Subgroups — C₆₇×#19754 #30699005

Order 17152.

728 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	C₆₇×#3184	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3182	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3141	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1719	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1718	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1717	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1711	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
10	#533×C₆₇	C₂²	1	1	yes	characteristic
11	#532×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#500×C₆₇	C₂²	1	2	yes	—
14	#498×C₆₇	C₂²	1	1	yes	characteristic
15	#493×C₆₇	C₂²	1	1	yes	characteristic
16	#492×C₆₇	C₂²	1	2	yes	—
17	#491×C₆₇	—	2	2	no	—
18	#489×C₆₇	—	2	4	no	—
19	#487×C₆₇	C₂²	1	2	yes	—
20	#487×C₆₇	C₂²	1	2	yes	—
21	#487×C₆₇	C₂²	1	2	yes	—
22	#487×C₆₇	C₂²	1	2	yes	—
23	#487×C₆₇	C₂²	1	2	yes	—
24	#486×C₆₇	C₂²	1	2	yes	—
25	#486×C₆₇	C₂²	1	2	yes	—
26	#486×C₆₇	C₂²	1	2	yes	—
27	#486×C₆₇	C₂²	1	2	yes	—
28	#469×C₆₇	—	2	2	no	—
29	#468×C₆₇	—	2	4	no	—
30	#442×C₆₇	—	2	2	no	—
31	#440×C₆₇	—	2	4	no	—
32	#419×C₆₇	—	2	2	no	—
33	#417×C₆₇	—	2	4	no	—
34	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
36	#322×C₆₇	C₂²	1	2	yes	—
37	#322×C₆₇	C₂²	1	1	yes	characteristic
38	C₈²×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂²×Q₈×C₆₇	—	2	1	no	—
41	C₂×SD₁₆×C₆₇	—	2	4	no	—
42	C₂×D₈×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
44	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
45	C₄⋊Q₈×C₆₇	—	2	2	no	—
46	C₄⋊Q₈×C₆₇	—	2	1	no	—
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
50	#127×C₆₇	C₂³	1	2	yes	—
51	#125×C₆₇	—	2	1	no	—
52	#124×C₆₇	—	2	2	no	—
53	C₂²⋊Q₈×C₆₇	—	2	2	no	—
54	#121×C₆₇	—	2	2	no	—
55	#121×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	#117×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	#107×C₆₇	—	2	4	no	—
63	#106×C₆₇	—	2	2	no	—
64	#103×C₆₇	—	2	4	no	—
65	#103×C₆₇	—	2	4	no	—
66	#103×C₆₇	—	2	4	no	—
67	#103×C₆₇	—	2	4	no	—
68	#103×C₆₇	—	2	4	no	—
69	#102×C₆₇	—	2	4	no	—
70	#102×C₆₇	—	2	4	no	—
71	#102×C₆₇	—	2	4	no	—
72	#102×C₆₇	—	2	4	no	—
73	#97×C₆₇	D₄	1	4	yes	—
74	#97×C₆₇	D₄	1	2	yes	—
75	C₄×C₈×C₆₇	C₂³	1	2	yes	—
76	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂³	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	#41×C₆₇	—	4	8	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	4	4	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	4	2	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	SD₁₆×C₆₇	—	4	8	no	—
94	D₈×C₆₇	—	4	4	no	—
95	M₁₆×C₆₇	—	4	2	no	—
96	M₁₆×C₆₇	—	4	4	no	—
97	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	4	2	no	—
108	C₄⋊C₄×C₆₇	—	4	2	no	—
109	C₂²⋊C₄×C₆₇	—	4	1	no	—
110	C₂²⋊C₄×C₆₇	—	4	2	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₄²×C₆₇	—	4	1	no	—
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	2	yes	—
115	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₇	—	4	2	no	—
117	C₂³×C₆₇	#127	1	1	yes	characteristic
118	Q₈×C₆₇	—	4	4	no	—
119	Q₈×C₆₇	—	4	4	no	—
120	Q₈×C₆₇	—	4	4	no	—
121	Q₈×C₆₇	—	8	2	no	—
122	D₄×C₆₇	—	4	4	no	—
123	D₄×C₆₇	—	4	4	no	—
124	D₄×C₆₇	—	4	4	no	—
125	D₄×C₆₇	—	8	4	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	8	4	no	—
132	C₂×C₄×C₆₇	—	8	2	no	—
133	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#127	1	2	yes	—
135	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
137	C₈×C₆₇	—	2	2	no	—
138	C₈×C₆₇	—	2	4	no	—
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	2	4	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	8	2	no	—
144	C₂²×C₆₇	—	8	2	no	—
145	C₂²×C₆₇	—	8	2	no	—
146	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	8	2	no	—
150	C₄×C₆₇	—	8	2	no	—
151	C₄×C₆₇	—	8	2	no	—
152	C₄×C₆₇	#575	1	2	yes	—
153	C₄×C₆₇	#575	1	4	yes	—
154	#19754	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₆₇	—	4	1	no	—
156	C₂×C₆₇	—	16	2	no	—
157	C₂×C₆₇	#3153	1	2	yes	—
158	C₂×C₆₇	#3153	1	1	yes	characteristic
159	#3184	C₂×C₆₇	1	1	yes	characteristic
160	#3182	C₂×C₆₇	1	2	yes	complemented
