Subgroups — #31011321

Order 17152.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570721	C₂	1	2	yes	maximal
2	#22562490	C₂	1	2	yes	maximal, complemented
3	#22562487	C₂	1	2	yes	maximal, complemented
4	#22562486	C₂	1	2	yes	maximal, complemented
5	#22562483	C₂	1	2	yes	maximal, complemented
6	#22561993	C₂	1	1	yes	maximal, characteristic
7	#22561991	C₂	1	2	yes	maximal
8	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic
10	#16863949	—	2	2	no	—
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16863800	C₂²	1	4	yes	—
13	#16863797	C₂²	1	4	yes	—
14	#16863796	—	2	2	no	—
15	#16863793	—	2	2	no	—
16	#16863434	C₂²	1	2	yes	—
17	#16863432	—	2	2	no	—
18	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
19	#16863224	C₂²	1	2	yes	—
20	#16863223	C₂²	1	4	yes	—
21	#16863222	C₂²	1	2	yes	—
22	#16863221	C₂²	1	4	yes	—
23	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
24	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	#427×C₆₇	C₂²	1	2	yes	—
29	#367×C₆₇	C₂²	1	2	yes	—
30	#366×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
32	#14524042	—	2	4	no	—
33	#14524034	—	2	4	no	—
34	#14524020	—	4	4	no	—
35	C₄×Dic₁₃₄	—	4	4	no	—
36	#14524002	—	2	2	no	—
37	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
38	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
39	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
41	#14523977	C₂³	1	4	yes	—
42	#14523976	—	2	4	no	—
43	#14523975	—	2	2	no	—
44	#14523974	—	2	4	no	—
45	C₈×Dic₆₇	D₄	1	1	yes	characteristic
46	C₈×Dic₆₇	D₄	1	1	yes	characteristic
47	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
48	C₈×Dic₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	#117×C₆₇	—	2	2	no	—
51	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
52	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
54	#107×C₆₇	C₂³	1	4	yes	—
55	#106×C₆₇	—	2	2	no	—
56	C₂×Dic₁₃₄	—	4	4	no	—
57	C₂²×Dic₆₇	D₈	1	2	yes	—
58	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	—
59	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₆₇	—	2	4	no	—
62	C₄⋊Dic₆₇	—	4	4	no	—
63	Dic₆₇⋊C₄	—	4	4	no	—
64	Dic₆₇⋊C₄	—	4	4	no	—
65	C₂²⋊Dic₆₇	—	4	2	no	—
66	C₄×Dic₆₇	—	4	2	no	—
67	C₄×Dic₆₇	D₈	1	2	yes	—
68	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
69	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
70	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
71	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₂²⋊C₄×C₆₇	—	4	2	no	—
79	C₄²×C₆₇	—	4	2	no	—
80	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
81	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	—	4	4	no	—
84	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₆₇	—	2	4	no	—
89	C₂×Dic₆₇	—	4	4	no	—
90	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
91	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
92	Dic₁₃₄	—	8	8	no	—
93	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
94	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
95	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
96	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
97	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	#500	1	2	yes	—
100	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
101	C₄×C₆₇	—	8	4	no	—
102	C₄×C₆₇	#500	1	2	yes	—
103	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
105	Dic₆₇	—	2	2	no	—
106	Dic₆₇	—	8	4	no	—
107	Dic₆₇	#508	1	4	yes	—
108	#33726	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	—	2	2	no	—
110	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#508	1	2	yes	—
112	#3160	—	67	2	no	—
113	#2254	—	67	2	no	—
114	#2253	—	67	2	no	—
115	#2253	—	67	2	no	—
116	#2252	—	67	2	no	—
117	#2158	—	67	1	no	—
118	#2158	—	67	2	no	—
