Subgroups — C₆₇×#10151 #30689402

Order 17152.

544 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3229	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3138	C₂	1	1	yes	maximal, characteristic
3	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1966	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1545	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	#533×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
11	#498×C₆₇	—	2	1	no	—
12	#490×C₆₇	—	2	2	no	—
13	#487×C₆₇	—	2	1	no	—
14	#486×C₆₇	—	2	2	no	—
15	#486×C₆₇	—	2	1	no	—
16	#486×C₆₇	—	2	1	no	—
17	#486×C₆₇	—	2	2	no	—
18	#483×C₆₇	—	2	2	no	—
19	#475×C₆₇	—	2	2	no	—
20	#431×C₆₇	C₂²	1	1	yes	characteristic
21	#423×C₆₇	C₂²	1	1	yes	characteristic
22	#418×C₆₇	—	2	2	no	—
23	C₂×#102×C₆₇	—	2	2	no	—
24	C₂×#101×C₆₇	—	2	2	no	—
25	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#331×C₆₇	—	2	2	no	—
28	Q₈⋊C₈×C₆₇	—	2	2	no	—
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×D₄×C₆₇	—	2	1	no	—
31	C₂²×D₄×C₆₇	—	2	1	no	—
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂×M₁₆×C₆₇	—	2	1	no	—
34	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
35	C₂²×C₈×C₆₇	—	2	1	no	—
36	C₄⋊Q₈×C₆₇	—	2	1	no	—
37	C₄⋊Q₈×C₆₇	—	2	1	no	—
38	C₄⋊Q₈×C₆₇	—	2	1	no	—
39	C₄⋊Q₈×C₆₇	—	2	1	no	—
40	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
42	#127×C₆₇	—	2	1	no	—
43	#127×C₆₇	—	2	1	no	—
44	#127×C₆₇	—	2	1	no	—
45	#127×C₆₇	D₄	1	2	yes	—
46	#125×C₆₇	—	2	1	no	—
47	C₂²⋊Q₈×C₆₇	—	2	1	no	—
48	C₂²⋊Q₈×C₆₇	—	2	1	no	—
49	C₄×Q₈×C₆₇	—	2	1	no	—
50	C₄×Q₈×C₆₇	—	2	1	no	—
51	#117×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#106×C₆₇	—	2	4	no	—
56	C₄⋊C₈×C₆₇	—	2	1	no	—
57	C₄⋊C₈×C₆₇	—	2	1	no	—
58	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
59	#103×C₆₇	—	2	4	no	—
60	#102×C₆₇	—	2	8	no	—
61	#102×C₆₇	—	2	4	no	—
62	#102×C₆₇	—	2	4	no	—
63	#101×C₆₇	—	4	4	no	—
64	C₂²⋊C₈×C₆₇	—	2	1	no	—
65	#97×C₆₇	—	2	1	no	—
66	C₄×C₈×C₆₇	—	2	1	no	—
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	—	2	1	no	—
69	C₄×C₈×C₆₇	D₄	1	2	yes	—
70	C₂⁴×C₆₇	—	4	1	no	—
71	C₂×Q₈×C₆₇	—	2	1	no	—
72	C₂×Q₈×C₆₇	—	2	1	no	—
73	C₂×Q₈×C₆₇	—	4	1	no	—
74	C₂×Q₈×C₆₇	—	4	1	no	—
75	C₂×Q₈×C₆₇	—	4	2	no	—
76	C₂×Q₈×C₆₇	—	4	2	no	—
77	C₂×D₄×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	—	2	2	no	—
79	C₂×D₄×C₆₇	—	2	2	no	—
80	C₂×D₄×C₆₇	—	4	4	no	—
81	C₂×D₄×C₆₇	—	4	1	no	—
82	C₂×D₄×C₆₇	—	4	1	no	—
83	C₂×D₄×C₆₇	—	4	1	no	—
84	C₂×D₄×C₆₇	—	4	2	no	—
85	C₂×D₄×C₆₇	—	4	2	no	—
86	C₂×D₄×C₆₇	—	8	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
91	M₁₆×C₆₇	—	2	2	no	—
92	M₁₆×C₆₇	—	2	2	no	—
93	M₁₆×C₆₇	—	4	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	4	2	no	—
98	C₂×C₈×C₆₇	—	4	1	no	—
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	2	1	no	—
101	C₄⋊C₄×C₆₇	—	2	1	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₄⋊C₄×C₆₇	—	2	1	no	—
105	C₄⋊C₄×C₆₇	—	4	1	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄⋊C₄×C₆₇	—	4	1	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₂²⋊C₄×C₆₇	—	4	1	no	—
110	C₄²×C₆₇	—	4	1	no	—
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	—	8	2	no	—
117	C₂³×C₆₇	—	8	2	no	—
118	C₂³×C₆₇	—	8	2	no	—
119	C₂³×C₆₇	#121	1	1	yes	characteristic
120	Q₈×C₆₇	—	4	2	no	—
121	Q₈×C₆₇	—	4	2	no	—
122	Q₈×C₆₇	—	8	2	no	—
123	Q₈×C₆₇	—	8	1	no	—
124	Q₈×C₆₇	—	8	1	no	—
125	D₄×C₆₇	—	4	4	no	—
126	D₄×C₆₇	—	4	4	no	—
127	D₄×C₆₇	—	8	2	no	—
128	D₄×C₆₇	—	8	2	no	—
129	D₄×C₆₇	—	8	2	no	—
130	D₄×C₆₇	—	8	1	no	—
131	D₄×C₆₇	—	8	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	—	4	1	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	4	no	—
138	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
139	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
142	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#135	1	2	yes	—
145	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
146	C₈×C₆₇	—	4	2	no	—
147	C₈×C₆₇	—	4	2	no	—
148	C₈×C₆₇	—	4	2	no	—
149	C₈×C₆₇	—	8	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	—	8	2	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	—	16	4	no	—
156	C₂²×C₆₇	#392	1	1	yes	characteristic
157	C₂²×C₆₇	#463	1	2	yes	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	—	2	1	no	—
160	C₄×C₆₇	—	8	1	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	—	8	1	no	—
163	C₄×C₆₇	#449	1	2	yes	—
164	C₄×C₆₇	#454	1	2	yes	—
165	C₄×C₆₇	#471	1	2	yes	—
166	C₄×C₆₇	#495	1	1	yes	characteristic
