Subgroups — #30774383

Order 17152.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22560088	C₂	1	1	yes	maximal, characteristic
3	#22558324	C₂	1	2	yes	maximal
4	#22558311	C₂	1	2	yes	maximal
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863953	—	2	1	no	—
7	#16863801	—	2	1	no	—
8	#16863798	—	2	1	no	—
9	#16863798	C₄	1	2	yes	complemented
10	#16863794	C₄	1	2	yes	complemented
11	C₂×#14523979	C₂²	1	1	yes	characteristic
12	#16863434	—	2	1	no	—
13	#16863434	C₂²	1	2	yes	—
14	#16863434	C₂²	1	1	yes	characteristic
15	#16863351	—	2	1	no	—
16	#16863156	—	2	1	no	—
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#364×C₆₇	—	2	1	no	—
20	#364×C₆₇	—	2	1	no	—
21	C₂×#11761612	—	2	1	no	—
22	#14524043	—	2	2	no	—
23	#14524035	—	2	2	no	—
24	#14524022	—	4	2	no	—
25	C₄×Dic₁₃₄	—	4	2	no	—
26	#14524002	—	2	1	no	—
27	#14524002	—	2	2	no	—
28	#14523979	—	2	1	no	—
29	#14523979	Q₈	1	2	yes	complemented
30	#14523977	—	2	2	no	—
31	#14523977	—	2	2	no	—
32	#14523977	—	2	1	no	—
33	#14523975	—	2	2	no	—
34	#14523975	C₂×C₄	1	4	yes	—
35	#14523975	Q₈	1	2	yes	complemented
36	#14523953	—	4	1	no	—
37	C₈×Dic₆₇	—	4	1	no	—
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	M₃₂×C₆₇	—	2	2	no	—
42	M₃₂×C₆₇	—	2	2	no	—
43	C₂×C₁₆×C₆₇	—	2	2	no	—
44	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂×Dic₁₃₄	—	4	2	no	—
49	#11761612	—	2	2	no	—
50	#11761612	—	4	1	no	—
51	C₂×C₆₇⋊C₈	—	4	1	no	—
52	C₄⋊Dic₆₇	—	2	4	no	—
53	C₄⋊Dic₆₇	—	2	2	no	—
54	C₄⋊Dic₆₇	—	4	2	no	—
55	Dic₆₇⋊C₄	—	4	4	no	—
56	C₂²⋊Dic₆₇	—	4	1	no	—
57	C₂²⋊Dic₆₇	—	4	2	no	—
58	C₄×Dic₆₇	—	4	1	no	—
59	C₄×Dic₆₇	—	4	2	no	—
60	C₂²×C₄×C₆₇	—	2	1	no	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
65	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
67	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
73	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
74	C₄²×C₆₇	—	2	1	no	—
75	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₁₆×C₆₇	—	4	2	no	—
78	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₆₇	—	2	1	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	4	2	no	—
82	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×Dic₆₇	—	4	4	no	—
89	C₂×Dic₆₇	—	4	2	no	—
90	Dic₁₃₄	—	8	4	no	—
91	C₈×C₆₇	#103	1	1	yes	characteristic
92	C₈×C₆₇	#103	1	1	yes	characteristic
93	C₈×C₆₇	#103	1	1	yes	characteristic
94	C₈×C₆₇	#103	1	1	yes	characteristic
95	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
96	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₆₇	#124	1	1	yes	characteristic
98	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
99	C₆₇⋊C₈	—	4	2	no	—
100	C₂²×C₆₇	—	4	1	no	—
101	C₂²×C₆₇	#418	1	1	yes	characteristic
102	C₂²×C₆₇	#476	1	1	yes	characteristic
103	C₂²×C₆₇	#477	1	1	yes	characteristic
104	C₄×C₆₇	—	2	2	no	—
105	C₄×C₆₇	—	4	1	no	—
106	C₄×C₆₇	#391	1	1	yes	characteristic
107	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
108	C₄×C₆₇	#476	1	1	yes	characteristic
109	C₄×C₆₇	#477	1	1	yes	characteristic
110	Dic₆₇	—	8	4	no	—
111	Dic₆₇	—	8	2	no	—
112	#13271	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	—	2	1	no	—
114	C₂×C₆₇	—	2	1	no	—
115	C₂×C₆₇	—	2	1	no	—
116	C₂×C₆₇	#1987	1	1	yes	characteristic
117	#3160	—	67	1	no	—
118	C₂×#364	D₆₇	1	1	yes	characteristic
119	#1772	—	67	1	no	—
120	#1389	—	67	2	no	—
121	#1384	—	67	2	no	—
122	C₆₇	#13271	1	1	yes	characteristic, complemented, Sylow
123	#533	—	134	1	no	—
124	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
125	#500	—	134	1	no	—
126	#499	—	67	2	no	—
127	#499	—	134	1	no	—
128	#498	—	67	2	no	—
