Subgroups — C₆₇×#10798 #30690049

Order 17152.

870 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#336×C₆₇	C₂	1	2	yes	maximal
3	C₂×#336×C₆₇	C₂	1	2	yes	maximal
4	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₄×#95×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
7	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#97×C₆₇	—	2	4	no	—
12	C₂×#97×C₆₇	C₄	1	4	yes	complemented
13	C₂×#95×C₆₇	C₄	1	4	yes	—
14	#336×C₆₇	—	2	4	no	—
15	#336×C₆₇	—	2	4	no	—
16	#336×C₆₇	C₄	1	4	yes	complemented
17	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	—	2	2	no	—
21	C₂²×C₈×C₆₇	—	2	2	no	—
22	C₂²×C₈×C₆₇	—	2	2	no	—
23	C₂²×C₈×C₆₇	—	2	1	no	—
24	C₂²×C₈×C₆₇	—	2	1	no	—
25	C₂²×C₈×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	—	2	2	no	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	D₄	1	4	yes	—
35	C₂²×C₈×C₆₇	D₄	1	2	yes	—
36	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₆₇	—	2	2	no	—
38	C₂×C₄²×C₆₇	—	2	2	no	—
39	C₂×C₄²×C₆₇	—	2	2	no	—
40	C₂×C₄²×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	#97×C₆₇	—	2	8	no	—
48	#95×C₆₇	—	2	8	no	—
49	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	—	2	4	no	—
51	C₂²×C₄×C₆₇	—	2	2	no	—
52	C₂²×C₄×C₆₇	—	2	2	no	—
53	C₂²×C₄×C₆₇	—	2	4	no	—
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	2	4	no	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
61	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	#41	1	2	yes	—
68	C₂²×C₄×C₆₇	#41	1	2	yes	—
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	#41	1	2	yes	—
71	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
72	C₂×C₈×C₆₇	—	2	2	no	—
73	C₂×C₈×C₆₇	—	2	2	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	4	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	4	2	no	—
84	C₂×C₈×C₆₇	—	4	2	no	—
85	C₂×C₈×C₆₇	—	4	2	no	—
86	C₂×C₈×C₆₇	—	4	2	no	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
88	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
89	C₄²×C₆₇	—	2	2	no	—
90	C₄²×C₆₇	—	2	2	no	—
91	C₄²×C₆₇	—	2	4	no	—
92	C₄²×C₆₇	—	4	2	no	—
93	C₄²×C₆₇	—	4	2	no	—
94	C₄²×C₆₇	—	4	2	no	—
95	C₄²×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
97	C₂³×C₆₇	#117	1	2	yes	—
98	C₂³×C₆₇	C₄×D₄	1	2	yes	—
99	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
100	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
101	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
102	C₂³×C₆₇	#131	1	4	yes	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	8	no	—
107	C₂×C₄×C₆₇	—	2	8	no	—
108	C₂×C₄×C₆₇	—	2	8	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	4	4	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#117	1	2	yes	—
122	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#121	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	#123	1	2	yes	—
128	C₂×C₄×C₆₇	#123	1	2	yes	—
129	C₂×C₄×C₆₇	#124	1	2	yes	—
130	C₂×C₄×C₆₇	#125	1	2	yes	—
131	C₂×C₄×C₆₇	#126	1	2	yes	—
132	C₂×C₄×C₆₇	#126	1	2	yes	—
133	C₈×C₆₇	—	2	4	no	—
134	C₈×C₆₇	—	2	4	no	—
135	C₈×C₆₇	—	4	4	no	—
136	C₂²×C₆₇	#376	1	1	yes	characteristic
137	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₆₇	#387	1	2	yes	—
140	C₂²×C₆₇	#388	1	2	yes	—
141	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
142	C₂²×C₆₇	#405	1	2	yes	—
143	C₂²×C₆₇	#432	1	4	yes	—
144	C₂²×C₆₇	#433	1	4	yes	—
145	C₂²×C₆₇	#435	1	4	yes	—
146	C₂²×C₆₇	#435	1	4	yes	—
147	C₂²×C₆₇	#436	1	4	yes	—
148	C₂²×C₆₇	#446	1	4	yes	—
149	C₄×C₆₇	—	4	8	no	—
150	C₄×C₆₇	—	4	4	no	—
151	C₄×C₆₇	#380	1	1	yes	characteristic
152	C₄×C₆₇	#381	1	1	yes	characteristic
153	C₄×C₆₇	#382	1	1	yes	characteristic
154	C₄×C₆₇	#383	1	1	yes	characteristic
155	C₄×C₆₇	#387	1	2	yes	—
156	C₄×C₆₇	#388	1	2	yes	—
157	#10798	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₆₇	#1434	1	1	yes	characteristic
159	C₂×C₆₇	#1745	1	1	yes	characteristic
160	C₂×C₆₇	#1746	1	1	yes	characteristic
161	C₂×C₆₇	#1753	1	2	yes	—
162	C₂×C₆₇	#1771	1	2	yes	—
163	C₂×C₆₇	#1917	1	4	yes	—
164	C₂×C₆₇	#1924	1	4	yes	—
165	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
166	C₂×#336	C₂×C₆₇	1	2	yes	—
