Subgroups — #30761522

Order 17152.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 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	C₂×#16863167	C₂	1	2	yes	maximal
3	#22559936	C₂	1	2	yes	maximal
4	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×#338×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂²×#11761612	C₂²	1	1	yes	characteristic
8	C₂×#14524002	C₂²	1	1	yes	characteristic
9	#16863354	—	2	2	no	—
10	#16863351	C₄	1	4	yes	—
11	M₁₆×Dic₆₇	—	2	2	no	—
12	#16863167	—	2	4	no	—
13	#16863161	C₄	1	8	yes	complemented
14	C₆₇⋊#338	—	2	2	no	—
15	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	#338×C₆₇	—	2	2	no	—
18	C₂×#11761612	—	2	1	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	C₂×C₄	1	4	yes	—
26	C₂×#11761612	C₂×C₄	1	2	yes	—
27	C₂×#11761612	C₂×C₄	1	4	yes	—
28	C₂²×C₆₇⋊C₈	—	2	1	no	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	#14524002	—	2	1	no	—
31	#14524002	—	2	1	no	—
32	#14524002	C₂×C₄	1	2	yes	—
33	#14524002	D₄	1	2	yes	—
34	C₂×C₄⋊Dic₆₇	—	2	1	no	—
35	C₂×C₂²⋊Dic₆₇	Q₈	1	2	yes	—
36	C₂×C₄×Dic₆₇	—	2	1	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	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
49	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
52	C₂²×Dic₆₇	—	2	2	no	—
53	#11761612	—	2	2	no	—
54	#11761612	—	2	4	no	—
55	#11761612	—	2	4	no	—
56	#11761612	—	2	4	no	—
57	#11761612	—	4	2	no	—
58	#11761612	—	4	2	no	—
59	#11761612	C₄²	1	4	yes	—
60	C₂×C₆₇⋊C₈	—	2	2	no	—
61	C₂×C₆₇⋊C₈	—	2	4	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₄⋊Dic₆₇	—	2	2	no	—
66	C₄⋊Dic₆₇	—	2	2	no	—
67	C₂²⋊Dic₆₇	—	2	2	no	—
68	C₂²⋊Dic₆₇	C₄⋊C₄	1	4	yes	—
69	C₄×Dic₆₇	—	2	2	no	—
70	C₄×Dic₆₇	—	2	2	no	—
71	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
82	M₁₆×C₆₇	—	2	4	no	—
83	M₁₆×C₆₇	—	2	4	no	—
84	M₁₆×C₆₇	—	4	2	no	—
85	M₁₆×C₆₇	—	4	2	no	—
86	M₁₆×C₆₇	—	4	2	no	—
87	M₁₆×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
94	C₂³×C₆₇	—	8	1	no	—
95	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
97	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₆₇	#125	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#117	1	2	yes	—
104	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#117	1	2	yes	—
108	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#123	1	2	yes	—
114	C₂×C₄×C₆₇	#123	1	2	yes	—
115	C₂×C₄×C₆₇	#124	1	2	yes	—
116	C₂×C₄×C₆₇	#126	1	2	yes	—
117	C₂×C₄×C₆₇	#126	1	2	yes	—
118	C₂×Dic₆₇	—	2	4	no	—
119	C₂×Dic₆₇	—	8	2	no	—
120	C₈×C₆₇	—	2	4	no	—
121	C₈×C₆₇	—	4	4	no	—
122	C₆₇⋊C₈	—	2	4	no	—
123	C₆₇⋊C₈	—	4	4	no	—
124	C₂²×C₆₇	—	4	1	no	—
125	C₂²×C₆₇	—	4	1	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₆₇	#380	1	1	yes	characteristic
131	C₂²×C₆₇	#382	1	1	yes	characteristic
132	C₂²×C₆₇	#387	1	2	yes	—
133	C₂²×C₆₇	#387	1	2	yes	—
134	C₄×C₆₇	#376	1	1	yes	characteristic
135	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₆₇	#381	1	1	yes	characteristic
137	C₄×C₆₇	#383	1	1	yes	characteristic
138	C₄×C₆₇	#388	1	2	yes	—
139	C₄×C₆₇	#388	1	2	yes	—
140	Dic₆₇	—	4	4	no	—
141	#10812	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	2	1	no	—
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	#1434	1	1	yes	characteristic
148	#2873	—	67	1	no	—
149	C₂×#344	—	67	2	no	—
150	#1738	—	67	2	no	—
151	C₂×#338	—	67	1	no	—
152	C₂×#338	D₆₇	1	1	yes	characteristic
153	C₆₇	#10812	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	67	1	no	—
155	C₂²×M₁₆	—	67	2	no	—
156	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
157	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
158	C₂×#117	—	67	1	no	—
159	#405	—	67	4	no	—
160	#405	—	134	2	no	—
161	C₄×M₁₆	—	134	2	no	—
162	#344	—	134	4	no	—
