Subgroups — #30856198

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	#22560359	C₂	1	2	yes	maximal, complemented
3	#22560334	C₂	1	2	yes	maximal, complemented
4	#22560314	C₂	1	2	yes	maximal
5	#22560255	C₂	1	1	yes	maximal, characteristic
6	#22560091	C₂	1	2	yes	maximal, complemented
7	#22560012	C₂	1	2	yes	maximal
8	#22559965	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1813	C₂	1	1	yes	maximal, characteristic
10	C₂×#14524050	C₂²	1	1	yes	characteristic
11	#41×Dic₆₇	C₂²	1	1	yes	characteristic
12	#16863434	C₂²	1	2	yes	—
13	#16863370	C₂²	1	2	yes	—
14	#16863354	C₄	1	2	yes	complemented
15	#16863354	C₂²	1	4	yes	—
16	#16863354	C₂²	1	1	yes	characteristic
17	#16863351	C₄	1	2	yes	complemented
18	#16863351	C₂²	1	1	yes	characteristic
19	M₁₆×Dic₆₇	C₄	1	2	yes	complemented
20	M₁₆×Dic₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₆₇	C₄	1	2	yes	complemented
22	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
23	#16863168	C₂²	1	4	yes	—
24	#16863157	C₂²	1	4	yes	—
25	#16863153	C₂²	1	4	yes	—
26	#16863153	C₂²	1	2	yes	—
27	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#108×C₆₇	C₂²	1	2	yes	—
29	#413×C₆₇	C₂²	1	2	yes	—
30	#338×C₆₇	C₂²	1	2	yes	—
31	#14524050	—	2	2	no	—
32	#14524050	—	2	2	no	—
33	C₂×#11761612	C₂×C₄	1	2	yes	—
34	C₂×#11761612	C₂³	1	1	yes	characteristic
35	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
37	Q₈×Dic₆₇	—	2	1	no	—
38	D₄×Dic₆₇	—	2	1	no	—
39	D₄×Dic₆₇	—	2	2	no	—
40	#14524002	C₂×C₄	1	2	yes	—
41	#14524002	C₂³	1	4	yes	—
42	#14524002	C₂³	1	1	yes	characteristic
43	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
45	#14523977	—	2	2	no	—
46	#14523975	—	2	2	no	—
47	#14523958	—	2	4	no	—
48	#14523953	—	2	4	no	—
49	#14523953	C₂×C₄	1	2	yes	—
50	#14523953	C₂×C₄	1	2	yes	—
51	#14523953	C₂×C₄	1	2	yes	—
52	C₈×Dic₆₇	—	2	4	no	—
53	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
54	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
55	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
56	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
57	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
61	#131×C₆₇	—	2	2	no	—
62	#131×C₆₇	—	2	2	no	—
63	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
65	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	#108×C₆₇	—	2	2	no	—
69	#108×C₆₇	—	2	2	no	—
70	#101×C₆₇	—	2	2	no	—
71	#100×C₆₇	—	2	2	no	—
72	C₂²×Dic₆₇	—	2	2	no	—
73	C₂²×Dic₆₇	—	2	1	no	—
74	#11761612	—	2	2	no	—
75	#11761612	—	2	4	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₆₇⋊C₈	#41	1	2	yes	—
82	C₂×C₆₇⋊C₈	#41	1	2	yes	—
83	C₄⋊Dic₆₇	—	2	2	no	—
84	C₄⋊Dic₆₇	—	2	1	no	—
85	C₄⋊Dic₆₇	—	2	4	no	—
86	C₂²⋊Dic₆₇	—	2	2	no	—
87	C₂²⋊Dic₆₇	—	2	1	no	—
88	C₂²⋊Dic₆₇	—	2	4	no	—
89	C₄×Dic₆₇	—	2	2	no	—
90	C₄×Dic₆₇	—	2	1	no	—
91	C₄×Dic₆₇	—	2	4	no	—
92	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
94	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
95	#41×C₆₇	—	2	2	no	—
96	#41×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
98	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
99	C₂×D₄×C₆₇	#41	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	M₁₆×C₆₇	—	2	2	no	—
