Subgroups — C₆₇×#10146 #30689397

Order 17152.

690 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3206	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3151	C₂	1	1	yes	maximal, characteristic
3	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
9	C₂²×SD₁₆×C₆₇	—	2	2	no	—
10	#532×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	#496×C₆₇	—	2	2	no	—
14	#493×C₆₇	—	2	1	no	—
15	#492×C₆₇	—	2	2	no	—
16	#492×C₆₇	—	2	1	no	—
17	#492×C₆₇	—	2	2	no	—
18	#492×C₆₇	—	2	1	no	—
19	#487×C₆₇	—	2	1	no	—
20	#461×C₆₇	—	2	2	no	—
21	#448×C₆₇	—	2	2	no	—
22	#431×C₆₇	C₂²	1	1	yes	characteristic
23	#423×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#343×C₆₇	—	2	2	no	—
27	#331×C₆₇	—	2	2	no	—
28	D₄⋊C₈×C₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	—	2	1	no	—
30	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂²×D₄×C₆₇	—	2	1	no	—
34	AGL(1,ℤ/8)×C₆₇	—	4	4	no	—
35	AGL(1,ℤ/8)×C₆₇	—	4	4	no	—
36	C₂×SD₁₆×C₆₇	—	2	8	no	—
37	C₂×SD₁₆×C₆₇	—	2	4	no	—
38	C₂×SD₁₆×C₆₇	—	2	4	no	—
39	C₂×SD₁₆×C₆₇	—	4	8	no	—
40	C₂×D₈×C₆₇	—	2	4	no	—
41	C₂×M₁₆×C₆₇	—	2	1	no	—
42	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
43	C₂²×C₈×C₆₇	—	2	1	no	—
44	C₄⋊Q₈×C₆₇	—	2	1	no	—
45	C₄⋊Q₈×C₆₇	—	2	1	no	—
46	C₄⋊Q₈×C₆₇	—	2	1	no	—
47	C₄⋊Q₈×C₆₇	—	2	1	no	—
48	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
49	#127×C₆₇	—	2	1	no	—
50	#127×C₆₇	—	2	1	no	—
51	#127×C₆₇	—	2	1	no	—
52	#127×C₆₇	D₄	1	1	yes	characteristic
53	#127×C₆₇	D₄	1	2	yes	—
54	#124×C₆₇	—	2	1	no	—
55	#121×C₆₇	—	2	1	no	—
56	#121×C₆₇	—	2	1	no	—
57	C₄×D₄×C₆₇	—	2	1	no	—
58	C₄×D₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₆₇	—	2	1	no	—
63	C₄⋊C₈×C₆₇	—	2	1	no	—
64	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
65	#103×C₆₇	—	2	4	no	—
66	C₂²⋊C₈×C₆₇	—	2	1	no	—
67	#97×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	—	2	1	no	—
69	C₄×C₈×C₆₇	—	2	1	no	—
70	C₄×C₈×C₆₇	—	2	1	no	—
71	C₄×C₈×C₆₇	D₄	1	2	yes	—
72	C₂⁴×C₆₇	—	4	1	no	—
73	#41×C₆₇	—	4	2	no	—
74	#41×C₆₇	—	4	2	no	—
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂×Q₈×C₆₇	—	2	1	no	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	—	4	4	no	—
79	C₂×D₄×C₆₇	—	2	1	no	—
80	C₂×D₄×C₆₇	—	2	1	no	—
81	C₂×D₄×C₆₇	—	2	2	no	—
82	C₂×D₄×C₆₇	—	2	2	no	—
83	C₂×D₄×C₆₇	—	2	1	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	4	4	no	—
86	C₂×D₄×C₆₇	—	4	1	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	1	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	8	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	4	1	no	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
98	SD₁₆×C₆₇	—	4	16	no	—
99	SD₁₆×C₆₇	—	4	8	no	—
100	SD₁₆×C₆₇	—	8	4	no	—
101	D₈×C₆₇	—	4	8	no	—
102	M₁₆×C₆₇	—	2	2	no	—
103	M₁₆×C₆₇	—	4	2	no	—
104	M₁₆×C₆₇	SD₁₆	1	4	yes	complemented
105	C₂×C₈×C₆₇	—	2	4	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	4	1	no	—
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	2	1	no	—
112	C₄⋊C₄×C₆₇	—	4	1	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₄⋊C₄×C₆₇	—	4	1	no	—
115	C₄⋊C₄×C₆₇	—	4	1	no	—
116	C₂²⋊C₄×C₆₇	—	4	1	no	—
117	C₂²⋊C₄×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂³×C₆₇	—	4	1	no	—
123	C₂³×C₆₇	—	4	2	no	—
124	C₂³×C₆₇	—	8	2	no	—
125	C₂³×C₆₇	—	8	2	no	—
126	C₂³×C₆₇	—	8	2	no	—
