Subgroups — C₆₇×#18994 #30698245

Order 17152.

936 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#563×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3234	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3153	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1660	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1657	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1528	C₂	1	2	yes	maximal, complemented
8	C₂×#137×C₆₇	—	2	1	no	—
9	C₂×AGL(1,ℤ/8)×C₆₇	—	2	1	no	—
10	#563×C₆₇	—	2	4	no	—
11	#563×C₆₇	—	2	2	no	—
12	#563×C₆₇	C₂²	1	4	yes	—
13	#532×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
15	C₂×#126×C₆₇	—	2	2	no	—
16	C₂×#125×C₆₇	C₂²	1	2	yes	—
17	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#123×C₆₇	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
20	#501×C₆₇	C₂²	1	4	yes	complemented
21	#497×C₆₇	C₂²	1	2	yes	—
22	#496×C₆₇	C₂²	1	2	yes	—
23	#495×C₆₇	—	2	2	no	—
24	#490×C₆₇	C₂²	1	4	yes	complemented
25	#481×C₆₇	—	2	2	no	—
26	#480×C₆₇	—	2	2	no	—
27	#478×C₆₇	—	2	4	no	—
28	#432×C₆₇	C₂²	1	2	yes	—
29	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
30	#330×C₆₇	C₂²	1	8	yes	complemented
31	#329×C₆₇	C₂²	1	4	yes	—
32	C₂×#41×C₆₇	—	2	1	no	—
33	C₂²×Q₈×C₆₇	—	2	1	no	—
34	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	—	2	1	no	—
37	#137×C₆₇	—	4	4	no	—
38	AGL(1,ℤ/8)×C₆₇	—	4	4	no	—
39	C₂×Q₁₆×C₆₇	—	2	2	no	—
40	C₂×SD₁₆×C₆₇	—	2	2	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×D₈×C₆₇	—	2	2	no	—
43	C₂×M₁₆×C₆₇	—	2	1	no	—
44	C₄⋊Q₈×C₆₇	—	2	4	no	—
45	C₄⋊Q₈×C₆₇	—	2	4	no	—
46	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	#127×C₆₇	C₂³	1	1	yes	characteristic
48	#126×C₆₇	—	2	8	no	—
49	#126×C₆₇	—	2	8	no	—
50	#125×C₆₇	—	2	2	no	—
51	#125×C₆₇	—	2	2	no	—
52	#125×C₆₇	D₄	1	4	yes	—
53	#125×C₆₇	C₂³	1	4	yes	—
54	#124×C₆₇	—	2	2	no	—
55	#124×C₆₇	—	2	1	no	—
56	#124×C₆₇	D₄	1	2	yes	—
57	#124×C₆₇	C₂³	1	2	yes	—
58	#123×C₆₇	—	2	4	no	—
59	#123×C₆₇	—	2	4	no	—
60	C₂²⋊Q₈×C₆₇	—	2	8	no	—
61	C₂²⋊Q₈×C₆₇	—	2	8	no	—
62	#121×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	1	no	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	—	2	4	no	—
75	#106×C₆₇	—	2	4	no	—
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	#102×C₆₇	—	2	4	no	—
78	#102×C₆₇	—	2	4	no	—
79	C₂²⋊C₈×C₆₇	—	2	2	no	—
80	#97×C₆₇	C₂³	1	4	yes	—
81	#97×C₆₇	C₂³	1	2	yes	—
82	C₄×C₈×C₆₇	—	2	1	no	—
83	C₂⁴×C₆₇	—	2	1	no	—
84	#41×C₆₇	—	4	2	no	—
85	#41×C₆₇	—	4	2	no	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	—	2	1	no	—
88	C₂×Q₈×C₆₇	—	2	1	no	—
89	C₂×Q₈×C₆₇	—	4	2	no	—
90	C₂×Q₈×C₆₇	—	8	2	no	—
91	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	4	2	no	—
96	C₂×D₄×C₆₇	—	4	1	no	—
97	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	4	no	—
102	C₂²×C₄×C₆₇	—	4	1	no	—
103	C₂²×C₄×C₆₇	—	8	2	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	Q₁₆×C₆₇	—	4	4	no	—
107	SD₁₆×C₆₇	—	4	4	no	—
108	SD₁₆×C₆₇	—	4	4	no	—
109	D₈×C₆₇	—	4	4	no	—
110	M₁₆×C₆₇	—	2	4	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	4	no	—
113	C₄⋊C₄×C₆₇	—	2	8	no	—
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	4	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
122	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
123	C₂²⋊C₄×C₆₇	—	2	8	no	—
124	C₂²⋊C₄×C₆₇	—	2	4	no	—
125	C₂²⋊C₄×C₆₇	—	2	4	no	—
126	C₂²⋊C₄×C₆₇	—	2	4	no	—
127	C₂²⋊C₄×C₆₇	—	4	2	no	—
128	C₄²×C₆₇	—	2	4	no	—
