₂³×C₆₇	—	4	8	no	—
77	C₂³×C₆₇	—	16	4	no	—
78	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
79	D₄×C₆₇	—	2	8	no	—
80	D₄×C₆₇	—	4	4	no	—
81	D₄×C₆₇	—	4	8	no	—
82	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#135	1	2	yes	—
85	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
86	C₂×Dic₆₇	—	4	4	no	—
87	C₂×Dic₆₇	—	16	8	no	—
88	C₂×Dic₆₇	C₂×D₈	1	4	yes	—
89	C₈×C₆₇	C₄×D₄	1	2	yes	—
90	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
92	C₆₇⋊C₈	—	2	2	no	—
93	C₂²×C₆₇	—	2	2	no	—
94	C₂²×C₆₇	—	4	8	no	—
95	C₂²×C₆₇	—	8	8	no	—
96	C₂²×C₆₇	—	16	8	no	—
97	C₂²×C₆₇	C₄×D₈	1	2	yes	—
98	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
99	C₄×C₆₇	C₄×D₈	1	2	yes	—
100	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
101	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
102	Dic₆₇	—	2	4	no	—
103	Dic₆₇	—	8	4	no	—
104	#33456	—	67	1	no	maximal, Sylow
105	C₂×C₆₇	—	2	2	no	—
106	C₂×C₆₇	—	8	8	no	—
107	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
108	C₂×C₆₇	#3415	1	2	yes	—
109	C₂²×D₁₆	D₆₇	1	1	yes	characteristic
110	#2948	—	67	2	no	—
111	#2177	—	67	4	no	—
112	C₄×D₁₆	—	67	4	no	—
113	#2158	—	67	1	no	—
114	#2140	—	67	2	no	—
115	#2108	—	67	1	no	—
116	C₆₇	#33456	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₈	C₂×D₆₇	1	2	yes	—
118	#518	—	134	2	no	—
119	C₂×D₁₆	Dic₆₇	1	8	yes	complemented
120	C₂×D₁₆	C₂×D₆₇	1	2	yes	—
121	C₂×D₁₆	C₂×D₆₇	1	2	yes	—
122	C₂×M₃₂	—	67	1	no	—
123	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
124	#442	—	134	4	no	—
125	C₄×D₈	—	67	8	no	—
126	#427	—	67	2	no	—
127	#418	—	134	2	no	—
128	#405	—	67	1	no	—
129	#367	—	67	2	no	—
130	#366	—	67	2	no	—
131	#357	—	67	8	no	—
132	#346	—	67	2	no	—
133	C₄×C₁₆	—	67	2	no	—
134	C₂²×D₄	—	2	2	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	C₂×Dic₆₇	1	8	yes	complemented
138	C₂×D₈	C₂²×D₆₇	1	4	yes	—
139	C₂×M₁₆	—	67	1	no	—
140	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
141	C₄×D₄	—	268	4	no	—
142	C₄×D₄	—	134	8	no	—
143	#117	—	67	1	no	—
144	#117	—	134	2	no	—
145	C₂×C₂²⋊C₄	—	268	4	no	—
146	D₁₆	C₂×Dic₆₇	1	8	yes	complemented
147	D₁₆	C₂×Dic₆₇	1	8	yes	complemented
148	M₃₂	—	134	2	no	—
149	C₂×C₁₆	—	67	2	no	—
150	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
153	#107	—	67	4	no	—
154	#106	—	134	2	no	—
155	#102	—	134	8	no	—
156	#97	—	67	2	no	—
157	C₄×C₈	—	67	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	268	8	no	—
164	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
165	D₈	—	2	8	no	—
166	D₈	—	2	4	no	—
167	D₈	C₂²×Dic₆₇	1	8	yes	—
168	M₁₆	—	134	2	no	—
169	C₂×C₈	—	67	2	no	—
170	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
171	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
172	C₂×C₈	#11761627	1	2	yes	—
173	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
174	C₄⋊C₄	—	67	2	no	—
175	C₄⋊C₄	—	134	4	no	—
176	C₂²⋊C₄	—	268	2	no	—
177	C₂²⋊C₄	—	268	8	no	—
178	C₂²⋊C₄	—	67	2	no	—
179	C₄²	—	268	2	no	—
180	C₄²	—	67	2	no	—
181	C₁₆	—	134	2	no	—
182	C₁₆	C₂²×Dic₆₇	1	2	yes	—
183	C₁₆	C₂²×Dic₆₇	1	2	yes	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	16	4	no	—
187	C₂³	D₈×D₆₇	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	D₄	—	4	4	no	—
190	D₄	—	4	8	no	—
191	C₂×C₄	—	268	4	no	—
192	C₂×C₄	—	1072	8	no	—
193	C₂×C₄	—	67	4	no	—
194	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
195	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
196	C₂×C₄	#14524062	1	2	yes	—
197	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
198	C₈	—	134	2	no	—
199	C₈	D₄×Dic₆₇	1	2	yes	—
200	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
201	C₈	C₂×#11761627	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	D₈×Dic₆₇	1	2	yes	—
207	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
208	C₄	—	536	4	no	—
209	C₄	—	134	4	no	—
210	C₄	D₈×Dic₆₇	1	2	yes	—
211	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
212	C₄	C₂×#14524062	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×D₈×Dic₆₇	1	1	yes	characteristic
216	C₂	#22573346	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting, prime radical	C₂²×D₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×D₁₆	D₆₇
hypocenter	C₆₇	#33456
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. #22569061	C₂	semidirect
2. #22562085	C₂	semidirect
3. D₁₆×Dic₆₇	C₂	semidirect
4. #22561993	C₂	semidirect
5. #22561925	C₂	semidirect
6. #22561805	C₂	semidirect
11. D₈×Dic₆₇	C₂²	semidirect
15. #16863224	C₂²	semidirect
16. #16863222	C₂²	semidirect
17. #16863197	C₂²	semidirect
18. #16863171	C₂²	semidirect
19. C₁₆×Dic₆₇	C₂²	semidirect
21. C₂×D₁₆×C₆₇	C₄	semidirect
41. C₂×D₈×C₆₇	C₂×C₄	semidirect
44. D₁₆×C₆₇	C₂×C₄	semidirect
45. D₁₆×C₆₇	C₂×C₄	semidirect
46. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
116. C₆₇	#33456	semidirect
119. C₂×D₁₆	Dic₆₇	semidirect
137. C₂×D₈	C₂×Dic₆₇	semidirect
146. D₁₆	C₂×Dic₆₇	semidirect
147. D₁₆	C₂×Dic₆₇	semidirect
150. C₂×C₁₆	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569061	2
#22562085	2
D₁₆×Dic₆₇	2
#22561993	2
#22561925	2
#22561805	2
C₂²×D₁₆×C₆₇	2
#33456	67