Subgroups — C₆₇×#10149 #30689400

Order 17152.

528 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3230	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×#1997	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	#500×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	#484×C₆₇	—	2	2	no	—
19	#477×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	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	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	#107×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	C₂²⋊C₈×C₆₇	—	2	1	no	—
64	#97×C₆₇	—	2	1	no	—
65	C₄×C₈×C₆₇	—	2	1	no	—
66	C₄×C₈×C₆₇	—	2	1	no	—
67	C₄×C₈×C₆₇	—	2	1	no	—
68	C₄×C₈×C₆₇	D₄	1	2	yes	—
69	C₂⁴×C₆₇	—	4	1	no	—
70	C₂×Q₈×C₆₇	—	2	1	no	—
71	C₂×Q₈×C₆₇	—	2	1	no	—
72	C₂×Q₈×C₆₇	—	4	1	no	—
73	C₂×Q₈×C₆₇	—	4	1	no	—
74	C₂×Q₈×C₆₇	—	8	2	no	—
75	C₂×D₄×C₆₇	—	2	2	no	—
76	C₂×D₄×C₆₇	—	2	2	no	—
77	C₂×D₄×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	—	4	4	no	—
79	C₂×D₄×C₆₇	—	4	1	no	—
80	C₂×D₄×C₆₇	—	4	1	no	—
81	C₂×D₄×C₆₇	—	4	1	no	—
82	C₂×D₄×C₆₇	—	4	2	no	—
83	C₂×D₄×C₆₇	—	4	2	no	—
84	C₂×D₄×C₆₇	—	8	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
89	M₁₆×C₆₇	—	2	2	no	—
90	M₁₆×C₆₇	—	2	2	no	—
91	M₁₆×C₆₇	—	4	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	4	2	no	—
96	C₂×C₈×C₆₇	—	4	1	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	1	no	—
99	C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	1	no	—
102	C₄⋊C₄×C₆₇	—	2	1	no	—
103	C₄⋊C₄×C₆₇	—	4	1	no	—
104	C₄⋊C₄×C₆₇	—	4	1	no	—
105	C₄⋊C₄×C₆₇	—	4	1	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₂²⋊C₄×C₆₇	—	4	1	no	—
108	C₄²×C₆₇	—	4	1	no	—
109	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	—	8	2	no	—
115	C₂³×C₆₇	—	8	2	no	—
116	C₂³×C₆₇	—	8	2	no	—
117	C₂³×C₆₇	#121	1	1	yes	characteristic
118	Q₈×C₆₇	—	4	2	no	—
119	Q₈×C₆₇	—	4	2	no	—
120	Q₈×C₆₇	—	8	2	no	—
121	Q₈×C₆₇	—	8	2	no	—
122	D₄×C₆₇	—	4	4	no	—
123	D₄×C₆₇	—	4	4	no	—
124	D₄×C₆₇	—	8	2	no	—
125	D₄×C₆₇	—	8	2	no	—
126	D₄×C₆₇	—	8	2	no	—
127	D₄×C₆₇	—	8	1	no	—
128	D₄×C₆₇	—	8	1	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	8	2	no	—
135	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
136	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
137	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
139	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#135	1	2	yes	—
142	C₂×C₄×C₆₇	#137	1	2	yes	—
143	C₈×C₆₇	—	4	2	no	—
144	C₈×C₆₇	—	4	2	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₈×C₆₇	—	8	1	no	—
147	C₂²×C₆₇	—	2	1	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	—	8	2	no	—
152	C₂²×C₆₇	—	16	4	no	—
153	C₂²×C₆₇	#392	1	1	yes	characteristic
154	C₂²×C₆₇	#464	1	2	yes	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	8	1	no	—
158	C₄×C₆₇	—	8	2	no	—
159	C₄×C₆₇	—	8	1	no	—
160	C₄×C₆₇	#452	1	2	yes	—
161	C₄×C₆₇	#454	1	2	yes	—
162	C₄×C₆₇	#472	1	2	yes	—
163	C₄×C₆₇	#495	1	1	yes	characteristic
164	C₄×C₆₇	#497	1	1	yes	characteristic
165	#10149	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂×C₆₇	—	2	2	no	—
167	C₂×C₆₇	—	16	2	no	—
168	C₂×C₆₇	#2003	1	1	yes	characteristic
169	C₂×C₆₇	#2015	1	1	yes	characteristic
170	C₂×C₆₇	#2018	1	1	yes	characteristic
171	#3230	C₂×C₆₇	1	1	yes	characteristic, complemented
172	#3138	C₂×C₆₇	1	1	yes	characteristic
173	C₂×#486	C₂×C₆₇	1	1	yes	characteristic
174	#1997	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#1545	C₂×C₆₇	1	1	yes	characteristic
176	#1493	C₂×C₆₇	1	1	yes	characteristic, complemented
