Subgroups — C₆₇×#19721 #30698972

Order 17152.

798 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3227	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3207	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3184	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3152	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1720	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1715	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1709	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1705	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	—	2	2	no	—
13	#532×C₆₇	C₂²	1	1	yes	characteristic
14	#532×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	#498×C₆₇	C₂²	1	1	yes	characteristic
17	#498×C₆₇	C₂²	1	2	yes	—
18	#498×C₆₇	C₂²	1	2	yes	—
19	#497×C₆₇	—	2	2	no	—
20	#494×C₆₇	C₂²	1	1	yes	characteristic
21	#494×C₆₇	C₂²	1	1	yes	characteristic
22	#493×C₆₇	C₂²	1	1	yes	characteristic
23	#493×C₆₇	C₂²	1	2	yes	—
24	#492×C₆₇	C₂²	1	1	yes	characteristic
25	#492×C₆₇	C₂²	1	1	yes	characteristic
26	#492×C₆₇	C₂²	1	2	yes	—
27	#487×C₆₇	C₂²	1	2	yes	—
28	#486×C₆₇	C₂²	1	2	yes	—
29	#486×C₆₇	C₂²	1	2	yes	—
30	#486×C₆₇	C₂²	1	2	yes	—
31	#486×C₆₇	C₂²	1	2	yes	—
32	#481×C₆₇	—	2	4	no	—
33	#476×C₆₇	—	2	4	no	—
34	#469×C₆₇	—	2	2	no	—
35	#463×C₆₇	—	2	4	no	—
36	#449×C₆₇	—	2	4	no	—
37	#442×C₆₇	—	2	2	no	—
38	#431×C₆₇	C₂²	1	2	yes	—
39	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	#322×C₆₇	C₂²	1	2	yes	—
41	#322×C₆₇	C₂²	1	1	yes	characteristic
42	C₈²×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	2	no	—
45	C₂²×Q₈×C₆₇	—	2	1	no	—
46	#137×C₆₇	—	4	8	no	—
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×SD₁₆×C₆₇	—	2	2	no	—
50	C₂×SD₁₆×C₆₇	—	2	2	no	—
51	C₂×SD₁₆×C₆₇	—	2	4	no	—
52	C₂×D₈×C₆₇	—	2	2	no	—
53	C₂×D₈×C₆₇	—	2	4	no	—
54	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
55	C₄⋊Q₈×C₆₇	—	2	2	no	—
56	C₄⋊Q₈×C₆₇	—	2	1	no	—
57	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
61	#127×C₆₇	C₂³	1	2	yes	—
62	#127×C₆₇	C₂³	1	1	yes	characteristic
63	#124×C₆₇	—	2	2	no	—
64	#124×C₆₇	—	2	1	no	—
65	#121×C₆₇	—	2	2	no	—
66	#121×C₆₇	—	2	2	no	—
67	#121×C₆₇	—	2	2	no	—
68	C₄×D₄×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	—	2	2	no	—
70	C₄×D₄×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	#106×C₆₇	—	2	2	no	—
74	#106×C₆₇	—	2	4	no	—
75	#106×C₆₇	—	2	4	no	—
76	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
77	#103×C₆₇	—	2	4	no	—
78	#102×C₆₇	—	2	4	no	—
79	#102×C₆₇	—	2	4	no	—
80	#102×C₆₇	—	2	4	no	—
81	#102×C₆₇	—	2	4	no	—
82	C₂²⋊C₈×C₆₇	D₄	1	4	yes	—
83	#97×C₆₇	D₄	1	2	yes	—
84	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₇	C₂³	1	2	yes	—
86	C₄×C₈×C₆₇	C₂³	1	2	yes	—
87	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
88	#41×C₆₇	—	4	4	no	—
89	#41×C₆₇	—	8	4	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×Q₈×C₆₇	—	2	1	no	—
93	C₂×Q₈×C₆₇	—	2	1	no	—
94	C₂×Q₈×C₆₇	—	4	4	no	—
95	C₂×Q₈×C₆₇	—	4	1	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	4	1	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	—	4	1	no	—
105	C₂²×C₄×C₆₇	—	4	2	no	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₆₇	—	4	4	no	—
109	Q₁₆×C₆₇	—	4	4	no	—
110	SD₁₆×C₆₇	—	4	4	no	—
111	SD₁₆×C₆₇	—	4	4	no	—
112	SD₁₆×C₆₇	—	8	4	no	—
113	D₈×C₆₇	—	4	4	no	—
114	D₈×C₆₇	—	8	4	no	—
115	M₁₆×C₆₇	—	2	4	no	—
116	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
117	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
119	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	4	2	no	—
126	C₂²⋊C₄×C₆₇	—	4	2	no	—
127	C₂²⋊C₄×C₆₇	—	4	2	no	—