127	C₂³×C₆₇	#121	1	1	yes	characteristic
128	Q₈×C₆₇	—	4	4	no	—
129	Q₈×C₆₇	—	4	2	no	—
130	Q₈×C₆₇	—	4	4	no	—
131	D₄×C₆₇	—	4	4	no	—
132	D₄×C₆₇	—	4	2	no	—
133	D₄×C₆₇	—	4	4	no	—
134	D₄×C₆₇	—	4	2	no	—
135	D₄×C₆₇	—	4	2	no	—
136	D₄×C₆₇	—	8	2	no	—
137	D₄×C₆₇	—	8	2	no	—
138	D₄×C₆₇	—	8	2	no	—
139	D₄×C₆₇	—	8	2	no	—
140	D₄×C₆₇	—	8	1	no	—
141	D₄×C₆₇	—	8	1	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	4	1	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	8	1	no	—
147	C₂×C₄×C₆₇	—	8	1	no	—
148	C₂×C₄×C₆₇	—	8	2	no	—
149	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
154	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
155	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
156	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
157	C₈×C₆₇	—	2	4	no	—
158	C₈×C₆₇	—	2	4	no	—
159	C₈×C₆₇	—	4	2	no	—
160	C₈×C₆₇	—	8	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	8	2	no	—
164	C₂²×C₆₇	—	8	1	no	—
165	C₂²×C₆₇	—	8	1	no	—
166	C₂²×C₆₇	—	8	2	no	—
167	C₂²×C₆₇	—	8	1	no	—
168	C₂²×C₆₇	—	8	2	no	—
169	C₂²×C₆₇	—	16	4	no	—
170	C₂²×C₆₇	#392	1	1	yes	characteristic
171	C₂²×C₆₇	#460	1	2	yes	—
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	8	2	no	—
175	C₄×C₆₇	—	8	1	no	—
176	C₄×C₆₇	#450	1	2	yes	—
177	C₄×C₆₇	#453	1	2	yes	—
178	C₄×C₆₇	#471	1	2	yes	—
179	C₄×C₆₇	#492	1	1	yes	characteristic
180	C₄×C₆₇	#496	1	1	yes	characteristic
181	#10146	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂×C₆₇	—	2	2	no	—
183	C₂×C₆₇	—	16	1	no	—
184	C₂×C₆₇	—	16	2	no	—
185	C₂×C₆₇	#2002	1	1	yes	characteristic
186	C₂×C₆₇	#2014	1	1	yes	characteristic
187	C₂×C₆₇	#2019	1	1	yes	characteristic
188	#3206	C₂×C₆₇	1	1	yes	characteristic, complemented
189	#3151	C₂×C₆₇	1	1	yes	characteristic
190	C₂×#492	C₂×C₆₇	1	1	yes	characteristic, complemented
191	#1997	C₂×C₆₇	1	1	yes	characteristic, complemented
192	#1545	C₂×C₆₇	1	1	yes	characteristic, complemented
193	#1492	C₂×C₆₇	1	1	yes	characteristic, complemented
194	#1278	C₂×C₆₇	1	1	yes	characteristic, complemented
195	C₆₇	#10146	1	1	yes	characteristic, direct product factor, Sylow
196	C₂×AGL(1,ℤ/8)	—	2	2	no	—
197	C₂²×SD₁₆	—	2	2	no	—
198	#532	C₂²×C₆₇	1	1	yes	characteristic
199	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
200	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
201	#496	—	2	2	no	—
202	#493	—	2	1	no	—
203	#492	—	2	1	no	—
204	#492	—	2	2	no	—
205	#492	—	2	1	no	—
206	#492	—	2	2	no	—
207	#487	—	2	1	no	—
208	#461	—	2	2	no	—
209	#448	—	2	2	no	—
210	#431	C₂²×C₆₇	1	1	yes	characteristic
211	#423	C₂²×C₆₇	1	1	yes	characteristic
212	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
214	#343	—	2	2	no	—
215	#331	—	2	2	no	—
216	D₄⋊C₈	—	2	2	no	—
217	C₂×#41	—	2	1	no	—
218	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
219	C₂²×D₄	—	2	1	no	—
220	C₂²×D₄	—	2	1	no	—
221	C₂²×D₄	—	2	1	no	—
222	AGL(1,ℤ/8)	—	4	4	no	—
223	AGL(1,ℤ/8)	—	4	4	no	—
224	C₂×SD₁₆	—	2	8	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×SD₁₆	—	2	4	no	—
227	C₂×SD₁₆	—	4	8	no	—
228	C₂×D₈	—	2	4	no	—
229	C₂×M₁₆	—	2	1	no	—
230	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
231	C₂²×C₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
237	#127	—	2	1	no	—
238	#127	—	2	1	no	—
239	#127	—	2	1	no	—
240	#127	D₄×C₆₇	1	2	yes	—
