Subgroups — C₆₇×#19712 #30698963

Order 17152.

704 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3228	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3210	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1722	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1720	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1717	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	#533×C₆₇	C₂²	1	2	yes	—
11	#532×C₆₇	C₂²	1	2	yes	—
12	#500×C₆₇	C₂²	1	1	yes	characteristic
13	#500×C₆₇	C₂²	1	1	yes	characteristic
14	#500×C₆₇	C₂²	1	2	yes	—
15	#498×C₆₇	C₂²	1	2	yes	—
16	#494×C₆₇	C₂²	1	1	yes	characteristic
17	#494×C₆₇	C₂²	1	2	yes	—
18	#493×C₆₇	C₂²	1	1	yes	characteristic
19	#492×C₆₇	C₂²	1	2	yes	—
20	#487×C₆₇	C₂²	1	2	yes	—
21	#487×C₆₇	C₂²	1	2	yes	—
22	#486×C₆₇	C₂²	1	2	yes	—
23	#486×C₆₇	C₂²	1	2	yes	—
24	#486×C₆₇	C₂²	1	2	yes	—
25	#486×C₆₇	C₂²	1	2	yes	—
26	#482×C₆₇	—	2	4	no	—
27	#478×C₆₇	—	2	4	no	—
28	#467×C₆₇	—	2	2	no	—
29	#466×C₆₇	—	2	2	no	—
30	#464×C₆₇	—	2	4	no	—
31	#452×C₆₇	—	2	4	no	—
32	C₄×Q₁₆×C₆₇	—	2	2	no	—
33	C₄×D₈×C₆₇	—	2	2	no	—
34	#431×C₆₇	C₂²	1	1	yes	characteristic
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#322×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	2	yes	—
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	2	2	no	—
41	C₂×Q₁₆×C₆₇	—	2	4	no	—
42	C₂×SD₁₆×C₆₇	—	2	4	no	—
43	C₂×D₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	#127×C₆₇	C₂³	1	2	yes	—
49	#125×C₆₇	—	2	2	no	—
50	#124×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	2	2	no	—
52	C₂²⋊Q₈×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	2	2	no	—
54	#121×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	#117×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#107×C₆₇	—	2	2	no	—
62	#107×C₆₇	—	2	2	no	—
63	#107×C₆₇	—	2	4	no	—
64	#106×C₆₇	—	2	4	no	—
65	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
66	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
67	#103×C₆₇	—	2	4	no	—
68	#103×C₆₇	—	2	4	no	—
69	#102×C₆₇	—	2	4	no	—
70	#102×C₆₇	—	2	4	no	—
71	#102×C₆₇	—	2	4	no	—
72	#102×C₆₇	—	2	4	no	—
73	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
74	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
75	C₄×C₈×C₆₇	D₄	1	2	yes	—
76	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂³	1	2	yes	—
78	C₄×C₈×C₆₇	C₂³	1	2	yes	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	#41×C₆₇	—	8	4	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	4	2	no	—
88	C₂²×C₄×C₆₇	—	4	2	no	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₆₇	—	4	4	no	—
93	Q₁₆×C₆₇	—	8	4	no	—
94	SD₁₆×C₆₇	—	8	4	no	—
95	D₈×C₆₇	—	4	4	no	—
96	C₂×C₈×C₆₇	—	4	2	no	—
97	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	4	2	no	—
108	C₄⋊C₄×C₆₇	—	4	2	no	—
109	C₂²⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	—	4	2	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₄²×C₆₇	—	4	2	no	—
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₇	—	4	2	no	—
118	C₂³×C₆₇	#127	1	1	yes	characteristic
119	Q₈×C₆₇	—	4	4	no	—
120	Q₈×C₆₇	—	4	4	no	—
121	Q₈×C₆₇	—	4	4	no	—
122	D₄×C₆₇	—	4	4	no	—
123	D₄×C₆₇	—	4	4	no	—
124	D₄×C₆₇	—	4	4	no	—
125	D₄×C₆₇	—	8	4	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	8	4	no	—
133	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
135	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₆₇	—	2	2	no	—
140	C₈×C₆₇	—	2	2	no	—
141	C₈×C₆₇	—	2	4	no	—
142	C₈×C₆₇	—	2	4	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	8	2	no	—