128	C₂²⋊C₄×C₆₇	—	4	2	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	—	4	1	no	—
133	C₂³×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	#127	1	1	yes	characteristic
135	Q₈×C₆₇	—	4	2	no	—
136	Q₈×C₆₇	—	4	2	no	—
137	Q₈×C₆₇	—	4	2	no	—
138	Q₈×C₆₇	—	4	4	no	—
139	Q₈×C₆₇	—	8	2	no	—
140	D₄×C₆₇	—	4	4	no	—
141	D₄×C₆₇	—	4	2	no	—
142	D₄×C₆₇	—	4	4	no	—
143	D₄×C₆₇	—	4	4	no	—
144	D₄×C₆₇	—	4	4	no	—
145	D₄×C₆₇	—	8	2	no	—
146	D₄×C₆₇	—	8	4	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	4	1	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	4	1	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	8	2	no	—
154	C₂×C₄×C₆₇	—	8	4	no	—
155	C₂×C₄×C₆₇	—	8	2	no	—
156	C₂×C₄×C₆₇	#127	1	2	yes	—
157	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
160	C₈×C₆₇	—	2	2	no	—
161	C₈×C₆₇	—	2	4	no	—
162	C₈×C₆₇	—	2	4	no	—
163	C₈×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
164	C₂²×C₆₇	—	2	1	no	—
165	C₂²×C₆₇	—	2	2	no	—
166	C₂²×C₆₇	—	8	1	no	—
167	C₂²×C₆₇	—	8	2	no	—
168	C₂²×C₆₇	—	8	2	no	—
169	C₂²×C₆₇	—	8	2	no	—
170	C₂²×C₆₇	—	8	2	no	—
171	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	—	2	2	no	—
174	C₄×C₆₇	—	8	1	no	—
175	C₄×C₆₇	—	8	1	no	—
176	C₄×C₆₇	—	8	1	no	—
177	C₄×C₆₇	—	8	2	no	—
178	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₄×C₆₇	#577	1	4	yes	—
180	#19721	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₆₇	—	4	1	no	—
182	C₂×C₆₇	—	16	1	no	—
183	C₂×C₆₇	—	16	2	no	—
184	C₂×C₆₇	#3151	1	1	yes	characteristic
185	C₂×C₆₇	#3154	1	2	yes	—
186	#3227	C₂×C₆₇	1	2	yes	complemented
187	#3207	C₂×C₆₇	1	2	yes	complemented
188	#3184	C₂×C₆₇	1	1	yes	characteristic, complemented
189	#3152	C₂×C₆₇	1	1	yes	characteristic, complemented
190	#1721	C₂×C₆₇	1	1	yes	characteristic, complemented
191	#1720	C₂×C₆₇	1	2	yes	complemented
192	#1716	C₂×C₆₇	1	1	yes	characteristic, complemented
193	#1715	C₂×C₆₇	1	2	yes	complemented
194	#1709	C₂×C₆₇	1	1	yes	characteristic, complemented
195	#1705	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#1453	C₂×C₆₇	1	1	yes	characteristic, complemented
197	C₆₇	#19721	1	1	yes	characteristic, direct product factor, Sylow
198	C₂×#137	—	2	2	no	—
199	#532	C₂²×C₆₇	1	1	yes	characteristic
200	#532	C₂²×C₆₇	1	2	yes	—
201	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
202	#498	C₂²×C₆₇	1	1	yes	characteristic
203	#498	C₂²×C₆₇	1	2	yes	—
204	#498	C₂²×C₆₇	1	2	yes	—
205	#497	—	2	2	no	—
206	#494	C₂²×C₆₇	1	1	yes	characteristic
207	#494	C₂²×C₆₇	1	1	yes	characteristic
208	#493	C₂²×C₆₇	1	1	yes	characteristic
209	#493	C₂²×C₆₇	1	2	yes	—
210	#492	C₂²×C₆₇	1	1	yes	characteristic
211	#492	C₂²×C₆₇	1	1	yes	characteristic
212	#492	C₂²×C₆₇	1	2	yes	—
213	#487	C₂²×C₆₇	1	2	yes	—
214	#486	C₂²×C₆₇	1	2	yes	—
215	#486	C₂²×C₆₇	1	2	yes	—
216	#486	C₂²×C₆₇	1	2	yes	—
217	#486	C₂²×C₆₇	1	2	yes	—
218	#481	—	2	4	no	—
219	#476	—	2	4	no	—
220	#469	—	2	2	no	—
221	#463	—	2	4	no	—
222	#449	—	2	4	no	—
223	#442	—	2	2	no	—
224	#431	C₂²×C₆₇	1	2	yes	—
225	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
226	#322	C₂²×C₆₇	1	2	yes	—
227	#322	C₂²×C₆₇	1	1	yes	characteristic
228	C₈²	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×#41	—	2	1	no	—
230	C₂×#41	—	2	2	no	—
231	C₂²×Q₈	—	2	1	no	—
232	#137	—	4	8	no	—
233	C₂×Q₁₆	—	2	2	no	—
234	C₂×Q₁₆	—	2	2	no	—
235	C₂×SD₁₆	—	2	2	no	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×SD₁₆	—	2	4	no	—
238	C₂×D₈	—	2	2	no	—
239	C₂×D₈	—	2	4	no	—
240	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
241	C₄⋊Q₈	—	2	2	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
244	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
