Subgroups — C₆₇×#17890 #30697141

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2965	C₂	1	2	yes	maximal, complemented
2	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1594	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1593	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1592	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1476	C₂	1	2	yes	maximal, complemented
8	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₆₇	C₄	1	1	yes	characteristic
10	C₄×#41×C₆₇	C₂²	1	2	yes	—
11	#446×C₆₇	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₆₇	C₄	1	2	yes	—
15	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
16	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
17	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
18	#436×C₆₇	C₂²	1	4	yes	—
19	#435×C₆₇	—	2	4	no	—
20	#431×C₆₇	—	2	2	no	—
21	#431×C₆₇	C₂²	1	2	yes	—
22	#427×C₆₇	C₄	1	1	yes	characteristic
23	#416×C₆₇	C₄	1	2	yes	—
24	C₄×M₁₆×C₆₇	—	2	2	no	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	#335×C₆₇	C₂²	1	2	yes	—
29	#334×C₆₇	C₂²	1	2	yes	—
30	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
31	D₄⋊C₈×C₆₇	C₂²	1	4	yes	—
32	#322×C₆₇	C₂²	1	2	yes	—
33	#322×C₆₇	C₂²	1	2	yes	—
34	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
35	#135×C₆₇	—	2	4	no	—
36	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₆₇	—	2	4	no	—
40	C₂²×C₈×C₆₇	—	2	1	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	D₄	1	2	yes	—
44	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
46	C₄×D₄×C₆₇	—	2	2	no	—
47	C₄×D₄×C₆₇	—	2	2	no	—
48	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₆₇	C₂³	1	2	yes	—
50	#117×C₆₇	—	2	2	no	—
51	#117×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
55	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
56	#106×C₆₇	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
59	#103×C₆₇	C₂×C₄	1	2	yes	—
60	#103×C₆₇	C₂×C₄	1	2	yes	—
61	#102×C₆₇	C₂×C₄	1	2	yes	—
62	#102×C₆₇	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈×C₆₇	—	2	2	no	—
64	C₂²⋊C₈×C₆₇	—	2	4	no	—
65	#97×C₆₇	—	2	4	no	—
66	C₄×C₈×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	D₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂³	1	2	yes	—
72	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	2	yes	—
75	#41×C₆₇	—	2	4	no	—
76	#41×C₆₇	—	2	4	no	—
77	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂×D₄×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	Q₁₆×C₆₇	—	2	2	no	—
87	SD₁₆×C₆₇	—	2	4	no	—
88	D₈×C₆₇	—	2	2	no	—
89	M₁₆×C₆₇	—	4	8	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	—	4	2	no	—
95	C₂×C₈×C₆₇	M₁₆	1	4	yes	complemented
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	#41	1	1	yes	characteristic
114	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
117	Q₈×C₆₇	—	2	2	no	—
118	Q₈×C₆₇	—	2	2	no	—
119	D₄×C₆₇	—	2	2	no	—
120	D₄×C₆₇	—	2	2	no	—
121	D₄×C₆₇	—	4	4	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	4	4	no	—
129	C₂×C₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
134	C₂×C₄×C₆₇	#131	1	2	yes	—
135	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₈×C₆₇	—	2	1	no	—
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	4	no	—
141	C₈×C₆₇	—	4	2	no	—
142	C₈×C₆₇	—	4	4	no	—
143	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
144	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	4	2	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	4	2	no	—
150	C₂²×C₆₇	#436	1	2	yes	—
151	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	4	2	no	—
155	C₄×C₆₇	—	4	2	no	—
156	C₄×C₆₇	—	4	2	no	—
157	C₄×C₆₇	#436	1	2	yes	—
158	C₄×C₆₇	#444	1	2	yes	—
159	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
160	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
161	C₄×C₆₇	#575	1	2	yes	—
162	#17890	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₆₇	—	2	2	no	—
164	C₂×C₆₇	—	8	2	no	—
165	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
166	C₂×C₆₇	#2943	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
168	#2965	C₂×C₆₇	1	2	yes	complemented
169	C₄×#135	C₂×C₆₇	1	1	yes	characteristic
170	#1594	C₂×C₆₇	1	2	yes	complemented
171	#1593	C₂×C₆₇	1	2	yes	complemented
172	#1592	C₂×C₆₇	1	4	yes	complemented
173	#1564	C₂×C₆₇	1	1	yes	characteristic, complemented
174	#1476	C₂×C₆₇	1	2	yes	complemented
175	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