103	M₁₆×C₆₇	—	2	4	no	—
104	M₁₆×C₆₇	—	2	4	no	—
105	M₁₆×C₆₇	—	2	4	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	4	no	—
109	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₆₇	#41	1	2	yes	—
113	C₂×C₈×C₆₇	#41	1	2	yes	—
114	C₂³×C₆₇	#117	1	2	yes	—
115	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	2	no	—
117	D₄×C₆₇	—	2	4	no	—
118	D₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	#117	1	2	yes	—
122	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
125	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₆₇	—	2	1	no	—
127	C₂×Dic₆₇	—	2	2	no	—
128	C₂×Dic₆₇	—	2	1	no	—
129	C₂×Dic₆₇	—	4	2	no	—
130	C₂×Dic₆₇	—	4	4	no	—
131	C₂×Dic₆₇	—	4	4	no	—
132	C₈×C₆₇	—	2	2	no	—
133	C₈×C₆₇	—	4	4	no	—
134	C₈×C₆₇	#117	1	1	yes	characteristic
135	C₈×C₆₇	#117	1	1	yes	characteristic
136	C₈×C₆₇	#117	1	2	yes	—
137	C₆₇⋊C₈	—	2	2	no	—
138	C₆₇⋊C₈	#117	1	1	yes	characteristic
139	C₆₇⋊C₈	#117	1	1	yes	characteristic
140	C₆₇⋊C₈	#117	1	2	yes	—
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	4	1	no	—
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
145	C₂²×C₆₇	#529	1	2	yes	—
146	C₄×C₆₇	—	2	2	no	—
147	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
148	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
149	C₄×C₆₇	#529	1	2	yes	—
150	Dic₆₇	—	2	2	no	—
151	Dic₆₇	—	4	2	no	—
152	Dic₆₇	—	4	1	no	—
153	Dic₆₇	—	8	4	no	—
154	#20787	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	2	no	—
157	C₂×C₆₇	—	4	2	no	—
158	C₂×C₆₇	#2283	1	1	yes	characteristic
159	C₄×#131	—	67	1	no	—
160	#1841	—	67	2	no	—
161	#1833	—	67	2	no	—
162	#1827	—	67	2	no	—
163	#1813	D₆₇	1	1	yes	characteristic
164	#1812	—	67	1	no	—
165	#1772	—	67	2	no	—
166	#1757	—	67	2	no	—
167	#1747	—	67	2	no	—
168	C₆₇	#20787	1	1	yes	characteristic, complemented, Sylow
169	C₂×#131	—	67	1	no	—
170	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
171	C₄×#41	—	67	1	no	—
172	C₂×#108	C₂×D₆₇	1	2	yes	—
173	#427	—	67	2	no	—
174	#413	C₂×D₆₇	1	2	yes	—
175	#410	—	67	2	no	—
176	#405	—	67	2	no	—
177	#405	—	67	1	no	—
178	#405	—	67	2	no	—
179	#405	—	67	1	no	—
180	#405	—	67	4	no	—
181	C₄×M₁₆	—	67	2	no	—
182	C₄×M₁₆	—	67	1	no	—
183	C₂×C₄×C₈	—	67	2	no	—
184	C₂×C₄×C₈	—	67	1	no	—
185	#344	—	67	4	no	—
186	#339	—	67	4	no	—
187	#338	C₂×D₆₇	1	2	yes	—
188	H(3,ℤ/4)	—	67	4	no	—
189	H(3,ℤ/4)	—	67	2	no	—
190	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
191	#131	—	2	2	no	—
192	#131	—	2	2	no	—
193	#131	—	134	2	no	—
194	#131	—	134	2	no	—
195	C₂×M₁₆	—	67	2	no	—
196	C₂×M₁₆	—	67	1	no	—
197	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
198	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
199	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
200	C₂²×C₈	—	67	1	no	—
201	C₂²×C₈	—	67	2	no	—
202	C₂²×C₈	C₄×D₆₇	1	2	yes	—
203	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
204	C₄×Q₈	—	134	1	no	—
205	C₄×D₄	—	134	1	no	—
206	C₄×D₄	—	134	2	no	—
207	#117	—	67	2	no	—
208	#117	—	67	1	no	—
209	#117	—	67	4	no	—
210	C₂×C₄²	—	67	2	no	—
211	C₂×C₄²	—	67	1	no	—
212	#108	—	2	2	no	—
213	#108	—	2	2	no	—