129	C₂×#108	—	67	1	no	—
130	#427	—	67	2	no	—
131	#427	—	67	1	no	—
132	#427	—	134	1	no	—
133	#405	—	134	1	no	—
134	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
135	#364	—	2	1	no	—
136	#364	—	2	1	no	—
137	#339	—	134	1	no	—
138	C₂×M₁₆	—	134	1	no	—
139	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
141	C₄⋊Q₈	—	134	2	no	—
142	#125	—	134	2	no	—
143	C₂²⋊Q₈	—	268	2	no	—
144	C₄×Q₈	—	268	2	no	—
145	#117	—	134	2	no	—
146	#117	—	134	1	no	—
147	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#108	—	67	2	no	—
152	#108	—	134	1	no	—
153	#107	—	134	2	no	—
154	#107	—	134	2	no	—
155	#107	—	134	1	no	—
156	#106	—	67	4	no	—
157	#106	—	67	2	no	—
158	#106	—	134	2	no	—
159	#97	—	268	1	no	—
160	C₄×C₈	—	268	1	no	—
161	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
162	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
165	C₂×Q₈	—	268	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
168	M₁₆	—	268	1	no	—
169	M₁₆	—	134	2	no	—
170	C₂×C₈	—	268	1	no	—
171	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
172	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	#11761617	1	1	yes	characteristic
174	C₂×C₈	#11761617	1	1	yes	characteristic
175	C₂×C₈	Dic₂₆₈	1	1	yes	characteristic
176	C₂×C₈	Dic₂₆₈	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
180	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
181	C₂×C₈	#11761627	1	1	yes	characteristic
182	C₂×C₈	#11761627	1	1	yes	characteristic
183	C₄⋊C₄	—	268	4	no	—
184	C₄⋊C₄	—	268	2	no	—
185	C₄⋊C₄	—	134	4	no	—
186	C₄⋊C₄	—	134	2	no	—
187	C₂²⋊C₄	—	268	2	no	—
188	C₂²⋊C₄	—	268	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	268	1	no	—
191	C₄²	—	268	2	no	—
192	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
193	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
194	C₁₆	—	4	2	no	—
195	C₂³	#14524071	1	1	yes	characteristic
196	Q₈	—	536	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	268	4	no	—
201	C₂×C₄	—	268	2	no	—
202	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#14524023	1	1	yes	characteristic
204	C₂×C₄	#14524023	1	1	yes	characteristic
205	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
206	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
207	C₂×C₄	#14524071	1	1	yes	characteristic
208	C₈	—	268	2	no	—
209	C₈	#14523968	1	1	yes	characteristic
210	C₈	#14523968	1	1	yes	characteristic
211	C₈	#14523968	1	1	yes	characteristic
212	C₈	#14523968	1	1	yes	characteristic
213	C₈	#14524008	1	1	yes	characteristic
214	C₈	#14524008	1	1	yes	characteristic
215	C₈	#14524030	1	1	yes	characteristic
216	C₈	#14524041	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#16863402	1	1	yes	characteristic
219	C₂²	#16863687	1	1	yes	characteristic
220	C₂²	#16863692	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	536	4	no	—
224	C₄	—	536	2	no	—
225	C₄	#16863301	1	1	yes	characteristic
226	C₄	C₂×#14523968	1	1	yes	characteristic
227	C₄	#16863687	1	1	yes	characteristic
228	C₄	#16863692	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#22561156	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863687
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂×C₆₇	#1987
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#13271
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
9. #16863798	C₄	semidirect
10. #16863794	C₄	semidirect
29. #14523979	Q₈	semidirect
35. #14523975	Q₈	semidirect
122. C₆₇	#13271	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22560088	2
#22558324	2
#22558311	2
C₂×#364×C₆₇	2
#13271	67