Subgroups — #30761517

Order 17152.

644 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568627	C₂	1	1	yes	maximal, characteristic
2	#22559942	C₂	1	2	yes	maximal
3	#22559919	C₂	1	2	yes	maximal
4	C₂×#16863153	C₂	1	1	yes	maximal, characteristic
5	C₂×H(3,ℤ/4)×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	1	yes	characteristic
7	C₂×#14524002	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	1	yes	characteristic
9	#16863354	—	2	2	no	—
10	#16863351	C₄	1	4	yes	complemented
11	M₁₆×Dic₆₇	—	2	2	no	—
12	#16863163	—	2	4	no	—
13	#16863156	C₄	1	8	yes	complemented
14	#16863153	—	2	2	no	—
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	2	yes	—
17	H(3,ℤ/4)×C₆₇	—	2	2	no	—
18	C₂×#11761612	—	2	1	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	C₂×C₄	1	2	yes	—
21	C₂×#11761612	C₂×C₄	1	4	yes	—
22	C₂²×C₆₇⋊C₈	—	2	1	no	—
23	#14524002	—	2	1	no	—
24	#14524002	—	2	2	no	—
25	#14524002	—	2	1	no	—
26	#14524002	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	C₂×C₄	1	4	yes	—
29	#14524002	C₂×C₄	1	2	yes	—
30	#14524002	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₄⋊Dic₆₇	—	2	2	no	—
33	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
34	C₂×C₂²⋊Dic₆₇	D₄	1	2	yes	—
35	C₂×C₄×Dic₆₇	—	2	1	no	—
36	C₂×C₄×Dic₆₇	—	2	2	no	—
37	#14523953	—	2	4	no	—
38	C₈×Dic₆₇	—	2	4	no	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
43	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	#117×C₆₇	—	2	2	no	—
46	#117×C₆₇	—	2	2	no	—
47	#117×C₆₇	—	2	2	no	—
48	#117×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
52	C₂²×Dic₆₇	—	2	2	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	#11761612	—	2	2	no	—
55	#11761612	—	2	4	no	—
56	#11761612	C₄²	1	4	yes	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₄⋊Dic₆₇	—	2	2	no	—
60	C₄⋊Dic₆₇	—	2	2	no	—
61	C₄⋊Dic₆₇	—	2	4	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₂²⋊Dic₆₇	—	2	2	no	—
64	C₂²⋊Dic₆₇	—	2	4	no	—
65	C₂²⋊Dic₆₇	—	4	2	no	—
66	C₂²⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₄×Dic₆₇	—	2	2	no	—
69	C₄×Dic₆₇	—	2	4	no	—
70	C₄×Dic₆₇	—	4	2	no	—
71	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	M₁₆×C₆₇	—	2	4	no	—
84	M₁₆×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₆₇	—	4	2	no	—
89	C₄⋊C₄×C₆₇	—	4	2	no	—
90	C₂²⋊C₄×C₆₇	—	4	2	no	—
91	C₂²⋊C₄×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	—	8	1	no	—
95	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₇	C₄×D₄	1	2	yes	—
97	C₂³×C₆₇	#121	1	1	yes	characteristic
98	C₂³×C₆₇	#121	1	1	yes	characteristic
99	C₂³×C₆₇	#124	1	2	yes	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	8	4	no	—
102	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	2	yes	—
106	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#117	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#123	1	2	yes	—
116	C₂×C₄×C₆₇	#123	1	2	yes	—
117	C₂×C₄×C₆₇	#125	1	2	yes	—
118	C₂×C₄×C₆₇	#126	1	2	yes	—
119	C₂×C₄×C₆₇	#126	1	2	yes	—
120	C₂×Dic₆₇	—	2	4	no	—
121	C₂×Dic₆₇	—	4	4	no	—
122	C₂×Dic₆₇	—	8	2	no	—
123	C₂×Dic₆₇	—	8	4	no	—
124	C₈×C₆₇	—	2	4	no	—
125	C₆₇⋊C₈	—	2	4	no	—
126	C₂²×C₆₇	—	4	1	no	—
127	C₂²×C₆₇	—	4	2	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	—	4	1	no	—
130	C₂²×C₆₇	—	4	1	no	—
131	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₆₇	#380	1	1	yes	characteristic
133	C₂²×C₆₇	#382	1	1	yes	characteristic
134	C₂²×C₆₇	#388	1	2	yes	—
135	C₂²×C₆₇	#388	1	2	yes	—
136	C₄×C₆₇	—	8	4	no	—
137	C₄×C₆₇	#376	1	1	yes	characteristic
138	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₆₇	#381	1	1	yes	characteristic
140	C₄×C₆₇	#383	1	1	yes	characteristic
141	C₄×C₆₇	#387	1	2	yes	—
142	C₄×C₆₇	#387	1	2	yes	—
143	Dic₆₇	—	4	4	no	—
144	Dic₆₇	—	8	4	no	—
145	#10811	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	#1434	1	1	yes	characteristic
152	#2873	—	67	1	no	—
153	#1740	—	67	2	no	—
154	#1733	—	67	2	no	—
155	C₂×H(3,ℤ/4)	—	67	1	no	—
156	C₂×H(3,ℤ/4)	D₆₇	1	1	yes	characteristic
157	C₆₇	#10811	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	67	1	no	—
159	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
160	C₂×#117	—	67	1	no	—
161	C₂×#117	—	67	2	no	—
