Subgroups — C₆₇×#33157 #30712408

Order 17152.

730 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2629	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1989	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1948	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1920	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1919	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1853	C₂	1	2	yes	maximal
8	C₆₇×#1825	C₂	1	2	yes	maximal
9	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#106×C₆₇	C₂²	1	2	yes	—
15	C₂×#106×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
18	#398×C₆₇	—	2	2	no	—
19	#396×C₆₇	—	2	2	no	—
20	#389×C₆₇	C₂²	1	2	yes	—
21	#384×C₆₇	C₂²	1	1	yes	characteristic
22	#380×C₆₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
27	#340×C₆₇	C₂²	1	2	yes	—
28	#340×C₆₇	C₂²	1	2	yes	—
29	#340×C₆₇	C₂²	1	2	yes	—
30	#340×C₆₇	C₂²	1	2	yes	—
31	#340×C₆₇	C₂²	1	2	yes	—
32	#340×C₆₇	C₂²	1	2	yes	—
33	#336×C₆₇	C₂²	1	1	yes	characteristic
34	#336×C₆₇	C₂²	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
39	#125×C₆₇	—	2	4	no	—
40	#117×C₆₇	—	2	2	no	—
41	#117×C₆₇	—	2	4	no	—
42	#117×C₆₇	Q₈	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
53	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
56	C₂×C₄²×C₆₇	—	2	1	no	—
57	C₂×C₄²×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#106×C₆₇	—	2	4	no	—
62	#106×C₆₇	—	2	4	no	—
63	C₄⋊C₈×C₆₇	—	2	4	no	—
64	C₂²⋊C₈×C₆₇	—	2	4	no	—
65	#95×C₆₇	—	2	2	no	—
66	#95×C₆₇	—	2	2	no	—
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	—	2	2	no	—
69	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	8	2	no	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₄⋊C₄×C₆₇	—	4	4	no	—
98	C₄⋊C₄×C₆₇	—	8	2	no	—
99	C₄⋊C₄×C₆₇	—	8	2	no	—
100	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
101	C₄⋊C₄×C₆₇	#41	1	4	yes	—
102	C₂²⋊C₄×C₆₇	—	2	4	no	—
103	C₂²⋊C₄×C₆₇	—	2	4	no	—
104	C₂²⋊C₄×C₆₇	—	8	2	no	—
105	C₄²×C₆₇	—	2	4	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	8	2	no	—
108	C₄²×C₆₇	—	8	2	no	—
109	C₄²×C₆₇	#41	1	4	yes	—
110	C₂³×C₆₇	—	2	1	no	—
111	C₂³×C₆₇	—	2	1	no	—
112	C₂³×C₆₇	—	2	2	no	—
113	C₂³×C₆₇	—	2	2	no	—
114	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
115	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	4	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	4	2	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	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	8	4	no	—
138	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	#123	1	4	yes	—
140	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
143	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
144	C₈×C₆₇	—	4	2	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₂²×C₆₇	—	2	4	no	—
147	C₂²×C₆₇	—	2	2	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	4	1	no	—
150	C₂²×C₆₇	—	4	1	no	—
151	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
153	C₂²×C₆₇	#525	1	1	yes	characteristic
154	C₂²×C₆₇	#577	1	2	yes	—
155	C₂²×C₆₇	#577	1	2	yes	—
156	C₄×C₆₇	—	2	4	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	4	4	no	—
159	C₄×C₆₇	—	4	2	no	—
160	C₄×C₆₇	—	4	4	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	—	8	2	no	—
163	C₄×C₆₇	#529	1	2	yes	—
164	C₄×C₆₇	#533	1	2	yes	—
165	#33157	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂×C₆₇	—	4	2	no	—
167	C₂×C₆₇	#2855	1	1	yes	characteristic
168	C₂×C₆₇	#3084	1	2	yes	—
169	C₂×C₆₇	#3099	1	2	yes	—
170	C₂×C₆₇	#3123	1	1	yes	characteristic
171	C₂×C₆₇	#3126	1	1	yes	characteristic
172	#2629	C₂×C₆₇	1	1	yes	characteristic
173	#2274	C₂×C₆₇	1	1	yes	characteristic
174	#1989	C₂×C₆₇	1	2	yes	complemented
175	#1948	C₂×C₆₇	1	2	yes	complemented
176	#1920	C₂×C₆₇	1	2	yes	complemented
177	#1919	C₂×C₆₇	1	2	yes	complemented
