Subgroups — C₆₇×#19746 #30698997

Order 17152.

770 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3182	C₂	1	2	yes	maximal
3	C₆₇×#3165	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3152	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1722	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1714	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1705	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1455	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	—	2	2	no	—
13	#533×C₆₇	C₂²	1	2	yes	—
14	#532×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	#501×C₆₇	—	2	4	no	—
17	#500×C₆₇	C₂²	1	2	yes	—
18	#500×C₆₇	C₂²	1	2	yes	—
19	#500×C₆₇	C₂²	1	2	yes	—
20	#498×C₆₇	C₂²	1	2	yes	—
21	#498×C₆₇	C₂²	1	2	yes	—
22	#498×C₆₇	C₂²	1	1	yes	characteristic
23	#497×C₆₇	—	2	2	no	—
24	#494×C₆₇	C₂²	1	1	yes	characteristic
25	#494×C₆₇	C₂²	1	1	yes	characteristic
26	#494×C₆₇	C₂²	1	1	yes	characteristic
27	#492×C₆₇	C₂²	1	1	yes	characteristic
28	#492×C₆₇	C₂²	1	1	yes	characteristic
29	#492×C₆₇	C₂²	1	2	yes	—
30	#487×C₆₇	C₂²	1	2	yes	—
31	#487×C₆₇	C₂²	1	2	yes	—
32	#486×C₆₇	C₂²	1	2	yes	—
33	#470×C₆₇	—	2	2	no	—
34	#468×C₆₇	—	2	4	no	—
35	#441×C₆₇	—	2	2	no	—
36	#440×C₆₇	—	2	4	no	—
37	#428×C₆₇	—	2	4	no	—
38	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
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	C₂²×Q₈×C₆₇	—	2	1	no	—
45	#137×C₆₇	—	4	8	no	—
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	2	2	no	—
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
53	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	C₄⋊Q₈×C₆₇	—	2	1	no	—
56	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
61	#127×C₆₇	C₂³	1	1	yes	characteristic
62	#125×C₆₇	—	2	2	no	—
63	#124×C₆₇	—	2	1	no	—
64	C₂²⋊Q₈×C₆₇	—	2	2	no	—
65	C₂²⋊Q₈×C₆₇	—	2	2	no	—
66	#121×C₆₇	—	2	2	no	—
67	C₄×Q₈×C₆₇	—	2	2	no	—
68	C₄×Q₈×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	—	2	2	no	—
70	#117×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	—	2	4	no	—
74	#107×C₆₇	—	2	4	no	—
75	#107×C₆₇	—	2	4	no	—
76	#106×C₆₇	—	2	2	no	—
77	#106×C₆₇	—	2	4	no	—
78	#106×C₆₇	—	2	4	no	—
79	#103×C₆₇	—	2	4	no	—
80	#103×C₆₇	—	2	4	no	—
81	#102×C₆₇	—	2	4	no	—
82	#97×C₆₇	D₄	1	2	yes	—
83	#97×C₆₇	D₄	1	4	yes	—
84	C₄×C₈×C₆₇	C₂³	1	2	yes	—
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₆₇	C₂³	1	2	yes	—
88	#41×C₆₇	—	4	4	no	—
89	C₂×Q₈×C₆₇	—	2	2	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	1	no	—
98	C₂×D₄×C₆₇	—	4	1	no	—
99	C₂²×C₄×C₆₇	—	4	1	no	—
100	C₂²×C₄×C₆₇	—	4	1	no	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₆₇	—	4	4	no	—
104	Q₁₆×C₆₇	—	4	4	no	—
105	Q₁₆×C₆₇	—	4	4	no	—
106	SD₁₆×C₆₇	—	4	4	no	—
107	SD₁₆×C₆₇	—	4	4	no	—
108	SD₁₆×C₆₇	—	4	8	no	—
109	M₁₆×C₆₇	—	2	4	no	—
110	M₁₆×C₆₇	—	2	8	no	—
111	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	4	2	no	—
123	C₄⋊C₄×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	—	4	2	no	—
125	C₂²⋊C₄×C₆₇	—	4	2	no	—
126	C₂²⋊C₄×C₆₇	—	4	2	no	—
127	C₄²×C₆₇	—	4	2	no	—
128	C₄²×C₆₇	C₂×D₄	1	2	yes	—
129	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₂³×C₆₇	—	4	1	no	—
132	C₂³×C₆₇	#127	1	1	yes	characteristic
133	Q₈×C₆₇	—	4	2	no	—
134	Q₈×C₆₇	—	4	2	no	—
135	Q₈×C₆₇	—	4	2	no	—
136	Q₈×C₆₇	—	4	4	no	—