162	C₂×#117	C₂×D₆₇	1	2	yes	—
163	#405	—	67	4	no	—
164	#405	—	134	2	no	—
165	C₄×M₁₆	—	134	2	no	—
166	#342	—	134	4	no	—
167	#339	—	67	8	no	—
168	H(3,ℤ/4)	—	2	2	no	—
169	H(3,ℤ/4)	—	134	2	no	—
170	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	67	4	no	—
174	C₂×M₁₆	—	67	2	no	—
175	C₂×M₁₆	—	134	1	no	—
176	C₂×M₁₆	—	134	2	no	—
177	C₂×M₁₆	D₂₆₈	1	2	yes	—
178	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
179	C₂²×C₈	—	134	1	no	—
180	C₂²×C₈	C₄×D₆₇	1	2	yes	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	67	2	no	—
186	#117	—	67	2	no	—
187	#117	—	67	4	no	—
188	#117	—	134	2	no	—
189	#117	—	134	2	no	—
190	#117	—	134	2	no	—
191	#117	—	134	1	no	—
192	#117	—	134	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	134	2	no	—
195	C₂×C₄⋊C₄	—	134	1	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	67	2	no	—
198	C₂×C₂²⋊C₄	—	134	2	no	—
199	C₂×C₄²	—	134	2	no	—
200	C₂×C₄²	—	134	1	no	—
201	C₂×C₄²	C₄×D₆₇	1	4	yes	—
202	#97	—	134	4	no	—
203	C₄×C₈	—	134	4	no	—
204	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	268	2	no	—
207	C₂²×C₄	—	134	2	no	—
208	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
212	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	#11761627	1	1	yes	characteristic
214	C₂²×C₄	#11761627	1	2	yes	—
215	C₂²×C₄	#11761627	1	2	yes	—
216	C₂²×C₄	#11761628	1	2	yes	—
217	C₂²×C₄	#11761629	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	67	4	no	—
221	M₁₆	—	134	2	no	—
222	M₁₆	—	134	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	134	2	no	—
226	C₂×C₈	—	134	4	no	—
227	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	268	2	no	—
231	C₄⋊C₄	—	134	4	no	—
232	C₄⋊C₄	—	134	2	no	—
233	C₄⋊C₄	—	134	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	268	2	no	—
237	C₂²⋊C₄	—	67	4	no	—
238	C₂²⋊C₄	—	134	4	no	—
239	C₂²⋊C₄	—	134	2	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	268	2	no	—
243	C₄²	—	134	4	no	—
244	C₄²	—	134	2	no	—
245	C₄²	—	134	2	no	—
246	C₂³	—	8	1	no	—
247	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
248	C₂³	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
249	C₂³	#14524014	1	1	yes	characteristic
250	C₂³	#14524018	1	1	yes	characteristic
251	C₂³	#14524029	1	2	yes	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	—	268	4	no	—
255	C₂×C₄	—	536	2	no	—
256	C₂×C₄	—	536	4	no	—
257	C₂×C₄	—	134	4	no	—
258	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	#14523999	1	1	yes	characteristic
261	C₂×C₄	#14523999	1	2	yes	—
262	C₂×C₄	#14523999	1	1	yes	characteristic
263	C₂×C₄	#14524000	1	1	yes	characteristic
264	C₂×C₄	#14524001	1	1	yes	characteristic
265	C₂×C₄	#14524001	1	2	yes	—
266	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
267	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
268	C₂×C₄	#14524008	1	2	yes	—
269	C₂×C₄	#14524019	1	1	yes	characteristic
270	C₂×C₄	#14524023	1	1	yes	characteristic
271	C₂×C₄	#14524024	1	2	yes	—
272	C₂×C₄	#14524027	1	2	yes	—
273	C₂×C₄	#14524033	1	2	yes	—
274	C₂×C₄	#14524036	1	2	yes	—
275	C₂×C₄	#14524038	1	2	yes	—
276	C₈	—	2	4	no	—
277	C₈	—	134	4	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	#16863247	1	1	yes	characteristic
284	C₂²	#16863253	1	1	yes	characteristic
285	C₂²	#16863260	1	1	yes	characteristic
286	C₂²	#16863284	1	2	yes	—
287	C₂²	#16863288	1	2	yes	—
288	C₄	—	8	4	no	—
289	C₄	—	268	4	no	—
290	C₄	—	536	4	no	—
291	C₄	#16863241	1	1	yes	characteristic
292	C₄	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
293	C₄	#16863257	1	1	yes	characteristic
294	C₄	#16863263	1	1	yes	characteristic
295	C₄	#16863277	1	2	yes	—
296	C₄	#16863281	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#22558433	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×H(3,ℤ/4)×C₆₇	C₂
socle	C₂×C₆₇	#1434
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₆₇
hypocenter	C₆₇	#10811
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
10. #16863351	C₄	semidirect
13. #16863156	C₄	semidirect
157. C₆₇	#10811	semidirect

Maximal subgroups

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

subgroup	index
#22568627	2
#22559942	2
#22559919	2
C₂×#16863153	2
C₂×H(3,ℤ/4)×C₆₇	2
#10811	67