214	#107	—	134	2	no	—
215	#106	—	134	2	no	—
216	#101	—	2	2	no	—
217	#100	—	2	2	no	—
218	#99	—	134	4	no	—
219	#97	—	67	2	no	—
220	#97	—	67	2	no	—
221	#97	—	67	2	no	—
222	#97	—	134	4	no	—
223	C₄×C₈	—	67	1	no	—
224	C₄×C₈	—	67	1	no	—
225	C₄×C₈	—	67	2	no	—
226	C₄×C₈	—	67	2	no	—
227	C₄×C₈	—	67	1	no	—
228	C₄×C₈	—	67	1	no	—
229	C₄×C₈	—	67	2	no	—
230	C₄×C₈	—	134	4	no	—
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	#11761627	1	1	yes	characteristic
234	C₂×D₄	#11761627	1	1	yes	characteristic
235	C₂×D₄	#11761629	1	2	yes	—
236	C₂²×C₄	—	134	2	no	—
237	C₂²×C₄	—	134	1	no	—
238	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
239	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	134	4	no	—
245	M₁₆	—	134	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	67	1	no	—
250	C₂×C₈	—	67	1	no	—
251	C₂×C₈	—	67	2	no	—
252	C₂×C₈	—	67	2	no	—
253	C₂×C₈	—	67	2	no	—
254	C₂×C₈	—	134	2	no	—
255	C₂×C₈	—	134	2	no	—
256	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
259	C₂×C₈	#11761627	1	2	yes	—
260	C₂×C₈	#11761627	1	2	yes	—
261	C₄⋊C₄	—	134	4	no	—
262	C₄⋊C₄	—	134	2	no	—
263	C₄⋊C₄	—	134	1	no	—
264	C₂²⋊C₄	—	134	4	no	—
265	C₂²⋊C₄	—	134	2	no	—
266	C₂²⋊C₄	—	134	1	no	—
267	C₄²	—	67	1	no	—
268	C₄²	—	67	2	no	—
269	C₄²	—	67	1	no	—
270	C₄²	—	134	4	no	—
271	C₄²	—	134	2	no	—
272	C₄²	—	134	1	no	—
273	C₂³	#14524001	1	2	yes	—
274	C₂³	C₂×#11761629	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	268	2	no	—
281	C₂×C₄	—	268	4	no	—
282	C₂×C₄	—	268	4	no	—
283	C₂×C₄	—	134	2	no	—
284	C₂×C₄	—	134	1	no	—
285	C₂×C₄	—	134	1	no	—
286	C₂×C₄	#14524001	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
288	C₂×C₄	#41×D₆₇	1	2	yes	—
289	C₂×C₄	C₂×#11761627	1	2	yes	—
290	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	4	4	no	—
293	C₈	—	67	1	no	—
294	C₈	—	67	1	no	—
295	C₈	—	67	2	no	—
296	C₈	—	134	2	no	—
297	C₈	#14523999	1	1	yes	characteristic
298	C₈	#14523999	1	1	yes	characteristic
299	C₈	#14523999	1	2	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×#14524001	1	1	yes	characteristic
304	C₂²	#16863932	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	268	1	no	—
307	C₄	—	268	2	no	—
308	C₄	—	536	4	no	—
309	C₄	—	134	2	no	—
310	C₄	#117×D₆₇	1	1	yes	characteristic
311	C₄	C₂×#14523999	1	1	yes	characteristic
312	C₄	#16863927	1	2	yes	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	#22562604	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1813	C₂
socle	C₂×C₆₇	#2283
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1813	D₆₇
hypocenter	C₆₇	#20787
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1813

Complements

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

N	H ≅ G/N	type
2. #22560359	C₂	semidirect
3. #22560334	C₂	semidirect
6. #22560091	C₂	semidirect
8. #22559965	C₂	semidirect
14. #16863354	C₄	semidirect
17. #16863351	C₄	semidirect
19. M₁₆×Dic₆₇	C₄	semidirect
21. C₂×C₈×Dic₆₇	C₄	semidirect
168. C₆₇	#20787	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₆₇	2
#22560359	2
#22560334	2
#22560314	2
#22560255	2
#22560091	2
#22560012	2
#22559965	2
C₆₇×#1813	2
#20787	67