167	C₄×C₆₇	#496	1	1	yes	characteristic
168	#10151	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂×C₆₇	—	2	2	no	—
170	C₂×C₆₇	—	16	2	no	—
171	C₂×C₆₇	#2003	1	1	yes	characteristic
172	C₂×C₆₇	#2012	1	1	yes	characteristic
173	C₂×C₆₇	#2019	1	1	yes	characteristic
174	#3229	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#3138	C₂×C₆₇	1	1	yes	characteristic
176	C₂×#486	C₂×C₆₇	1	1	yes	characteristic
177	#1966	C₂×C₆₇	1	1	yes	characteristic, complemented
178	#1545	C₂×C₆₇	1	1	yes	characteristic
179	#1493	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#1278	C₂×C₆₇	1	1	yes	characteristic, complemented
181	C₆₇	#10151	1	1	yes	characteristic, direct product factor, Sylow
182	#533	C₂²×C₆₇	1	1	yes	characteristic
183	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
184	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
185	#498	—	2	1	no	—
186	#490	—	2	2	no	—
187	#487	—	2	1	no	—
188	#486	—	2	1	no	—
189	#486	—	2	1	no	—
190	#486	—	2	2	no	—
191	#486	—	2	2	no	—
192	#483	—	2	2	no	—
193	#475	—	2	2	no	—
194	#431	C₂²×C₆₇	1	1	yes	characteristic
195	#423	C₂²×C₆₇	1	1	yes	characteristic
196	#418	—	2	2	no	—
197	C₂×#102	—	2	2	no	—
198	C₂×#101	—	2	2	no	—
199	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
201	#331	—	2	2	no	—
202	Q₈⋊C₈	—	2	2	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
209	C₂²×C₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
215	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
216	#127	—	2	1	no	—
217	#127	—	2	1	no	—
218	#127	—	2	1	no	—
219	#127	D₄×C₆₇	1	2	yes	—
220	#125	—	2	1	no	—
221	C₂²⋊Q₈	—	2	1	no	—
222	C₂²⋊Q₈	—	2	1	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	2	1	no	—
225	#117	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
228	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
229	#106	—	2	4	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
233	#103	—	2	4	no	—
234	#102	—	2	8	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	#101	—	4	4	no	—
238	C₂²⋊C₈	—	2	1	no	—
239	#97	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	—	2	1	no	—
243	C₄×C₈	D₄×C₆₇	1	2	yes	—
244	C₂⁴	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	4	4	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	8	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
265	M₁₆	—	2	2	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	4	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
289	C₂³	—	4	2	no	—
290	C₂³	—	8	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	#121×C₆₇	1	1	yes	characteristic
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	8	1	no	—
298	Q₈	—	8	1	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	1	no	—
305	D₄	—	8	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
313	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
314	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
315	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
316	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
317	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
318	C₂×C₄	#135×C₆₇	1	2	yes	—
319	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
320	C₈	—	4	2	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₈	—	8	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	16	4	no	—
330	C₂²	#392×C₆₇	1	1	yes	characteristic
331	C₂²	#463×C₆₇	1	2	yes	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	1	no	—
337	C₄	#449×C₆₇	1	2	yes	—
338	C₄	#454×C₆₇	1	2	yes	—
339	C₄	#471×C₆₇	1	2	yes	—
340	C₄	#495×C₆₇	1	1	yes	characteristic
341	C₄	#496×C₆₇	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	16	2	no	—
344	C₂	C₆₇×#2003	1	1	yes	characteristic
345	C₂	C₆₇×#2012	1	1	yes	characteristic
346	C₂	C₆₇×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
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×#3229	C₂	semidirect
4. C₆₇×#1966	C₂	semidirect
6. C₆₇×#1493	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
168. #10151	C₆₇	direct (N × H)
174. #3229	C₂×C₆₇	semidirect
177. #1966	C₂×C₆₇	semidirect
179. #1493	C₂×C₆₇	semidirect
180. #1278	C₂×C₆₇	semidirect
181. C₆₇	#10151	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×#3229	2
C₆₇×#3138	2
C₂×#486×C₆₇	2
C₆₇×#1966	2
C₆₇×#1545	2
C₆₇×#1493	2
C₆₇×#1278	2
#10151	67

Subgroups — C67×#10151 #30689402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10151 #30689402