163	#341	—	67	8	no	—
164	#338	—	2	2	no	—
165	#338	—	134	2	no	—
166	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	67	4	no	—
176	C₂×M₁₆	—	67	2	no	—
177	C₂×M₁₆	—	67	4	no	—
178	C₂×M₁₆	—	134	2	no	—
179	C₂×M₁₆	—	134	2	no	—
180	C₂×M₁₆	—	134	2	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	1	no	—
184	C₂×M₁₆	—	134	2	no	—
185	C₂×M₁₆	D₂₆₈	1	2	yes	—
186	C₂×M₁₆	C₂²⋊D₆₇	1	2	yes	—
187	C₂²×C₈	—	134	2	no	—
188	C₂²×C₈	—	134	1	no	—
189	C₂²×C₈	C₄×D₆₇	1	4	yes	—
190	C₂²×C₈	C₄×D₆₇	1	2	yes	—
191	#117	—	67	2	no	—
192	#117	—	67	2	no	—
193	#117	—	134	1	no	—
194	#117	—	134	1	no	—
195	C₂×C₄⋊C₄	—	134	1	no	—
196	C₂×C₂²⋊C₄	—	67	2	no	—
197	C₂×C₄²	—	134	1	no	—
198	#97	—	134	4	no	—
199	C₄×C₈	—	134	4	no	—
200	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
201	C₂²×C₄	—	134	2	no	—
202	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
204	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
207	C₂²×C₄	#11761627	1	1	yes	characteristic
208	C₂²×C₄	#11761627	1	2	yes	—
209	C₂²×C₄	#11761627	1	2	yes	—
210	C₂²×C₄	#11761628	1	2	yes	—
211	C₂²×C₄	#11761629	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	268	2	no	—
219	M₁₆	—	268	2	no	—
220	M₁₆	—	67	4	no	—
221	M₁₆	—	134	4	no	—
222	M₁₆	—	134	4	no	—
223	M₁₆	—	134	2	no	—
224	M₁₆	—	134	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	268	2	no	—
231	C₂×C₈	—	268	2	no	—
232	C₂×C₈	—	134	4	no	—
233	C₂×C₈	—	134	2	no	—
234	C₂×C₈	—	134	4	no	—
235	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
236	C₄⋊C₄	—	134	2	no	—
237	C₄⋊C₄	—	134	2	no	—
238	C₂²⋊C₄	—	67	4	no	—
239	C₂²⋊C₄	—	134	2	no	—
240	C₄²	—	134	2	no	—
241	C₄²	—	134	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
244	C₂³	#14524008	1	2	yes	—
245	C₂³	#14524019	1	1	yes	characteristic
246	C₂³	#14524023	1	1	yes	characteristic
247	C₂³	#14524033	1	2	yes	—
248	C₂×C₄	—	536	2	no	—
249	C₂×C₄	—	134	4	no	—
250	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
252	C₂×C₄	#14523999	1	1	yes	characteristic
253	C₂×C₄	#14523999	1	2	yes	—
254	C₂×C₄	#14523999	1	1	yes	characteristic
255	C₂×C₄	#14524000	1	1	yes	characteristic
256	C₂×C₄	#14524001	1	1	yes	characteristic
257	C₂×C₄	#14524001	1	2	yes	—
258	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
259	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
260	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
261	C₂×C₄	#14524014	1	1	yes	characteristic
262	C₂×C₄	#14524018	1	1	yes	characteristic
263	C₂×C₄	#14524024	1	2	yes	—
264	C₂×C₄	#14524027	1	2	yes	—
265	C₂×C₄	#14524029	1	2	yes	—
266	C₂×C₄	#14524036	1	2	yes	—
267	C₂×C₄	#14524038	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	4	4	no	—
270	C₈	—	268	4	no	—
271	C₈	—	134	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	#16863247	1	1	yes	characteristic
278	C₂²	#16863253	1	1	yes	characteristic
279	C₂²	#16863260	1	1	yes	characteristic
280	C₂²	#16863277	1	2	yes	—
281	C₂²	#16863281	1	2	yes	—
282	C₄	—	268	4	no	—
283	C₄	#16863241	1	1	yes	characteristic
284	C₄	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
285	C₄	#16863257	1	1	yes	characteristic
286	C₄	#16863263	1	1	yes	characteristic
287	C₄	#16863284	1	2	yes	—
288	C₄	#16863288	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	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×#338×C₆₇	C₂
socle	C₂×C₆₇	#1434
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#338	D₆₇
hypocenter	C₆₇	#10812
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
13. #16863161	C₄	semidirect
153. C₆₇	#10812	semidirect

Maximal subgroups

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

subgroup	index
#22568627	2
C₂×#16863167	2
#22559936	2
C₂×C₆₇⋊#338	2
C₂×#338×C₆₇	2
#10812	67