245	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
246	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
247	#127	C₂³×C₆₇	1	2	yes	—
248	#127	C₂³×C₆₇	1	1	yes	characteristic
249	#124	—	2	2	no	—
250	#124	—	2	1	no	—
251	#121	—	2	2	no	—
252	#121	—	2	2	no	—
253	#121	—	2	2	no	—
254	C₄×D₄	—	2	2	no	—
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
259	#106	—	2	2	no	—
260	#106	—	2	4	no	—
261	#106	—	2	4	no	—
262	C₄⋊C₈	D₄×C₆₇	1	4	yes	—
263	#103	—	2	4	no	—
264	#102	—	2	4	no	—
265	#102	—	2	4	no	—
266	#102	—	2	4	no	—
267	#102	—	2	4	no	—
268	C₂²⋊C₈	D₄×C₆₇	1	4	yes	—
269	#97	D₄×C₆₇	1	2	yes	—
270	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₄×C₈	C₂³×C₆₇	1	2	yes	—
272	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
273	C₄×C₈	C₂³×C₆₇	1	2	yes	—
274	#41	—	4	4	no	—
275	#41	—	8	4	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	4	4	no	—
281	C₂×Q₈	—	4	1	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	2	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
293	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
294	Q₁₆	—	4	4	no	—
295	Q₁₆	—	4	4	no	—
296	SD₁₆	—	4	4	no	—
297	SD₁₆	—	4	4	no	—
298	SD₁₆	—	8	4	no	—
299	D₈	—	4	4	no	—
300	D₈	—	8	4	no	—
301	M₁₆	—	2	4	no	—
302	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
303	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
304	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
305	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
316	C₄²	C₂×D₄×C₆₇	1	2	yes	—
317	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
318	C₂³	—	4	1	no	—
319	C₂³	—	4	2	no	—
320	C₂³	#127×C₆₇	1	1	yes	characteristic
321	Q₈	—	4	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	8	2	no	—
326	D₄	—	4	4	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	4	no	—
329	D₄	—	4	4	no	—
330	D₄	—	4	4	no	—
331	D₄	—	8	2	no	—
332	D₄	—	8	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	4	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	8	2	no	—
340	C₂×C₄	—	8	4	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	#127×C₆₇	1	2	yes	—
343	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
346	C₈	—	2	2	no	—
347	C₈	—	2	4	no	—
348	C₈	—	2	4	no	—
349	C₈	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	2	no	—
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	2	no	—
364	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
365	C₄	#577×C₆₇	1	4	yes	—
366	C₂	—	4	1	no	—
367	C₂	—	16	1	no	—
368	C₂	—	16	2	no	—
369	C₂	C₆₇×#3151	1	1	yes	characteristic
370	C₂	C₆₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
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. C₆₇×#3227	C₂	semidirect
2. C₆₇×#3207	C₂	semidirect
3. C₆₇×#3184	C₂	semidirect
4. C₆₇×#3152	C₂	semidirect
5. C₆₇×#1721	C₂	semidirect
6. C₆₇×#1720	C₂	semidirect
7. C₆₇×#1716	C₂	semidirect
8. C₆₇×#1715	C₂	semidirect
9. C₆₇×#1709	C₂	semidirect
10. C₆₇×#1705	C₂	semidirect
11. C₆₇×#1453	C₂	semidirect
180. #19721	C₆₇	direct (N × H)
186. #3227	C₂×C₆₇	semidirect
187. #3207	C₂×C₆₇	semidirect
188. #3184	C₂×C₆₇	semidirect
189. #3152	C₂×C₆₇	semidirect
190. #1721	C₂×C₆₇	semidirect
191. #1720	C₂×C₆₇	semidirect
192. #1716	C₂×C₆₇	semidirect
193. #1715	C₂×C₆₇	semidirect
194. #1709	C₂×C₆₇	semidirect
195. #1705	C₂×C₆₇	semidirect
196. #1453	C₂×C₆₇	semidirect
197. C₆₇	#19721	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3227	2
C₆₇×#3207	2
C₆₇×#3184	2
C₆₇×#3152	2
C₆₇×#1721	2
C₆₇×#1720	2
C₆₇×#1716	2
C₆₇×#1715	2
C₆₇×#1709	2
C₆₇×#1705	2
C₆₇×#1453	2
#19721	67

Subgroups — C67×#19721 #30698972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19721 #30698972