Subgroups — #30761650

Order 17152.

1126 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863266	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	—	2	4	no	—
5	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	—
6	#16863266	—	2	4	no	—
7	#16863266	C₄	1	8	yes	complemented
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	16	no	—
15	C₂×C₄⋊Dic₆₇	—	2	8	no	—
16	C₂×C₄⋊Dic₆₇	—	2	4	no	—
17	C₂×C₄⋊Dic₆₇	—	2	8	no	—
18	C₂×C₄⋊Dic₆₇	—	4	8	no	—
19	C₂×C₄⋊Dic₆₇	—	4	8	no	—
20	C₂×C₄⋊Dic₆₇	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₆₇	D₄	1	2	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	—	2	4	no	—
24	C₂²×C₈×C₆₇	—	2	4	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₆₇	D₄	1	4	yes	—
27	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	D₄	1	4	yes	—
31	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
32	C₄×C₈×C₆₇	—	2	8	no	—
33	C₂²×Dic₆₇	—	2	8	no	—
34	C₂²×Dic₆₇	—	4	4	no	—
35	C₂²×Dic₆₇	—	4	4	no	—
36	C₂²×Dic₆₇	—	4	8	no	—
37	C₂²×Dic₆₇	—	4	8	no	—
38	C₄⋊Dic₆₇	—	2	16	no	—
39	C₄⋊Dic₆₇	—	4	32	no	—
40	C₄⋊Dic₆₇	—	4	8	no	—
41	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
43	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₆₇	D₈	1	4	yes	—
45	C₂²×C₄×C₆₇	SD₁₆	1	8	yes	—
46	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
47	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
53	C₂²×C₄×C₆₇	#41	1	4	yes	—
54	C₂×C₈×C₆₇	—	2	4	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	—	2	8	no	—
57	C₂×C₈×C₆₇	—	2	8	no	—
58	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	complemented
59	C₄²×C₆₇	C₄²	1	4	yes	—
60	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
62	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₆₇	#124	1	2	yes	—
65	C₂³×C₆₇	#127	1	1	yes	characteristic
66	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂³×C₆₇	C₂×D₈	1	2	yes	—
68	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
69	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₆₇	#95	1	8	yes	—
71	C₂×C₄×C₆₇	#102	1	16	yes	—
72	C₂×C₄×C₆₇	#103	1	16	yes	—
73	C₂×C₄×C₆₇	#106	1	8	yes	—
74	C₂×C₄×C₆₇	#107	1	8	yes	—
75	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₆₇	#117	1	2	yes	—
80	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
82	C₂×C₄×C₆₇	#121	1	4	yes	—
83	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
84	C₂×C₄×C₆₇	#125	1	2	yes	—
85	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
86	C₂×Dic₆₇	—	4	8	no	—
87	C₂×Dic₆₇	—	4	16	no	—
88	C₂×Dic₆₇	—	8	8	no	—
89	C₂×Dic₆₇	—	8	4	no	—
90	C₂×Dic₆₇	—	8	4	no	—
91	C₈×C₆₇	—	2	8	no	—
92	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
93	C₂²×C₆₇	#382	1	1	yes	characteristic
94	C₂²×C₆₇	#384	1	1	yes	characteristic
95	C₂²×C₆₇	#390	1	2	yes	—
96	C₂²×C₆₇	#391	1	2	yes	—
97	C₂²×C₆₇	C₂×#102	1	4	yes	—
98	C₂²×C₆₇	C₂×#103	1	4	yes	—
99	C₂²×C₆₇	C₂×#106	1	2	yes	—
100	C₂²×C₆₇	C₂×#107	1	2	yes	—
101	C₂²×C₆₇	#486	1	4	yes	—
102	C₂²×C₆₇	#487	1	4	yes	—
103	C₂²×C₆₇	#492	1	2	yes	—
104	C₂²×C₆₇	#493	1	1	yes	characteristic
105	C₂²×C₆₇	#494	1	1	yes	characteristic
106	C₂²×C₆₇	#498	1	2	yes	—
107	C₂²×C₆₇	#500	1	2	yes	—
108	C₄×C₆₇	#340	1	16	yes	complemented
109	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
110	C₄×C₆₇	#380	1	2	yes	—
111	C₄×C₆₇	#389	1	4	yes	—
112	Dic₆₇	—	8	8	no	—
113	#10841	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#340	1	2	yes	—
116	C₂×C₆₇	#1834	1	2	yes	—
117	C₂×C₆₇	#1844	1	1	yes	characteristic
118	C₂×C₆₇	#1845	1	1	yes	characteristic
119	C₂×C₆₇	#1956	1	2	yes	—
120	C₂×C₆₇	#1957	1	2	yes	—
121	C₂×C₆₇	#1990	1	4	yes	—
122	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
123	C₂×#384	—	67	2	no	—
124	C₂×#340	—	67	4	no	—
125	C₆₇	#10841	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₈	C₂×D₆₇	1	2	yes	—
127	C₂²×C₄⋊C₄	—	67	4	no	—
128	C₂²×C₄⋊C₄	—	134	4	no	—
129	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	4	no	—
131	C₂×C₄×C₈	Dic₆₇	1	4	yes	complemented
132	#384	—	67	8	no	—
133	#384	—	134	4	no	—
134	#340	—	134	8	no	—
135	C₂³×C₄	—	134	4	no	—
136	C₂³×C₄	D₂₆₈	1	2	yes	—
