Subgroups — #30745709

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562616	C₂	1	2	yes	maximal
2	#22558077	C₂	1	4	yes	maximal
3	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14524002	—	2	2	no	—
5	#16863266	—	2	2	no	—
6	#16863262	C₄	1	4	yes	complemented
7	#16863256	—	2	2	no	—
8	#16863256	C₂²	1	4	yes	—
9	C₄×C₄⋊Dic₆₇	—	2	2	no	—
10	C₄×C₄⋊Dic₆₇	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	#14524002	—	4	4	no	—
18	#14524002	—	4	4	no	—
19	C₂×C₄⋊Dic₆₇	—	2	4	no	—
20	C₂×C₄⋊Dic₆₇	—	2	4	no	—
21	C₂×C₄⋊Dic₆₇	—	2	4	no	—
22	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
23	C₂×C₄×Dic₆₇	—	2	4	no	—
24	C₆₇⋊#95	—	2	4	no	—
25	C₆₇⋊#95	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	—	2	4	no	—
39	C₂²⋊C₈×C₆₇	—	2	4	no	—
40	C₄×C₈×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	—	2	2	no	—
42	C₂²×Dic₆₇	—	2	8	no	—
43	C₄⋊Dic₆₇	—	4	8	no	—
44	C₄⋊Dic₆₇	—	4	8	no	—
45	C₄⋊Dic₆₇	—	4	8	no	—
46	C₂²⋊Dic₆₇	—	4	4	no	—
47	C₂²⋊Dic₆₇	—	4	4	no	—
48	C₄×Dic₆₇	—	4	8	no	—
49	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	—	4	2	no	—
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	D₈	1	2	yes	—
61	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
62	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₂×C₈×C₆₇	—	4	4	no	—
69	C₄²×C₆₇	C₄²	1	2	yes	—
70	C₄²×C₆₇	C₄²	1	1	yes	characteristic
71	C₄²×C₆₇	C₄²	1	1	yes	characteristic
72	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
77	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
78	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
79	C₂³×C₆₇	—	4	1	no	—
80	C₂³×C₆₇	—	4	1	no	—
81	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
84	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
85	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	1	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	4	1	no	—
94	C₂×C₄×C₆₇	#95	1	2	yes	—
95	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
96	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#95	1	2	yes	—
98	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#102	1	4	yes	—
101	C₂×C₄×C₆₇	#102	1	4	yes	—
102	C₂×C₄×C₆₇	#103	1	4	yes	—
103	C₂×C₄×C₆₇	#103	1	4	yes	—
104	C₂×C₄×C₆₇	#106	1	2	yes	—
105	C₂×C₄×C₆₇	#106	1	2	yes	—
106	C₂×C₄×C₆₇	#107	1	2	yes	—
107	C₂×C₄×C₆₇	#107	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×Dic₆₇	—	4	8	no	—
112	C₂×Dic₆₇	—	4	8	no	—
113	C₂×Dic₆₇	—	4	8	no	—
114	C₈×C₆₇	—	4	4	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	1	no	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	—	4	1	no	—
120	C₂²×C₆₇	—	4	2	no	—
121	C₂²×C₆₇	—	4	1	no	—
122	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
123	C₂²×C₆₇	C₂×#102	1	2	yes	—
124	C₂²×C₆₇	C₂×#103	1	2	yes	—
125	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
127	C₄×C₆₇	—	2	2	no	—
128	C₄×C₆₇	—	2	1	no	—
129	C₄×C₆₇	—	2	1	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	H(3,ℤ/4)	1	2	yes	—
132	C₄×C₆₇	#340	1	4	yes	—
133	C₄×C₆₇	#340	1	2	yes	—
134	C₄×C₆₇	#340	1	2	yes	—
135	C₄×C₆₇	#410	1	2	yes	—
136	Dic₆₇	—	8	8	no	—
137	#7981	—	67	1	no	maximal, Sylow
138	C₂×C₆₇	—	2	2	no	—
139	C₂×C₆₇	—	2	2	no	—
140	C₂×C₆₇	—	2	1	no	—
141	C₂×C₆₇	—	2	1	no	—
142	C₂×C₆₇	C₂×H(3,ℤ/4)	1	1	yes	characteristic
143	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
144	C₂×C₆₇	#1740	1	1	yes	characteristic
145	C₂×#431	D₆₇	1	1	yes	characteristic
146	#2285	—	67	2	no	—
147	#1289	—	67	4	no	—
148	C₆₇	#7981	1	1	yes	characteristic, complemented, Sylow
149	C₂×#117	—	134	2	no	—
150	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
151	#431	—	2	2	no	—
152	#431	—	2	2	no	—
153	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	complemented
154	C₂×C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
155	C₂×C₄×C₈	Dic₆₇	1	2	yes	complemented
156	#384	—	134	2	no	—
157	#382	—	67	4	no	—