241	#127	D₄×C₆₇	1	1	yes	characteristic
242	#124	—	2	1	no	—
243	#121	—	2	1	no	—
244	#121	—	2	1	no	—
245	C₄×D₄	—	2	1	no	—
246	C₄×D₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	2	1	no	—
249	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
253	#103	—	2	4	no	—
254	C₂²⋊C₈	—	2	1	no	—
255	#97	—	2	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	D₄×C₆₇	1	2	yes	—
260	C₂⁴	—	4	1	no	—
261	#41	—	4	2	no	—
262	#41	—	4	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	4	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	4	4	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	8	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
286	SD₁₆	—	4	16	no	—
287	SD₁₆	—	4	8	no	—
288	SD₁₆	—	8	4	no	—
289	D₈	—	4	8	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	4	2	no	—
292	M₁₆	SD₁₆×C₆₇	1	4	yes	complemented
293	C₂×C₈	—	2	4	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	4	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₂²⋊C₄	—	4	1	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
307	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂³	—	4	1	no	—
311	C₂³	—	4	2	no	—
312	C₂³	—	8	2	no	—
313	C₂³	—	8	2	no	—
314	C₂³	—	8	2	no	—
315	C₂³	#121×C₆₇	1	1	yes	characteristic
316	Q₈	—	4	4	no	—
317	Q₈	—	4	2	no	—
318	Q₈	—	4	4	no	—
319	D₄	—	4	4	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	2	no	—
327	D₄	—	8	2	no	—
328	D₄	—	8	1	no	—
329	D₄	—	8	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
338	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
339	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
340	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
341	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
342	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
344	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
345	C₈	—	2	4	no	—
346	C₈	—	2	4	no	—
347	C₈	—	4	2	no	—
348	C₈	—	8	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	16	4	no	—
358	C₂²	#392×C₆₇	1	1	yes	characteristic
359	C₂²	#460×C₆₇	1	2	yes	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	#450×C₆₇	1	2	yes	—
365	C₄	#453×C₆₇	1	2	yes	—
366	C₄	#471×C₆₇	1	2	yes	—
367	C₄	#492×C₆₇	1	1	yes	characteristic
368	C₄	#496×C₆₇	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	16	2	no	—
371	C₂	—	16	1	no	—
372	C₂	C₆₇×#2002	1	1	yes	characteristic
373	C₂	C₆₇×#2014	1	1	yes	characteristic
374	C₂	C₆₇×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₆₇×#3151	C₂
LCS	G ▷ C₂×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. C₆₇×#3206	C₂	semidirect
3. C₂×#492×C₆₇	C₂	semidirect
4. C₆₇×#1997	C₂	semidirect
5. C₆₇×#1545	C₂	semidirect
6. C₆₇×#1492	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
104. M₁₆×C₆₇	SD₁₆	semidirect
181. #10146	C₆₇	direct (N × H)
188. #3206	C₂×C₆₇	semidirect
190. C₂×#492	C₂×C₆₇	semidirect
191. #1997	C₂×C₆₇	semidirect
192. #1545	C₂×C₆₇	semidirect
193. #1492	C₂×C₆₇	semidirect
194. #1278	C₂×C₆₇	semidirect
195. C₆₇	#10146	direct (N × H)
292. M₁₆	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3206	2
C₆₇×#3151	2
C₂×#492×C₆₇	2
C₆₇×#1997	2
C₆₇×#1545	2
C₆₇×#1492	2
C₆₇×#1278	2
#10146	67

Subgroups — C67×#10146 #30689397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10146 #30689397