129	C₄²×C₆₇	C₂×D₄	1	2	yes	—
130	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	4	2	no	—
135	C₂³×C₆₇	—	4	1	no	—
136	C₂³×C₆₇	#120	1	1	yes	characteristic
137	Q₈×C₆₇	—	4	2	no	—
138	Q₈×C₆₇	—	4	1	no	—
139	Q₈×C₆₇	—	4	1	no	—
140	Q₈×C₆₇	—	8	1	no	—
141	Q₈×C₆₇	—	8	4	no	—
142	D₄×C₆₇	—	4	2	no	—
143	D₄×C₆₇	—	4	1	no	—
144	D₄×C₆₇	—	4	1	no	—
145	D₄×C₆₇	—	4	4	no	—
146	D₄×C₆₇	—	8	1	no	—
147	D₄×C₆₇	—	8	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	4	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	4	4	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	1	no	—
157	C₂×C₄×C₆₇	—	8	2	no	—
158	C₂×C₄×C₆₇	—	8	4	no	—
159	C₂×C₄×C₆₇	#120	1	2	yes	—
160	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₆₇	—	2	4	no	—
164	C₈×C₆₇	—	4	4	no	—
165	C₂²×C₆₇	—	2	2	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	4	no	—
168	C₂²×C₆₇	—	8	4	no	—
169	C₂²×C₆₇	—	8	1	no	—
170	C₂²×C₆₇	—	8	2	no	—
171	C₂²×C₆₇	#453	1	2	yes	—
172	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	4	4	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	—	4	4	no	—
180	C₄×C₆₇	—	8	1	no	—
181	#18994	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂×C₆₇	—	2	2	no	—
183	C₂×C₆₇	—	8	2	no	—
184	C₂×C₆₇	—	16	1	no	—
185	C₂×C₆₇	C₂×#453	1	1	yes	characteristic
186	C₂×C₆₇	#3020	1	2	yes	—
187	C₂×#563	C₂×C₆₇	1	1	yes	characteristic, complemented
188	#3234	C₂×C₆₇	1	2	yes	complemented
189	#3153	C₂×C₆₇	1	1	yes	characteristic
190	#1660	C₂×C₆₇	1	4	yes	complemented
191	#1657	C₂×C₆₇	1	4	yes	complemented
192	#1533	C₂×C₆₇	1	1	yes	characteristic, complemented
193	#1528	C₂×C₆₇	1	2	yes	complemented
194	C₆₇	#18994	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×#137	—	2	1	no	—
196	C₂×AGL(1,ℤ/8)	—	2	1	no	—
197	#563	—	2	4	no	—
198	#563	—	2	2	no	—
199	#563	C₂²×C₆₇	1	4	yes	—
200	#532	C₂²×C₆₇	1	1	yes	characteristic
201	C₂×C₄⋊Q₈	—	2	1	no	—
202	C₂×#126	—	2	2	no	—
203	C₂×#125	C₂²×C₆₇	1	2	yes	—
204	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
205	C₂×#123	—	2	1	no	—
206	C₂×C₂²⋊Q₈	—	2	2	no	—
207	#501	C₂²×C₆₇	1	4	yes	complemented
208	#497	C₂²×C₆₇	1	2	yes	—
209	#496	C₂²×C₆₇	1	2	yes	—
210	#495	—	2	2	no	—
211	#490	C₂²×C₆₇	1	4	yes	complemented
212	#481	—	2	2	no	—
213	#480	—	2	2	no	—
214	#478	—	2	4	no	—
215	#432	C₂²×C₆₇	1	2	yes	—
216	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
217	#330	C₂²×C₆₇	1	8	yes	complemented
218	#329	C₂²×C₆₇	1	4	yes	—
219	C₂×#41	—	2	1	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
222	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
223	C₂³×C₄	—	2	1	no	—
224	#137	—	4	4	no	—
225	AGL(1,ℤ/8)	—	4	4	no	—
226	C₂×Q₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×SD₁₆	—	2	2	no	—
229	C₂×D₈	—	2	2	no	—
230	C₂×M₁₆	—	2	1	no	—
231	C₄⋊Q₈	—	2	4	no	—
232	C₄⋊Q₈	—	2	4	no	—
233	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
234	#127	C₂³×C₆₇	1	1	yes	characteristic
235	#126	—	2	8	no	—
236	#126	—	2	8	no	—
237	#125	—	2	2	no	—
238	#125	—	2	2	no	—
239	#125	D₄×C₆₇	1	4	yes	—
240	#125	C₂³×C₆₇	1	4	yes	—
241	#124	—	2	2	no	—
242	#124	—	2	1	no	—
243	#124	D₄×C₆₇	1	2	yes	—
244	#124	C₂³×C₆₇	1	2	yes	—
245	#123	—	2	4	no	—
246	#123	—	2	4	no	—
247	C₂²⋊Q₈	—	2	8	no	—
248	C₂²⋊Q₈	—	2	8	no	—
249	#121	—	2	2	no	—
250	C₄×D₄	—	2	2	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	2	no	—
253	C₂×C₄⋊C₄	—	2	2	no	—
254	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