158	#380	—	67	4	no	—
159	#380	—	134	2	no	—
160	C₄×C₄⋊C₄	—	67	4	no	—
161	C₄×C₄⋊C₄	—	134	2	no	—
162	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
163	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
165	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
166	#117	—	268	4	no	—
167	#117	—	268	4	no	—
168	C₂×C₄⋊C₄	—	134	4	no	—
169	C₂×C₄⋊C₄	—	134	4	no	—
170	C₂×C₄⋊C₄	—	134	4	no	—
171	C₂×C₂²⋊C₄	—	134	4	no	—
172	C₂×C₄²	—	134	4	no	—
173	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
174	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₆₇	1	2	yes	—
176	C₂×C₄²	C₄×D₆₇	1	2	yes	—
177	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
181	C₄⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	#95	—	67	8	no	—
186	#95	—	134	4	no	—
187	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	134	8	no	—
191	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
192	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
193	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
194	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
195	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
199	C₂²×C₄	D₅₃₆	1	2	yes	—
200	C₂²×C₄	#11761617	1	4	yes	—
201	C₂²×C₄	Dic₂₆₈	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
205	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₄⋊C₄	—	268	8	no	—
209	C₄⋊C₄	—	268	8	no	—
210	C₄⋊C₄	—	268	8	no	—
211	C₂²⋊C₄	—	268	4	no	—
212	C₂²⋊C₄	—	268	4	no	—
213	C₄²	—	268	8	no	—
214	C₄²	C₄×Dic₆₇	1	2	yes	—
215	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
216	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
217	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
218	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
219	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
220	C₄²	Dic₆₇⋊C₄	1	2	yes	—
221	C₄²	Dic₆₇⋊C₄	1	2	yes	—
222	C₄²	Dic₆₇⋊C₄	1	2	yes	—
223	C₄²	Dic₆₇⋊C₄	1	2	yes	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
227	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
228	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
229	C₂³	C₂×#11761617	1	2	yes	—
230	C₂³	C₂×Dic₂₆₈	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	268	8	no	—
240	C₂×C₄	—	268	8	no	—
241	C₂×C₄	—	268	8	no	—
242	C₂×C₄	C₆₇⋊#95	1	2	yes	—
243	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
244	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
245	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
246	C₂×C₄	C₆₇⋊#95	1	2	yes	—
247	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
248	C₂×C₄	#14523965	1	4	yes	—
249	C₂×C₄	#14523965	1	4	yes	—
250	C₂×C₄	#14523968	1	4	yes	—
251	C₂×C₄	#14523968	1	4	yes	—
252	C₂×C₄	#14523975	1	2	yes	—
253	C₂×C₄	#14523975	1	2	yes	—
254	C₂×C₄	#14523977	1	2	yes	—
255	C₂×C₄	#14523977	1	2	yes	—
256	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
259	C₈	—	4	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
268	C₂²	C₂×#14523965	1	2	yes	—
269	C₂²	C₂×#14523968	1	2	yes	—
270	C₂²	C₂×#14523975	1	1	yes	characteristic
271	C₂²	C₂×#14523977	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	536	8	no	—
277	C₄	#16863153	1	2	yes	—
278	C₄	#16863159	1	4	yes	—
279	C₄	#16863159	1	2	yes	—
280	C₄	#16863159	1	2	yes	—
281	C₄	#16863370	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	C₂×#16863153	1	1	yes	characteristic
287	C₂	C₂×#16863159	1	1	yes	characteristic
288	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	#7981
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
6. #16863262	C₄	semidirect
10. C₄×C₄⋊Dic₆₇	C₄	semidirect
14. C₂×C₄⋊C₈×C₆₇	C₄	semidirect
16. C₂×C₄×C₈×C₆₇	C₄	semidirect
148. C₆₇	#7981	semidirect
153. C₂×C₄⋊C₈	Dic₆₇	semidirect
155. 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
#22562616	2
#22558077	2
C₂×#431×C₆₇	2
#7981	67