Subgroups — C₆₇×#21989 #30701240

Order 17152.

786 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1949	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1854	C₂	1	4	yes	maximal
6	C₆₇×#1847	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1836	C₂	1	2	yes	maximal, complemented
8	C₂×#117×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	#432×C₆₇	C₄	1	8	yes	complemented
11	C₂×#107×C₆₇	C₂²	1	4	yes	—
12	C₂×#106×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	#384×C₆₇	C₂²	1	2	yes	—
17	#383×C₆₇	—	2	1	no	—
18	#382×C₆₇	C₂²	1	1	yes	characteristic
19	#382×C₆₇	C₂²	1	1	yes	characteristic
20	#382×C₆₇	C₂²	1	2	yes	—
21	#381×C₆₇	—	2	1	no	—
22	#380×C₆₇	—	2	1	no	—
23	#380×C₆₇	—	2	1	no	—
24	#380×C₆₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
26	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
29	#376×C₆₇	—	2	1	no	—
30	#340×C₆₇	C₂²	1	8	yes	—
31	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
35	#117×C₆₇	—	4	2	no	—
36	#117×C₆₇	—	4	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	#107×C₆₇	—	4	4	no	—
53	#106×C₆₇	—	4	4	no	—
54	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
55	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
56	#97×C₆₇	C₂×C₄	1	8	yes	—
57	#95×C₆₇	—	2	2	no	—
58	#95×C₆₇	—	2	1	no	—
59	#95×C₆₇	—	2	1	no	—
60	#95×C₆₇	—	2	1	no	—
61	#95×C₆₇	—	2	1	no	—
62	#95×C₆₇	—	2	2	no	—
63	#95×C₆₇	—	2	2	no	—
64	#95×C₆₇	—	2	2	no	—
65	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	4	2	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
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	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	4	4	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	4	no	—
93	C₂²⋊C₄×C₆₇	—	4	4	no	—
94	C₄²×C₆₇	—	4	4	no	—
95	C₄²×C₆₇	—	4	4	no	—
96	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
97	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
98	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
99	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
101	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
102	C₂³×C₆₇	—	2	1	no	—
103	C₂³×C₆₇	—	2	1	no	—
104	C₂³×C₆₇	—	2	2	no	—
105	C₂³×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	4	4	no	—
126	C₂×C₄×C₆₇	—	4	4	no	—
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
135	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
136	C₈×C₆₇	—	2	8	no	—
137	C₂²×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	2	2	no	—
140	C₂²×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	—	4	2	no	—
142	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₆₇	#535	1	1	yes	characteristic
144	C₂²×C₆₇	#536	1	1	yes	characteristic
145	C₂²×C₆₇	#575	1	2	yes	—
146	C₂²×C₆₇	#575	1	2	yes	—
147	C₄×C₆₇	—	2	2	no	—
148	C₄×C₆₇	—	2	2	no	—
149	C₄×C₆₇	—	2	2	no	—
150	C₄×C₆₇	—	2	2	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	8	4	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	#520	1	2	yes	—
156	C₄×C₆₇	#544	1	2	yes	—
157	#21989	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₆₇	—	4	2	no	—
159	C₂×C₆₇	#2317	1	1	yes	characteristic
160	C₂×C₆₇	#2911	1	1	yes	characteristic
161	C₂×C₆₇	#2911	1	1	yes	characteristic
162	C₂×C₆₇	#3219	1	2	yes	—
163	C₂×C₆₇	#3240	1	2	yes	—
164	C₂×#432	C₂×C₆₇	1	1	yes	characteristic
165	#2285	C₂×C₆₇	1	1	yes	characteristic
166	#2283	C₂×C₆₇	1	1	yes	characteristic
167	#1949	C₂×C₆₇	1	4	yes	complemented
168	#1854	C₂×C₆₇	1	4	yes	—
169	#1847	C₂×C₆₇	1	2	yes	complemented
170	#1836	C₂×C₆₇	1	2	yes	complemented
171	C₆₇	#21989	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#117	C₂²×C₆₇	1	2	yes	—
173	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
174	#432	C₄×C₆₇	1	8	yes	complemented
175	C₂×#107	C₂²×C₆₇	1	4	yes	—
176	C₂×#106	C₂²×C₆₇	1	4	yes	—
177	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
178	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
179	C₂×#97	C₂²×C₆₇	1	2	yes	—
180	#384	C₂²×C₆₇	1	2	yes	—
181	#383	—	2	1	no	—
182	#382	C₂²×C₆₇	1	1	yes	characteristic
183	#382	C₂²×C₆₇	1	1	yes	characteristic
184	#382	C₂²×C₆₇	1	2	yes	—
185	#381	—	2	1	no	—
186	#380	—	2	1	no	—
187	#380	—	2	1	no	—
188	#380	C₂²×C₆₇	1	2	yes	—
189	C₄×C₄⋊C₄	—	2	1	no	—
190	C₄×C₄⋊C₄	—	2	1	no	—
191	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
192	C₄×C₂²⋊C₄	—	2	1	no	—
193	#376	—	2	1	no	—
194	#340	C₂²×C₆₇	1	8	yes	—
195	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
196	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
197	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
199	#117	—	4	2	no	—
200	#117	—	4	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
204	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
211	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
212	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
213	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
214	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
216	#107	—	4	4	no	—
217	#106	—	4	4	no	—
218	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
219	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
220	#97	C₂×C₄×C₆₇	1	8	yes	—
221	#95	—	2	2	no	—
222	#95	—	2	1	no	—
223	#95	—	2	1	no	—
224	#95	—	2	1	no	—
225	#95	—	2	1	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
238	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
239	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	C₂²×C₄×C₆₇	1	8	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₄²	—	4	4	no	—
259	C₄²	—	4	4	no	—
260	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
261	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
262	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
263	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
264	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
265	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
271	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
294	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
296	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
298	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
299	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
300	C₈	—	2	8	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
307	C₂²	#535×C₆₇	1	1	yes	characteristic
308	C₂²	#536×C₆₇	1	1	yes	characteristic
309	C₂²	#575×C₆₇	1	2	yes	—
310	C₂²	#575×C₆₇	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	8	4	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	#520×C₆₇	1	2	yes	—
320	C₄	#544×C₆₇	1	2	yes	—
321	C₂	—	4	2	no	—
322	C₂	C₆₇×#2317	1	1	yes	characteristic
323	C₂	C₆₇×#2911	1	1	yes	characteristic
324	C₂	C₆₇×#2911	1	1	yes	characteristic
325	C₂	C₆₇×#3219	1	2	yes	—
326	C₂	C₆₇×#3240	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1949	C₂	semidirect
6. C₆₇×#1847	C₂	semidirect
7. C₆₇×#1836	C₂	semidirect
10. #432×C₆₇	C₄	semidirect
157. #21989	C₆₇	direct (N × H)
167. #1949	C₂×C₆₇	semidirect
169. #1847	C₂×C₆₇	semidirect
170. #1836	C₂×C₆₇	semidirect
171. C₆₇	#21989	direct (N × H)
174. #432	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₆₇	2
C₆₇×#2285	2
C₆₇×#2283	2
C₆₇×#1949	2
C₆₇×#1854	2
C₆₇×#1847	2
C₆₇×#1836	2
#21989	67

Subgroups — C67×#21989 #30701240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21989 #30701240