161	#3141	C₂×C₆₇	1	2	yes	complemented
162	#3139	C₂×C₆₇	1	1	yes	characteristic, complemented
163	#1719	C₂×C₆₇	1	2	yes	complemented
164	#1718	C₂×C₆₇	1	2	yes	complemented
165	#1717	C₂×C₆₇	1	2	yes	complemented
166	#1711	C₂×C₆₇	1	2	yes	complemented
167	#1455	C₂×C₆₇	1	1	yes	characteristic, complemented
168	C₆₇	#19754	1	1	yes	characteristic, direct product factor, Sylow
169	#533	C₂²×C₆₇	1	1	yes	characteristic
170	#532	C₂²×C₆₇	1	2	yes	—
171	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
172	#500	C₂²×C₆₇	1	2	yes	—
173	#498	C₂²×C₆₇	1	1	yes	characteristic
174	#493	C₂²×C₆₇	1	1	yes	characteristic
175	#492	C₂²×C₆₇	1	2	yes	—
176	#491	—	2	2	no	—
177	#489	—	2	4	no	—
178	#487	C₂²×C₆₇	1	2	yes	—
179	#487	C₂²×C₆₇	1	2	yes	—
180	#487	C₂²×C₆₇	1	2	yes	—
181	#487	C₂²×C₆₇	1	2	yes	—
182	#487	C₂²×C₆₇	1	2	yes	—
183	#486	C₂²×C₆₇	1	2	yes	—
184	#486	C₂²×C₆₇	1	2	yes	—
185	#486	C₂²×C₆₇	1	2	yes	—
186	#486	C₂²×C₆₇	1	2	yes	—
187	#469	—	2	2	no	—
188	#468	—	2	4	no	—
189	#442	—	2	2	no	—
190	#440	—	2	4	no	—
191	#419	—	2	2	no	—
192	#417	—	2	4	no	—
193	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
194	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
195	#322	C₂²×C₆₇	1	2	yes	—
196	#322	C₂²×C₆₇	1	1	yes	characteristic
197	C₈²	C₂²×C₆₇	1	1	yes	characteristic
198	C₂×#41	—	2	2	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂×SD₁₆	—	2	4	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
203	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
208	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
209	#127	C₂³×C₆₇	1	2	yes	—
210	#125	—	2	1	no	—
211	#124	—	2	2	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	#121	—	2	2	no	—
214	#121	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	#117	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
221	#107	—	2	4	no	—
222	#106	—	2	2	no	—
223	#103	—	2	4	no	—
224	#103	—	2	4	no	—
225	#103	—	2	4	no	—
226	#103	—	2	4	no	—
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	#97	D₄×C₆₇	1	4	yes	—
233	#97	D₄×C₆₇	1	2	yes	—
234	C₄×C₈	C₂³×C₆₇	1	2	yes	—
235	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₆₇	1	2	yes	—
237	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
238	#41	—	4	8	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
252	SD₁₆	—	4	8	no	—
253	D₈	—	4	4	no	—
254	M₁₆	—	4	2	no	—
255	M₁₆	—	4	4	no	—
256	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
257	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
259	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
273	C₄²	C₂×D₄×C₆₇	1	2	yes	—
274	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
275	C₂³	—	4	2	no	—
276	C₂³	#127×C₆₇	1	1	yes	characteristic
277	Q₈	—	4	4	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	4	4	no	—
280	Q₈	—	8	2	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	8	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
293	C₂×C₄	#127×C₆₇	1	2	yes	—
294	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	4	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	#575×C₆₇	1	2	yes	—
312	C₄	#575×C₆₇	1	4	yes	—
313	C₂	—	4	1	no	—
314	C₂	—	16	2	no	—
315	C₂	C₆₇×#3153	1	2	yes	—
316	C₂	C₆₇×#3153	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3184	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3182	C₂	semidirect
3. C₆₇×#3141	C₂	semidirect
4. C₆₇×#3139	C₂	semidirect
5. C₆₇×#1719	C₂	semidirect
6. C₆₇×#1718	C₂	semidirect
7. C₆₇×#1717	C₂	semidirect
8. C₆₇×#1711	C₂	semidirect
9. C₆₇×#1455	C₂	semidirect
154. #19754	C₆₇	direct (N × H)
160. #3182	C₂×C₆₇	semidirect
161. #3141	C₂×C₆₇	semidirect
162. #3139	C₂×C₆₇	semidirect
163. #1719	C₂×C₆₇	semidirect
164. #1718	C₂×C₆₇	semidirect
165. #1717	C₂×C₆₇	semidirect
166. #1711	C₂×C₆₇	semidirect
167. #1455	C₂×C₆₇	semidirect
168. C₆₇	#19754	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3184	2
C₆₇×#3182	2
C₆₇×#3141	2
C₆₇×#3139	2
C₆₇×#1719	2
C₆₇×#1718	2
C₆₇×#1717	2
C₆₇×#1711	2
C₆₇×#1455	2
#19754	67

Subgroups — C67×#19754 #30699005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19754 #30699005