178	#1853	C₂×C₆₇	1	2	yes	—
179	#1825	C₂×C₆₇	1	2	yes	—
180	#1798	C₂×C₆₇	1	1	yes	characteristic
181	C₆₇	#33157	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
183	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
184	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
185	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
186	C₂×#106	C₂²×C₆₇	1	2	yes	—
187	C₂×#106	C₂²×C₆₇	1	2	yes	—
188	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
189	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
190	#398	—	2	2	no	—
191	#396	—	2	2	no	—
192	#389	C₂²×C₆₇	1	2	yes	—
193	#384	C₂²×C₆₇	1	1	yes	characteristic
194	#380	C₂²×C₆₇	1	2	yes	—
195	C₄×C₄⋊C₄	—	2	2	no	—
196	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
197	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
198	C₄×C₂²⋊C₄	—	2	2	no	—
199	#340	C₂²×C₆₇	1	2	yes	—
200	#340	C₂²×C₆₇	1	2	yes	—
201	#340	C₂²×C₆₇	1	2	yes	—
202	#340	C₂²×C₆₇	1	2	yes	—
203	#340	C₂²×C₆₇	1	2	yes	—
204	#340	C₂²×C₆₇	1	2	yes	—
205	#336	C₂²×C₆₇	1	1	yes	characteristic
206	#336	C₂²×C₆₇	1	1	yes	characteristic
207	C₂³×C₄	—	2	1	no	—
208	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
210	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
211	#125	—	2	4	no	—
212	#117	—	2	2	no	—
213	#117	—	2	4	no	—
214	#117	Q₈×C₆₇	1	4	yes	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
225	C₂×C₂²⋊C₄	—	2	1	no	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
228	C₂×C₄²	—	2	1	no	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	—	2	2	no	—
231	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
233	#106	—	2	4	no	—
234	#106	—	2	4	no	—
235	C₄⋊C₈	—	2	4	no	—
236	C₂²⋊C₈	—	2	4	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	8	2	no	—
250	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
253	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
254	C₂²×C₄	#41×C₆₇	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄⋊C₄	—	8	2	no	—
271	C₄⋊C₄	—	8	2	no	—
272	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
273	C₄⋊C₄	#41×C₆₇	1	4	yes	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	8	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	8	2	no	—
280	C₄²	—	8	2	no	—
281	C₄²	#41×C₆₇	1	4	yes	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	2	2	no	—
286	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
287	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
311	C₂×C₄	#123×C₆₇	1	4	yes	—
312	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
313	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
315	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
316	C₈	—	4	2	no	—
317	C₈	—	4	2	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
324	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
325	C₂²	#525×C₆₇	1	1	yes	characteristic
326	C₂²	#577×C₆₇	1	2	yes	—
327	C₂²	#577×C₆₇	1	2	yes	—
328	C₄	—	2	4	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	#529×C₆₇	1	2	yes	—
336	C₄	#533×C₆₇	1	2	yes	—
337	C₂	—	4	2	no	—
338	C₂	C₆₇×#2855	1	1	yes	characteristic
339	C₂	C₆₇×#3084	1	2	yes	—
340	C₂	C₆₇×#3099	1	2	yes	—
341	C₂	C₆₇×#3123	1	1	yes	characteristic
342	C₂	C₆₇×#3126	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1989	C₂	semidirect
4. C₆₇×#1948	C₂	semidirect
5. C₆₇×#1920	C₂	semidirect
6. C₆₇×#1919	C₂	semidirect
165. #33157	C₆₇	direct (N × H)
174. #1989	C₂×C₆₇	semidirect
175. #1948	C₂×C₆₇	semidirect
176. #1920	C₂×C₆₇	semidirect
177. #1919	C₂×C₆₇	semidirect
181. C₆₇	#33157	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×#2629	2
C₆₇×#2274	2
C₆₇×#1989	2
C₆₇×#1948	2
C₆₇×#1920	2
C₆₇×#1919	2
C₆₇×#1853	2
C₆₇×#1825	2
C₆₇×#1798	2
#33157	67

Subgroups — C67×#33157 #30712408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33157 #30712408