Subgroups — #30745706

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562607	C₂	1	2	yes	maximal
2	#22558077	C₂	1	4	yes	maximal
3	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863264	C₄	1	4	yes	complemented
5	#16863262	—	2	2	no	—
6	#16863259	—	2	2	no	—
7	#16863256	C₂²	1	4	yes	—
8	C₄×C₄⋊Dic₆₇	—	2	2	no	—
9	C₄×C₂²⋊Dic₆₇	—	2	2	no	—
10	#16863242	C₄	1	4	yes	complemented
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	#431×C₆₇	—	2	2	no	—
13	#431×C₆₇	—	2	2	no	—
14	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	complemented
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
17	C₂×C₄⋊Dic₆₇	—	2	4	no	—
18	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
19	C₂×C₄×Dic₆₇	—	2	4	no	—
20	C₆₇⋊#95	—	2	4	no	—
21	C₆₇⋊#95	—	2	4	no	—
22	C₆₇⋊#95	—	2	4	no	—
23	C₆₇⋊#95	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	2	yes	—
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
36	C₄⋊C₈×C₆₇	—	2	4	no	—
37	C₂²⋊C₈×C₆₇	—	2	4	no	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	C₂²×Dic₆₇	—	2	8	no	—
41	C₄⋊Dic₆₇	—	4	8	no	—
42	C₂²⋊Dic₆₇	—	4	8	no	—
43	C₄×Dic₆₇	—	4	8	no	—
44	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	—	2	4	no	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	—	4	2	no	—
48	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	—	4	4	no	—
62	C₄²×C₆₇	C₄²	1	2	yes	—
63	C₄²×C₆₇	C₄²	1	1	yes	characteristic
64	C₄²×C₆₇	C₄²	1	1	yes	characteristic
65	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
70	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
71	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂³×C₆₇	—	4	1	no	—
73	C₂³×C₆₇	—	4	1	no	—
74	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
75	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂³×C₆₇	#135	1	4	yes	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	4	no	—
85	C₂×C₄×C₆₇	—	4	2	no	—
86	C₂×C₄×C₆₇	—	4	1	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	1	no	—
89	C₂×C₄×C₆₇	#95	1	2	yes	—
90	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#95	1	2	yes	—
92	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×Dic₆₇	—	4	8	no	—
99	C₂×Dic₆₇	—	4	8	no	—
100	C₂×Dic₆₇	—	4	8	no	—
101	C₈×C₆₇	—	4	4	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	—	2	1	no	—
104	C₂²×C₆₇	—	2	2	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	—	4	4	no	—
107	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
108	C₂²×C₆₇	#416	1	4	yes	—
109	C₂²×C₆₇	#427	1	2	yes	—
110	C₄×C₆₇	—	2	2	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	—	2	1	no	—
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	H(3,ℤ/4)	1	2	yes	—
118	C₄×C₆₇	#410	1	2	yes	—
119	Dic₆₇	—	8	8	no	—
120	#7980	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	—	2	4	no	—
122	C₂×C₆₇	—	2	2	no	—
123	C₂×C₆₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
124	C₂×C₆₇	#1735	1	1	yes	characteristic
125	C₂×C₆₇	#1740	1	1	yes	characteristic
126	C₂×#431	D₆₇	1	1	yes	characteristic
127	#2283	—	67	2	no	—
128	#1289	—	67	4	no	—
129	C₆₇	#7980	1	1	yes	characteristic, complemented, Sylow
130	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
131	#431	—	2	2	no	—
132	#431	—	2	2	no	—
133	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	complemented
134	C₂×C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
135	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
136	#383	—	67	4	no	—
137	#382	—	134	2	no	—
138	#381	—	134	2	no	—
139	#380	—	67	4	no	—
140	C₄×C₄⋊C₄	—	134	2	no	—
141	C₄×C₂²⋊C₄	—	134	2	no	—
142	#376	—	67	4	no	—
143	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
144	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
145	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
147	C₂×C₄⋊C₄	—	134	4	no	—
148	C₂×C₂²⋊C₄	—	134	4	no	—
149	C₂×C₄²	—	134	4	no	—
150	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
151	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
152	C₂×C₄²	C₄×D₆₇	1	2	yes	—
153	C₂×C₄²	C₄×D₆₇	1	2	yes	—
154	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
158	C₄⋊C₈	—	2	4	no	—
159	C₂²⋊C₈	—	2	4	no	—
160	C₄×C₈	—	2	2	no	—
161	C₄×C₈	—	2	2	no	—
162	#95	—	67	8	no	—
163	#95	—	134	4	no	—
164	#95	—	134	4	no	—
165	#95	—	134	4	no	—
166	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	134	8	no	—
171	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
172	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
173	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
174	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
175	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
180	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
182	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	4	4	no	—
185	C₄⋊C₄	—	268	8	no	—
186	C₂²⋊C₄	—	268	8	no	—
187	C₄²	—	268	8	no	—
188	C₄²	C₄×Dic₆₇	1	2	yes	—
189	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
190	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
191	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
192	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
193	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
194	C₄²	Dic₆₇⋊C₄	1	2	yes	—
195	C₄²	Dic₆₇⋊C₄	1	2	yes	—
196	C₄²	Dic₆₇⋊C₄	1	2	yes	—
197	C₄²	Dic₆₇⋊C₄	1	2	yes	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
201	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
202	C₂³	#14524065	1	4	yes	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	268	8	no	—
216	C₂×C₄	—	268	8	no	—
217	C₂×C₄	—	268	8	no	—
218	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
219	C₂×C₄	C₆₇⋊#95	1	2	yes	—
220	C₂×C₄	C₆₇⋊#95	1	2	yes	—
221	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
222	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
223	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
226	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
234	C₂²	#16863393	1	4	yes	—
235	C₂²	#16863434	1	2	yes	—
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	2	no	—
243	C₄	—	536	8	no	—
244	C₄	#16863153	1	2	yes	—
245	C₄	#16863370	1	2	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	2	2	no	—
248	C₂	C₂×#16863153	1	1	yes	characteristic
249	C₂	#22559926	1	1	yes	characteristic
250	C₂	#22559942	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₆₇⋊#95
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	#7980
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
4. #16863264	C₄	semidirect
10. #16863242	C₄	semidirect
14. C₂×C₄⋊C₈×C₆₇	C₄	semidirect
16. C₂×C₄×C₈×C₆₇	C₄	semidirect
129. C₆₇	#7980	semidirect
133. C₂×C₄⋊C₈	Dic₆₇	semidirect
135. C₂×C₄×C₈	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22562607	2
#22558077	2
C₂×#431×C₆₇	2
#7980	67