177	#1278	C₂×C₆₇	1	1	yes	characteristic, complemented
178	C₆₇	#10149	1	1	yes	characteristic, direct product factor, Sylow
179	#533	C₂²×C₆₇	1	1	yes	characteristic
180	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
181	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
182	#500	—	2	1	no	—
183	#490	—	2	2	no	—
184	#487	—	2	1	no	—
185	#486	—	2	1	no	—
186	#486	—	2	1	no	—
187	#486	—	2	2	no	—
188	#486	—	2	2	no	—
189	#484	—	2	2	no	—
190	#477	—	2	2	no	—
191	#431	C₂²×C₆₇	1	1	yes	characteristic
192	#423	C₂²×C₆₇	1	1	yes	characteristic
193	#418	—	2	2	no	—
194	C₂×#102	—	2	2	no	—
195	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
197	#343	—	2	2	no	—
198	#331	—	2	2	no	—
199	Q₈⋊C₈	—	2	2	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
206	C₂²×C₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
212	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
213	#127	—	2	1	no	—
214	#127	—	2	1	no	—
215	#127	—	2	1	no	—
216	#127	D₄×C₆₇	1	2	yes	—
217	#125	—	2	1	no	—
218	C₂²⋊Q₈	—	2	1	no	—
219	C₂²⋊Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	1	no	—
222	#117	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
225	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
226	#107	—	2	4	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
230	#103	—	2	4	no	—
231	#102	—	2	8	no	—
232	#102	—	2	4	no	—
233	#102	—	2	4	no	—
234	C₂²⋊C₈	—	2	1	no	—
235	#97	—	2	1	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	D₄×C₆₇	1	2	yes	—
240	C₂⁴	—	4	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	8	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	4	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	8	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
259	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	—	8	2	no	—
286	C₂³	—	8	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	#121×C₆₇	1	1	yes	characteristic
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
307	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
308	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
309	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
310	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
311	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
312	C₂×C₄	#135×C₆₇	1	2	yes	—
313	C₂×C₄	#137×C₆₇	1	2	yes	—
314	C₈	—	4	2	no	—
315	C₈	—	4	2	no	—
316	C₈	—	4	2	no	—
317	C₈	—	8	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	4	no	—
324	C₂²	#392×C₆₇	1	1	yes	characteristic
325	C₂²	#464×C₆₇	1	2	yes	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	#452×C₆₇	1	2	yes	—
332	C₄	#454×C₆₇	1	2	yes	—
333	C₄	#472×C₆₇	1	2	yes	—
334	C₄	#495×C₆₇	1	1	yes	characteristic
335	C₄	#497×C₆₇	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₆₇×#2003	1	1	yes	characteristic
339	C₂	C₆₇×#2015	1	1	yes	characteristic
340	C₂	C₆₇×#2018	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×#3230	C₂	semidirect
4. C₆₇×#1997	C₂	semidirect
6. C₆₇×#1493	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
165. #10149	C₆₇	direct (N × H)
171. #3230	C₂×C₆₇	semidirect
174. #1997	C₂×C₆₇	semidirect
176. #1493	C₂×C₆₇	semidirect
177. #1278	C₂×C₆₇	semidirect
178. C₆₇	#10149	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×#3230	2
C₆₇×#3138	2
C₂×#486×C₆₇	2
C₆₇×#1997	2
C₆₇×#1545	2
C₆₇×#1493	2
C₆₇×#1278	2
#10149	67

Subgroups — C67×#10149 #30689400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10149 #30689400