Subgroups — #30761532

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
2	#22559947	C₂	1	2	yes	maximal
3	C₂×#16863161	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₇⋊#338	C₂	1	2	yes	maximal
5	C₂×#341×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	#16863351	—	2	2	no	—
10	#16863351	—	2	2	no	—
11	C₂×#14523953	C₄	1	2	yes	—
12	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
13	#16863164	—	2	4	no	—
14	#16863161	—	2	2	no	—
15	C₆₇⋊#338	C₄	1	8	yes	complemented
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#341×C₆₇	—	2	2	no	—
19	C₂×#11761612	—	2	1	no	—
20	C₂×#11761612	—	2	1	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	—	2	2	no	—
26	C₂×#11761612	—	2	2	no	—
27	C₂×#11761612	C₂×C₄	1	4	yes	—
28	C₂×#11761612	C₂×C₄	1	4	yes	—
29	C₂²×C₆₇⋊C₈	—	2	2	no	—
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
31	#14524002	—	2	1	no	—
32	#14524002	—	2	1	no	—
33	#14524002	—	2	1	no	—
34	#14524002	—	2	1	no	—
35	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
36	C₂×C₂²⋊Dic₆₇	Q₈	1	2	yes	—
37	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
38	#14523953	—	2	4	no	—
39	C₈×Dic₆₇	—	2	4	no	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	D₄	1	2	yes	—
53	C₂²×C₈×C₆₇	D₄	1	2	yes	—
54	C₂²×Dic₆₇	C₄⋊C₄	1	4	yes	—
55	#11761612	—	2	4	no	—
56	#11761612	—	2	4	no	—
57	#11761612	—	2	4	no	—
58	#11761612	—	2	4	no	—
59	#11761612	—	4	2	no	—
60	#11761612	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	2	2	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂×C₆₇⋊C₈	C₄²	1	4	yes	—
67	C₄⋊Dic₆₇	—	2	2	no	—
68	C₄⋊Dic₆₇	—	2	2	no	—
69	C₂²⋊Dic₆₇	—	2	2	no	—
70	C₂²⋊Dic₆₇	—	2	2	no	—
71	C₄×Dic₆₇	—	2	2	no	—
72	C₄×Dic₆₇	—	2	2	no	—
73	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	M₁₆×C₆₇	—	2	4	no	—
85	M₁₆×C₆₇	—	2	4	no	—
86	M₁₆×C₆₇	—	4	2	no	—
87	M₁₆×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	—	2	1	no	—
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	—	2	4	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	4	no	—
95	C₂×C₈×C₆₇	—	4	2	no	—
96	C₂×C₈×C₆₇	—	4	2	no	—
97	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
99	C₂³×C₆₇	—	4	2	no	—
100	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
102	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₆₇	#125	1	2	yes	—
105	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#117	1	2	yes	—
109	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#117	1	2	yes	—
112	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#123	1	2	yes	—
119	C₂×C₄×C₆₇	#123	1	2	yes	—
120	C₂×C₄×C₆₇	#124	1	2	yes	—
121	C₂×C₄×C₆₇	#126	1	2	yes	—
122	C₂×C₄×C₆₇	#126	1	2	yes	—
123	C₂×Dic₆₇	—	2	4	no	—
124	C₂×Dic₆₇	—	4	4	no	—
125	C₈×C₆₇	—	2	2	no	—
126	C₈×C₆₇	—	2	2	no	—
127	C₈×C₆₇	—	4	4	no	—
128	C₆₇⋊C₈	—	2	4	no	—
129	C₆₇⋊C₈	—	4	4	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	4	2	no	—
134	C₂²×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₆₇	#380	1	1	yes	characteristic
137	C₂²×C₆₇	#382	1	1	yes	characteristic
138	C₂²×C₆₇	#387	1	2	yes	—
139	C₂²×C₆₇	#387	1	2	yes	—
140	C₄×C₆₇	#376	1	1	yes	characteristic
141	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₄×C₆₇	#381	1	1	yes	characteristic
143	C₄×C₆₇	#383	1	1	yes	characteristic
144	C₄×C₆₇	#388	1	2	yes	—
145	C₄×C₆₇	#388	1	2	yes	—
146	Dic₆₇	—	4	4	no	—
147	#10815	—	67	1	no	maximal, Sylow
148	C₂×C₆₇	—	2	1	no	—
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	2	no	—
151	C₂×C₆₇	—	2	2	no	—
152	C₂×C₆₇	#1434	1	1	yes	characteristic
153	C₂×C₆₇	#1813	1	2	yes	—
154	C₂×#405	—	67	1	no	—
155	#1742	—	67	2	no	—
156	C₂×#341	—	67	1	no	—
157	C₂×#341	D₆₇	1	1	yes	characteristic
158	C₂×#338	—	67	2	no	—
159	C₆₇	#10815	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	67	1	no	—
161	C₂²×M₁₆	—	67	2	no	—
162	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
163	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
164	C₂×#117	—	67	1	no	—
165	#405	—	134	2	no	—
166	#405	—	134	2	no	—
167	C₂×#97	—	67	2	no	—