137	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	Dic₁₃₄	1	4	yes	—
141	C₂²×C₈	D₂₆₈	1	4	yes	—
142	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
143	C₂×C₄⋊C₄	—	268	8	no	—
144	C₂×C₄⋊C₄	—	268	8	no	—
145	C₂×C₄⋊C₄	—	67	8	no	—
146	C₂×C₄⋊C₄	—	134	16	no	—
147	C₂×C₄⋊C₄	—	134	4	no	—
148	C₂×C₄⋊C₄	—	134	8	no	—
149	C₂×C₄⋊C₄	—	134	8	no	—
150	C₂×C₄²	Dic₁₃₄	1	2	yes	—
151	C₂×C₄²	C₄×D₆₇	1	4	yes	—
152	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
153	C₂×C₄²	C₂²⋊D₆₇	1	4	yes	—
154	C₄×C₈	—	2	8	no	—
155	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
156	C₂²×C₄	—	268	4	no	—
157	C₂²×C₄	—	268	4	no	—
158	C₂²×C₄	—	268	8	no	—
159	C₂²×C₄	—	268	8	no	—
160	C₂²×C₄	—	134	8	no	—
161	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	8	yes	—
162	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
163	C₂²×C₄	D₅₃₆	1	4	yes	—
164	C₂²×C₄	#11761617	1	8	yes	—
165	C₂²×C₄	Dic₂₆₈	1	4	yes	—
166	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
167	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
169	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
170	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
171	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
172	C₂²×C₄	#11761628	1	4	yes	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	complemented
178	C₄⋊C₄	—	268	32	no	—
179	C₄⋊C₄	—	268	8	no	—
180	C₄⋊C₄	—	134	16	no	—
181	C₄²	C₄×Dic₆₇	1	4	yes	—
182	C₄²	C₂²⋊Dic₆₇	1	4	yes	—
183	C₄²	Dic₆₇⋊C₄	1	8	yes	—
184	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
185	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
186	C₂³	#14524031	1	2	yes	—
187	C₂³	#14524040	1	1	yes	characteristic
188	C₂³	#14524043	1	1	yes	characteristic
189	C₂³	C₂×D₅₃₆	1	2	yes	—
190	C₂³	C₂×#11761617	1	4	yes	—
191	C₂³	C₂×Dic₂₆₈	1	2	yes	—
192	C₂×C₄	—	268	8	no	—
193	C₂×C₄	—	268	16	no	—
194	C₂×C₄	—	536	4	no	—
195	C₂×C₄	—	536	4	no	—
196	C₂×C₄	—	536	8	no	—
197	C₂×C₄	C₆₇⋊#95	1	8	yes	—
198	C₂×C₄	#14523965	1	16	yes	—
199	C₂×C₄	#14523968	1	16	yes	—
200	C₂×C₄	#14523975	1	8	yes	—
201	C₂×C₄	#14523977	1	8	yes	—
202	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
203	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
204	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
206	C₂×C₄	#14524002	1	2	yes	—
207	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
208	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
209	C₂×C₄	#14524017	1	4	yes	—
210	C₂×C₄	#14524022	1	4	yes	—
211	C₂×C₄	#14524035	1	2	yes	—
212	C₂×C₄	#14524043	1	2	yes	—
213	C₈	—	2	8	no	—
214	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
215	C₂²	#16863262	1	1	yes	characteristic
216	C₂²	#16863266	1	1	yes	characteristic
217	C₂²	#16863298	1	2	yes	—
218	C₂²	#16863302	1	2	yes	—
219	C₂²	C₂×#14523965	1	4	yes	—
220	C₂²	C₂×#14523968	1	4	yes	—
221	C₂²	C₂×#14523975	1	2	yes	—
222	C₂²	C₂×#14523977	1	2	yes	—
223	C₂²	#16863743	1	4	yes	—
224	C₂²	#16863747	1	4	yes	—
225	C₂²	#16863769	1	2	yes	—
226	C₂²	#16863773	1	1	yes	characteristic
227	C₂²	#16863776	1	1	yes	characteristic
228	C₂²	#16863794	1	2	yes	—
229	C₂²	#16863801	1	2	yes	—
230	C₄	—	536	8	no	—
231	C₄	#16863159	1	16	yes	complemented
232	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
233	C₄	#16863256	1	2	yes	—
234	C₄	#16863294	1	4	yes	—
235	C₂	#22558439	1	1	yes	characteristic
236	C₂	C₂×#16863159	1	2	yes	—
237	C₂	#22560337	1	2	yes	—
238	C₂	#22560366	1	1	yes	characteristic
239	C₂	#22560369	1	1	yes	characteristic
240	C₂	#22561011	1	2	yes	—
241	C₂	#22561015	1	2	yes	—
242	C₂	#22561170	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	#10841
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
7. #16863266	C₄	semidirect
12. C₂×C₄×C₈×C₆₇	C₄	semidirect
58. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
108. C₄×C₆₇	#340	semidirect
125. C₆₇	#10841	semidirect
131. C₂×C₄×C₈	Dic₆₇	semidirect
177. C₂×C₈	C₄⋊Dic₆₇	semidirect
231. C₄	#16863159	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863266	2
C₂×#16863159	2
C₂²×C₄×C₈×C₆₇	2
#10841	67