255	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
256	C₂×C₂²⋊C₄	—	2	2	no	—
257	C₂×C₂²⋊C₄	—	2	1	no	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₄²	—	2	1	no	—
260	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
261	#107	—	2	4	no	—
262	#106	—	2	4	no	—
263	C₄⋊C₈	—	2	2	no	—
264	#102	—	2	4	no	—
265	#102	—	2	4	no	—
266	C₂²⋊C₈	—	2	2	no	—
267	#97	C₂³×C₆₇	1	4	yes	—
268	#97	C₂³×C₆₇	1	2	yes	—
269	C₄×C₈	—	2	1	no	—
270	C₂⁴	—	2	1	no	—
271	#41	—	4	2	no	—
272	#41	—	4	2	no	—
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	4	2	no	—
277	C₂×Q₈	—	8	2	no	—
278	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	—	4	1	no	—
284	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	8	2	no	—
291	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
293	Q₁₆	—	4	4	no	—
294	SD₁₆	—	4	4	no	—
295	SD₁₆	—	4	4	no	—
296	D₈	—	4	4	no	—
297	M₁₆	—	2	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	4	no	—
300	C₄⋊C₄	—	2	8	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	4	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
309	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
310	C₂²⋊C₄	—	2	8	no	—
311	C₂²⋊C₄	—	2	4	no	—
312	C₂²⋊C₄	—	2	4	no	—
313	C₂²⋊C₄	—	2	4	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₄²	—	2	4	no	—
316	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₄²	C₂×D₄×C₆₇	1	2	yes	—
318	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
319	C₂³	—	2	2	no	—
320	C₂³	—	4	4	no	—
321	C₂³	—	4	2	no	—
322	C₂³	—	4	1	no	—
323	C₂³	#120×C₆₇	1	1	yes	characteristic
324	Q₈	—	4	2	no	—
325	Q₈	—	4	1	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	8	1	no	—
328	Q₈	—	8	4	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	1	no	—
331	D₄	—	4	1	no	—
332	D₄	—	4	4	no	—
333	D₄	—	8	1	no	—
334	D₄	—	8	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	8	2	no	—
345	C₂×C₄	—	8	4	no	—
346	C₂×C₄	#120×C₆₇	1	2	yes	—
347	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
349	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	4	4	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	—	8	4	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	8	2	no	—
358	C₂²	#453×C₆₇	1	2	yes	—
359	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	2	no	—
363	C₄	—	2	2	no	—
364	C₄	—	4	4	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	4	no	—
367	C₄	—	8	1	no	—
368	C₂	—	2	2	no	—
369	C₂	—	8	2	no	—
370	C₂	—	16	1	no	—
371	C₂	C₂×#453×C₆₇	1	1	yes	characteristic
372	C₂	C₆₇×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3153	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
1. C₂×#563×C₆₇	C₂	semidirect
2. C₆₇×#3234	C₂	semidirect
4. C₆₇×#1660	C₂	semidirect
5. C₆₇×#1657	C₂	semidirect
6. C₆₇×#1533	C₂	semidirect
7. C₆₇×#1528	C₂	semidirect
20. #501×C₆₇	C₂²	semidirect
24. #490×C₆₇	C₂²	semidirect
30. #330×C₆₇	C₂²	semidirect
181. #18994	C₆₇	direct (N × H)
187. C₂×#563	C₂×C₆₇	semidirect
188. #3234	C₂×C₆₇	semidirect
190. #1660	C₂×C₆₇	semidirect
191. #1657	C₂×C₆₇	semidirect
192. #1533	C₂×C₆₇	semidirect
193. #1528	C₂×C₆₇	semidirect
194. C₆₇	#18994	direct (N × H)
207. #501	C₂²×C₆₇	semidirect
211. #490	C₂²×C₆₇	semidirect
217. #330	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#563×C₆₇	2
C₆₇×#3234	2
C₆₇×#3153	2
C₆₇×#1660	2
C₆₇×#1657	2
C₆₇×#1533	2
C₆₇×#1528	2
#18994	67

Subgroups — C67×#18994 #30698245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#18994 #30698245