168	C₂×C₄×C₈	—	67	2	no	—
169	#343	—	134	4	no	—
170	#341	—	2	2	no	—
171	#341	—	134	2	no	—
172	#338	—	67	8	no	—
173	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	67	4	no	—
180	C₂×M₁₆	—	67	4	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	2	no	—
184	C₂×M₁₆	—	134	2	no	—
185	C₂×M₁₆	—	134	2	no	—
186	C₂×M₁₆	—	134	1	no	—
187	C₂×M₁₆	—	134	1	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	67	2	no	—
194	C₂²×C₈	—	134	2	no	—
195	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
197	C₂²×C₈	D₂₆₈	1	2	yes	—
198	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
199	#117	—	134	1	no	—
200	#117	—	134	1	no	—
201	#117	—	134	1	no	—
202	#117	—	134	1	no	—
203	C₂×C₄⋊C₄	—	67	2	no	—
204	C₂×C₂²⋊C₄	—	67	2	no	—
205	C₂×C₄²	—	67	2	no	—
206	#97	—	134	4	no	—
207	C₄×C₈	—	134	4	no	—
208	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
209	C₂²×C₄	—	67	4	no	—
210	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
212	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
215	C₂²×C₄	#11761627	1	2	yes	—
216	C₂²×C₄	#11761627	1	1	yes	characteristic
217	C₂²×C₄	#11761627	1	2	yes	—
218	C₂²×C₄	#11761628	1	2	yes	—
219	C₂²×C₄	#11761629	1	1	yes	characteristic
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	268	2	no	—
225	M₁₆	—	268	2	no	—
226	M₁₆	—	134	4	no	—
227	M₁₆	—	134	4	no	—
228	M₁₆	—	134	4	no	—
229	M₁₆	—	134	4	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	268	2	no	—
240	C₂×C₈	—	268	2	no	—
241	C₂×C₈	—	67	4	no	—
242	C₂×C₈	—	134	4	no	—
243	C₂×C₈	—	134	2	no	—
244	C₂×C₈	—	134	2	no	—
245	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
246	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
247	C₄⋊C₄	—	134	2	no	—
248	C₄⋊C₄	—	134	2	no	—
249	C₂²⋊C₄	—	134	2	no	—
250	C₂²⋊C₄	—	134	2	no	—
251	C₄²	—	134	2	no	—
252	C₄²	—	134	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
255	C₂³	#14524008	1	2	yes	—
256	C₂³	#14524019	1	1	yes	characteristic
257	C₂³	#14524023	1	1	yes	characteristic
258	C₂³	#14524033	1	2	yes	—
259	C₂×C₄	—	268	4	no	—
260	C₂×C₄	—	134	4	no	—
261	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
262	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	#14523999	1	1	yes	characteristic
264	C₂×C₄	#14523999	1	2	yes	—
265	C₂×C₄	#14523999	1	1	yes	characteristic
266	C₂×C₄	#14524000	1	1	yes	characteristic
267	C₂×C₄	#14524001	1	2	yes	—
268	C₂×C₄	#14524001	1	1	yes	characteristic
269	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
270	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
271	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
272	C₂×C₄	#14524014	1	1	yes	characteristic
273	C₂×C₄	#14524018	1	1	yes	characteristic
274	C₂×C₄	#14524024	1	2	yes	—
275	C₂×C₄	#14524027	1	2	yes	—
276	C₂×C₄	#14524029	1	2	yes	—
277	C₂×C₄	#14524036	1	2	yes	—
278	C₂×C₄	#14524038	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	4	4	no	—
282	C₈	—	268	4	no	—
283	C₈	—	134	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#16863247	1	1	yes	characteristic
290	C₂²	#16863253	1	1	yes	characteristic
291	C₂²	#16863260	1	1	yes	characteristic
292	C₂²	#16863277	1	2	yes	—
293	C₂²	#16863281	1	2	yes	—
294	C₄	—	268	4	no	—
295	C₄	#16863241	1	1	yes	characteristic
296	C₄	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
297	C₄	#16863257	1	1	yes	characteristic
298	C₄	#16863263	1	1	yes	characteristic
299	C₄	#16863284	1	2	yes	—
300	C₄	#16863288	1	2	yes	—
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#22558433	1	1	yes	characteristic
306	C₂	#22560256	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524001
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#341×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#341	D₆₇
hypocenter	C₆₇	#10815
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
15. C₆₇⋊#338	C₄	semidirect
159. C₆₇	#10815	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863351	2
#22559947	2
C₂×#16863161	2
C₂×C₆₇⋊#338	2
C₂×#341×C₆₇	2
#10815	67