119	#2158	D₆₇	1	1	yes	characteristic
120	C₂×C₄×C₁₆	—	67	1	no	—
121	C₆₇	#33726	1	1	yes	characteristic, complemented, Sylow
122	#533	—	134	2	no	—
123	C₂²×C₁₆	—	67	1	no	—
124	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
125	#500	—	67	4	no	—
126	#499	—	67	4	no	—
127	#499	—	134	2	no	—
128	#498	—	134	2	no	—
129	#427	—	67	2	no	—
130	#427	—	134	2	no	—
131	#427	C₂×D₆₇	1	2	yes	—
132	C₂×C₄×C₈	—	67	1	no	—
133	#367	—	67	2	no	—
134	#367	—	67	4	no	—
135	#367	C₂×D₆₇	1	2	yes	—
136	#366	—	67	2	no	—
137	#366	—	67	4	no	—
138	#366	C₂×D₆₇	1	2	yes	—
139	C₄×C₁₆	—	67	1	no	—
140	C₄×C₁₆	—	67	1	no	—
141	C₄×C₁₆	—	67	1	no	—
142	C₄×C₁₆	—	67	1	no	—
143	C₂²×C₈	—	67	1	no	—
144	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
145	C₄⋊Q₈	—	134	4	no	—
146	#125	—	134	4	no	—
147	C₂²⋊Q₈	—	268	4	no	—
148	C₄×Q₈	—	268	4	no	—
149	#117	—	2	2	no	—
150	#117	—	134	2	no	—
151	C₂×C₄²	—	67	1	no	—
152	C₂×C₁₆	—	67	4	no	—
153	C₂×C₁₆	—	67	1	no	—
154	C₂×C₁₆	—	67	1	no	—
155	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
156	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
157	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
158	#107	—	67	4	no	—
159	#107	—	134	4	no	—
160	#107	C₂²×D₆₇	1	4	yes	—
161	#106	—	2	2	no	—
162	#106	—	134	4	no	—
163	#106	—	134	2	no	—
164	C₄×C₈	—	67	1	no	—
165	C₄×C₈	—	67	1	no	—
166	C₄×C₈	—	67	1	no	—
167	C₄×C₈	—	67	1	no	—
168	C₂×Q₈	—	268	4	no	—
169	C₂²×C₄	—	67	2	no	—
170	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
171	C₂×C₈	—	67	4	no	—
172	C₂×C₈	—	67	1	no	—
173	C₂×C₈	—	67	1	no	—
174	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
175	C₂×C₈	Q₈×D₆₇	1	2	yes	—
176	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
177	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	268	4	no	—
180	C₄⋊C₄	—	268	4	no	—
181	C₄⋊C₄	—	268	4	no	—
182	C₄⋊C₄	—	134	4	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	268	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	—	268	2	no	—
187	C₄²	—	67	2	no	—
188	C₄²	—	67	1	no	—
189	C₄²	—	67	1	no	—
190	C₁₆	—	67	2	no	—
191	C₁₆	—	67	2	no	—
192	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
193	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
194	C₂³	D₈×D₆₇	1	1	yes	characteristic
195	Q₈	—	536	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	268	4	no	—
198	C₂×C₄	—	67	2	no	—
199	C₂×C₄	—	67	2	no	—
200	C₂×C₄	—	134	4	no	—
201	C₂×C₄	#14524042	1	2	yes	—
202	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
203	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
204	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
205	C₈	—	67	2	no	—
206	C₈	—	67	2	no	—
207	C₈	#14524042	1	2	yes	—
208	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
209	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	#16863800	1	2	yes	—
212	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
213	C₄	—	8	4	no	—
214	C₄	—	536	4	no	—
215	C₄	—	67	4	no	—
216	C₄	—	134	2	no	—
217	C₄	#16863800	1	2	yes	—
218	C₄	C₂×#14524042	1	1	yes	characteristic
219	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#16863800	1	1	yes	characteristic
222	C₂	#508×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2158	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#2158	D₆₇
hypocenter	C₆₇	#33726
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #22562490	C₂	semidirect
3. #22562487	C₂	semidirect
4. #22562486	C₂	semidirect
5. #22562483	C₂	semidirect
38. C₂×C₆₇⋊C₁₆	Q₈	semidirect
51. C₂×C₁₆×C₆₇	Q₈	semidirect
121. C₆₇	#33726	semidirect
155. C₂×C₁₆	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570721	2
#22562490	2
#22562487	2
#22562486	2
#22562483	2
#22561993	2
#22561991	2
C₂×C₁₆×Dic₆₇	2
C₆₇×#2158	2
#33726	67