167	C₂×#336	C₂×C₆₇	1	2	yes	—
168	C₂×#336	C₂×C₆₇	1	1	yes	characteristic
169	C₄×#95	C₂×C₆₇	1	1	yes	characteristic
170	C₆₇	#10798	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
172	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
173	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
174	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
175	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
176	C₂×#97	—	2	4	no	—
177	C₂×#97	C₄×C₆₇	1	4	yes	complemented
178	C₂×#95	C₄×C₆₇	1	4	yes	—
179	#336	—	2	4	no	—
180	#336	—	2	4	no	—
181	#336	C₄×C₆₇	1	4	yes	complemented
182	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
183	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
184	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
197	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
198	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
199	C₂²×C₈	D₄×C₆₇	1	4	yes	—
200	C₂²×C₈	D₄×C₆₇	1	2	yes	—
201	C₂²×C₈	Q₈×C₆₇	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
211	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
212	#97	—	2	8	no	—
213	#95	—	2	8	no	—
214	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
226	C₂²×C₄	M₁₆×C₆₇	1	8	yes	—
227	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	#41×C₆₇	1	2	yes	—
233	C₂²×C₄	#41×C₆₇	1	2	yes	—
234	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
235	C₂²×C₄	#41×C₆₇	1	2	yes	—
236	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
253	C₂×C₈	C₄⋊C₄×C₆₇	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₂³	C₂×C₄²×C₆₇	1	1	yes	characteristic
262	C₂³	#117×C₆₇	1	2	yes	—
263	C₂³	C₄×D₄×C₆₇	1	2	yes	—
264	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
265	C₂³	C₄×Q₈×C₆₇	1	1	yes	characteristic
266	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
267	C₂³	#131×C₆₇	1	4	yes	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
282	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
283	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
284	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
285	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
286	C₂×C₄	#117×C₆₇	1	2	yes	—
287	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
288	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
289	C₂×C₄	C₄×Q₈×C₆₇	1	1	yes	characteristic
290	C₂×C₄	#121×C₆₇	1	2	yes	—
291	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
292	C₂×C₄	#123×C₆₇	1	2	yes	—
293	C₂×C₄	#123×C₆₇	1	2	yes	—
294	C₂×C₄	#124×C₆₇	1	2	yes	—
295	C₂×C₄	#125×C₆₇	1	2	yes	—
296	C₂×C₄	#126×C₆₇	1	2	yes	—
297	C₂×C₄	#126×C₆₇	1	2	yes	—
298	C₈	—	2	4	no	—
299	C₈	—	2	4	no	—
300	C₈	—	4	4	no	—
301	C₂²	#376×C₆₇	1	1	yes	characteristic
302	C₂²	C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
303	C₂²	C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
304	C₂²	#387×C₆₇	1	2	yes	—
305	C₂²	#388×C₆₇	1	2	yes	—
306	C₂²	C₄×M₁₆×C₆₇	1	2	yes	—
307	C₂²	#405×C₆₇	1	2	yes	—
308	C₂²	#432×C₆₇	1	4	yes	—
309	C₂²	#433×C₆₇	1	4	yes	—
310	C₂²	#435×C₆₇	1	4	yes	—
311	C₂²	#435×C₆₇	1	4	yes	—
312	C₂²	#436×C₆₇	1	4	yes	—
313	C₂²	#446×C₆₇	1	4	yes	—
314	C₄	—	4	4	no	—
315	C₄	—	4	8	no	—
316	C₄	#380×C₆₇	1	1	yes	characteristic
317	C₄	#381×C₆₇	1	1	yes	characteristic
318	C₄	#382×C₆₇	1	1	yes	characteristic
319	C₄	#383×C₆₇	1	1	yes	characteristic
320	C₄	#387×C₆₇	1	2	yes	—
321	C₄	#388×C₆₇	1	2	yes	—
322	C₂	C₆₇×#1434	1	1	yes	characteristic
323	C₂	C₆₇×#1745	1	1	yes	characteristic
324	C₂	C₆₇×#1746	1	1	yes	characteristic
325	C₂	C₆₇×#1753	1	2	yes	—
326	C₂	C₆₇×#1771	1	2	yes	—
327	C₂	C₆₇×#1917	1	4	yes	—
328	C₂	C₆₇×#1924	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator	C₂³	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×#97×C₆₇	C₄	semidirect
16. #336×C₆₇	C₄	semidirect
157. #10798	C₆₇	direct (N × H)
170. C₆₇	#10798	direct (N × H)
177. C₂×#97	C₄×C₆₇	semidirect
181. #336	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#97×C₆₇	2
C₂×#336×C₆₇	2
C₄×#95×C₆₇	2
#10798	67

Subgroups — C67×#10798 #30690049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10798 #30690049