137	Q₈×C₆₇	—	4	4	no	—
138	Q₈×C₆₇	—	8	2	no	—
139	D₄×C₆₇	—	4	2	no	—
140	D₄×C₆₇	—	4	4	no	—
141	D₄×C₆₇	—	8	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	4	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	8	2	no	—
150	C₂×C₄×C₆₇	—	8	2	no	—
151	C₂×C₄×C₆₇	#127	1	2	yes	—
152	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
155	C₈×C₆₇	—	2	2	no	—
156	C₈×C₆₇	—	2	4	no	—
157	C₈×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
158	C₈×C₆₇	#137	1	8	yes	complemented
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	8	1	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
164	C₄×C₆₇	—	2	1	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	8	1	no	—
167	C₄×C₆₇	—	8	1	no	—
168	C₄×C₆₇	—	8	1	no	—
169	C₄×C₆₇	—	8	2	no	—
170	C₄×C₆₇	—	8	2	no	—
171	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
172	C₄×C₆₇	C₂×#137	1	4	yes	—
173	#19746	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₂×C₆₇	—	4	1	no	—
175	C₂×C₆₇	—	16	1	no	—
176	C₂×C₆₇	#3151	1	1	yes	characteristic
177	C₂×C₆₇	#3152	1	2	yes	—
178	#3185	C₂×C₆₇	1	1	yes	characteristic, complemented
179	#3182	C₂×C₆₇	1	2	yes	—
180	#3165	C₂×C₆₇	1	2	yes	complemented
181	#3152	C₂×C₆₇	1	1	yes	characteristic
182	#1723	C₂×C₆₇	1	2	yes	complemented
183	#1722	C₂×C₆₇	1	1	yes	characteristic, complemented
184	#1716	C₂×C₆₇	1	1	yes	characteristic, complemented
185	#1714	C₂×C₆₇	1	2	yes	complemented
186	#1713	C₂×C₆₇	1	1	yes	characteristic, complemented
187	#1705	C₂×C₆₇	1	1	yes	characteristic, complemented
188	#1455	C₂×C₆₇	1	1	yes	characteristic, complemented
189	C₆₇	#19746	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×#137	—	2	2	no	—
191	#533	C₂²×C₆₇	1	2	yes	—
192	#532	C₂²×C₆₇	1	1	yes	characteristic
193	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
194	#501	—	2	4	no	—
195	#500	C₂²×C₆₇	1	2	yes	—
196	#500	C₂²×C₆₇	1	2	yes	—
197	#500	C₂²×C₆₇	1	2	yes	—
198	#498	C₂²×C₆₇	1	2	yes	—
199	#498	C₂²×C₆₇	1	2	yes	—
200	#498	C₂²×C₆₇	1	1	yes	characteristic
201	#497	—	2	2	no	—
202	#494	C₂²×C₆₇	1	1	yes	characteristic
203	#494	C₂²×C₆₇	1	1	yes	characteristic
204	#494	C₂²×C₆₇	1	1	yes	characteristic
205	#492	C₂²×C₆₇	1	1	yes	characteristic
206	#492	C₂²×C₆₇	1	1	yes	characteristic
207	#492	C₂²×C₆₇	1	2	yes	—
208	#487	C₂²×C₆₇	1	2	yes	—
209	#487	C₂²×C₆₇	1	2	yes	—
210	#486	C₂²×C₆₇	1	2	yes	—
211	#470	—	2	2	no	—
212	#468	—	2	4	no	—
213	#441	—	2	2	no	—
214	#440	—	2	4	no	—
215	#428	—	2	4	no	—
216	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
217	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
218	#322	C₂²×C₆₇	1	2	yes	—
219	#322	C₂²×C₆₇	1	1	yes	characteristic
220	C₈²	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×#41	—	2	1	no	—
222	C₂²×Q₈	—	2	1	no	—
223	#137	—	4	8	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×Q₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×SD₁₆	—	2	2	no	—
229	C₂×SD₁₆	—	2	4	no	—
230	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
231	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
232	C₄⋊Q₈	—	2	2	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
235	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
236	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