176	C₆₇	#17890	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×#135	C₄×C₆₇	1	1	yes	characteristic
178	C₄×#41	C₂²×C₆₇	1	2	yes	—
179	#446	C₂²×C₆₇	1	4	yes	—
180	C₄×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
181	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
182	C₄×SD₁₆	C₄×C₆₇	1	2	yes	—
183	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
184	C₄×D₈	C₄×C₆₇	1	1	yes	characteristic
185	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
186	#436	C₂²×C₆₇	1	4	yes	—
187	#435	—	2	4	no	—
188	#431	—	2	2	no	—
189	#431	C₂²×C₆₇	1	2	yes	—
190	#427	C₄×C₆₇	1	1	yes	characteristic
191	#416	C₄×C₆₇	1	2	yes	—
192	C₄×M₁₆	—	2	2	no	—
193	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
196	#335	C₂²×C₆₇	1	2	yes	—
197	#334	C₂²×C₆₇	1	2	yes	—
198	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
199	D₄⋊C₈	C₂²×C₆₇	1	4	yes	—
200	#322	C₂²×C₆₇	1	2	yes	—
201	#322	C₂²×C₆₇	1	2	yes	—
202	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
203	#135	—	2	4	no	—
204	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
205	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
206	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
207	C₂×M₁₆	—	2	4	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
211	C₂²×C₈	D₄×C₆₇	1	2	yes	—
212	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
213	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
217	C₄×D₄	C₂³×C₆₇	1	2	yes	—
218	#117	—	2	2	no	—
219	#117	C₂×C₄×C₆₇	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
222	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
223	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
224	#106	C₂×C₄×C₆₇	1	2	yes	—
225	C₄⋊C₈	—	2	4	no	—
226	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
227	#103	C₂×C₄×C₆₇	1	2	yes	—
228	#103	C₂×C₄×C₆₇	1	2	yes	—
229	#102	C₂×C₄×C₆₇	1	2	yes	—
230	#102	C₂×C₄×C₆₇	1	2	yes	—
231	C₂²⋊C₈	—	2	2	no	—
232	C₂²⋊C₈	—	2	4	no	—
233	#97	—	2	4	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
237	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
238	C₄×C₈	D₄×C₆₇	1	2	yes	—
239	C₄×C₈	C₂³×C₆₇	1	2	yes	—
240	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₆₇	1	2	yes	—
243	#41	—	2	4	no	—
244	#41	—	2	4	no	—
245	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
253	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	SD₁₆	—	2	4	no	—
256	D₈	—	2	2	no	—
257	M₁₆	—	4	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	M₁₆×C₆₇	1	4	yes	complemented
264	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
265	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
267	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
272	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₄²	#41×C₆₇	1	1	yes	characteristic
282	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	D₄	—	2	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
302	C₂×C₄	#131×C₆₇	1	2	yes	—
303	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	4	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	4	no	—
311	C₈	C₂×M₁₆×C₆₇	1	2	yes	—
312	C₈	C₂×M₁₆×C₆₇	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	#436×C₆₇	1	2	yes	—
319	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	#436×C₆₇	1	2	yes	—
326	C₄	#444×C₆₇	1	2	yes	—
327	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
328	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
329	C₄	#575×C₆₇	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	8	2	no	—
332	C₂	C₂×#436×C₆₇	1	1	yes	characteristic
333	C₂	C₆₇×#2943	1	1	yes	characteristic
334	C₂	C₂×#444×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2965	C₂	semidirect
3. C₆₇×#1594	C₂	semidirect
4. C₆₇×#1593	C₂	semidirect
5. C₆₇×#1592	C₂	semidirect
6. C₆₇×#1564	C₂	semidirect
7. C₆₇×#1476	C₂	semidirect
8. C₂×#322×C₆₇	C₂	semidirect
95. C₂×C₈×C₆₇	M₁₆	semidirect
162. #17890	C₆₇	direct (N × H)
168. #2965	C₂×C₆₇	semidirect
170. #1594	C₂×C₆₇	semidirect
171. #1593	C₂×C₆₇	semidirect
172. #1592	C₂×C₆₇	semidirect
173. #1564	C₂×C₆₇	semidirect
174. #1476	C₂×C₆₇	semidirect
175. C₂×#322	C₂×C₆₇	semidirect
176. C₆₇	#17890	direct (N × H)
263. C₂×C₈	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2965	2
C₄×#135×C₆₇	2
C₆₇×#1594	2
C₆₇×#1593	2
C₆₇×#1592	2
C₆₇×#1564	2
C₆₇×#1476	2
C₂×#322×C₆₇	2
#17890	67

Subgroups — C67×#17890 #30697141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17890 #30697141