237	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
238	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
239	#127	C₂³×C₆₇	1	1	yes	characteristic
240	#125	—	2	2	no	—
241	#124	—	2	1	no	—
242	C₂²⋊Q₈	—	2	2	no	—
243	C₂²⋊Q₈	—	2	2	no	—
244	#121	—	2	2	no	—
245	C₄×Q₈	—	2	2	no	—
246	C₄×Q₈	—	2	2	no	—
247	C₄×D₄	—	2	2	no	—
248	#117	—	2	2	no	—
249	C₂×C₄⋊C₄	—	2	2	no	—
250	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
251	#107	—	2	4	no	—
252	#107	—	2	4	no	—
253	#107	—	2	4	no	—
254	#106	—	2	4	no	—
255	#106	—	2	4	no	—
256	#106	—	2	2	no	—
257	#103	—	2	4	no	—
258	#103	—	2	4	no	—
259	#102	—	2	4	no	—
260	#97	D₄×C₆₇	1	4	yes	—
261	#97	D₄×C₆₇	1	2	yes	—
262	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₆₇	1	2	yes	—
264	C₄×C₈	C₂³×C₆₇	1	2	yes	—
265	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
266	#41	—	4	4	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×Q₈	—	2	1	no	—
272	C₂×Q₈	—	4	4	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
281	Q₁₆	—	4	4	no	—
282	Q₁₆	—	4	4	no	—
283	Q₁₆	—	4	4	no	—
284	SD₁₆	—	4	4	no	—
285	SD₁₆	—	4	4	no	—
286	SD₁₆	—	4	8	no	—
287	M₁₆	—	2	4	no	—
288	M₁₆	—	2	8	no	—
289	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
290	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
291	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
292	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₄²	—	4	2	no	—
306	C₄²	C₂×D₄×C₆₇	1	2	yes	—
307	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
309	C₂³	—	4	1	no	—
310	C₂³	#127×C₆₇	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	4	no	—
315	Q₈	—	4	4	no	—
316	Q₈	—	8	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	4	no	—
319	D₄	—	8	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	8	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	#127×C₆₇	1	2	yes	—
330	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	2	4	no	—
335	C₈	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
336	C₈	#137×C₆₇	1	8	yes	complemented
337	C₂²	—	2	1	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
350	C₄	C₂×#137×C₆₇	1	4	yes	—
351	C₂	—	4	1	no	—
352	C₂	—	16	1	no	—
353	C₂	C₆₇×#3151	1	1	yes	characteristic
354	C₂	C₆₇×#3152	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	#532×C₆₇	C₂²
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×#3185	C₂	semidirect
3. C₆₇×#3165	C₂	semidirect
5. C₆₇×#1723	C₂	semidirect
6. C₆₇×#1722	C₂	semidirect
7. C₆₇×#1716	C₂	semidirect
8. C₆₇×#1714	C₂	semidirect
9. C₆₇×#1713	C₂	semidirect
10. C₆₇×#1705	C₂	semidirect
11. C₆₇×#1455	C₂	semidirect
158. C₈×C₆₇	#137	semidirect
173. #19746	C₆₇	direct (N × H)
178. #3185	C₂×C₆₇	semidirect
180. #3165	C₂×C₆₇	semidirect
182. #1723	C₂×C₆₇	semidirect
183. #1722	C₂×C₆₇	semidirect
184. #1716	C₂×C₆₇	semidirect
185. #1714	C₂×C₆₇	semidirect
186. #1713	C₂×C₆₇	semidirect
187. #1705	C₂×C₆₇	semidirect
188. #1455	C₂×C₆₇	semidirect
189. C₆₇	#19746	direct (N × H)
336. C₈	#137×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3185	2
C₆₇×#3182	2
C₆₇×#3165	2
C₆₇×#3152	2
C₆₇×#1723	2
C₆₇×#1722	2
C₆₇×#1716	2
C₆₇×#1714	2
C₆₇×#1713	2
C₆₇×#1705	2
C₆₇×#1455	2
#19746	67

Subgroups — C67×#19746 #30698997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19746 #30698997