147	C₂²×C₆₇	—	8	2	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	—	8	2	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	#575	1	2	yes	—
157	C₄×C₆₇	#576	1	2	yes	—
158	C₄×C₆₇	#578	1	2	yes	—
159	#19712	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂×C₆₇	—	4	1	no	—
161	C₂×C₆₇	—	16	2	no	—
162	C₂×C₆₇	#3150	1	1	yes	characteristic
163	C₂×C₆₇	#3155	1	1	yes	characteristic
164	C₂×C₆₇	#3156	1	1	yes	characteristic
165	#3228	C₂×C₆₇	1	2	yes	complemented
166	#3210	C₂×C₆₇	1	2	yes	complemented
167	#3170	C₂×C₆₇	1	1	yes	characteristic, complemented
168	#3169	C₂×C₆₇	1	1	yes	characteristic
169	#1723	C₂×C₆₇	1	2	yes	complemented
170	#1722	C₂×C₆₇	1	2	yes	complemented
171	#1720	C₂×C₆₇	1	2	yes	complemented
172	#1717	C₂×C₆₇	1	2	yes	complemented
173	#1452	C₂×C₆₇	1	1	yes	characteristic, complemented
174	C₆₇	#19712	1	1	yes	characteristic, direct product factor, Sylow
175	#533	C₂²×C₆₇	1	2	yes	—
176	#532	C₂²×C₆₇	1	2	yes	—
177	#500	C₂²×C₆₇	1	1	yes	characteristic
178	#500	C₂²×C₆₇	1	1	yes	characteristic
179	#500	C₂²×C₆₇	1	2	yes	—
180	#498	C₂²×C₆₇	1	2	yes	—
181	#494	C₂²×C₆₇	1	1	yes	characteristic
182	#494	C₂²×C₆₇	1	2	yes	—
183	#493	C₂²×C₆₇	1	1	yes	characteristic
184	#492	C₂²×C₆₇	1	2	yes	—
185	#487	C₂²×C₆₇	1	2	yes	—
186	#487	C₂²×C₆₇	1	2	yes	—
187	#486	C₂²×C₆₇	1	2	yes	—
188	#486	C₂²×C₆₇	1	2	yes	—
189	#486	C₂²×C₆₇	1	2	yes	—
190	#486	C₂²×C₆₇	1	2	yes	—
191	#482	—	2	4	no	—
192	#478	—	2	4	no	—
193	#467	—	2	2	no	—
194	#466	—	2	2	no	—
195	#464	—	2	4	no	—
196	#452	—	2	4	no	—
197	C₄×Q₁₆	—	2	2	no	—
198	C₄×D₈	—	2	2	no	—
199	#431	C₂²×C₆₇	1	1	yes	characteristic
200	#431	C₂²×C₆₇	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
202	#322	C₂²×C₆₇	1	2	yes	—
203	#322	C₂²×C₆₇	1	2	yes	—
204	C₂×#41	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂²×C₈	D₄×C₆₇	1	2	yes	—
210	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
211	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
213	#127	C₂³×C₆₇	1	2	yes	—
214	#125	—	2	2	no	—
215	#124	—	2	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	#121	—	2	2	no	—
219	#121	—	2	2	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	#117	—	2	2	no	—
225	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	#107	—	2	2	no	—
228	#107	—	2	4	no	—
229	#106	—	2	4	no	—
230	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
231	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
232	#103	—	2	4	no	—
233	#103	—	2	4	no	—
234	#102	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
239	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
240	C₄×C₈	D₄×C₆₇	1	2	yes	—
241	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₆₇	1	2	yes	—
244	C₄×C₈	C₂³×C₆₇	1	2	yes	—
245	#41	—	8	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
257	Q₁₆	—	4	4	no	—
258	Q₁₆	—	8	4	no	—
259	SD₁₆	—	8	4	no	—
260	D₈	—	4	4	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
263	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
282	C₂³	—	4	2	no	—
283	C₂³	#127×C₆₇	1	1	yes	characteristic
284	Q₈	—	4	4	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	8	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
299	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
300	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	4	no	—
307	C₈	—	2	4	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	#575×C₆₇	1	2	yes	—
322	C₄	#576×C₆₇	1	2	yes	—
323	C₄	#578×C₆₇	1	2	yes	—
324	C₂	—	4	1	no	—
325	C₂	—	16	2	no	—
326	C₂	C₆₇×#3150	1	1	yes	characteristic
327	C₂	C₆₇×#3155	1	1	yes	characteristic
328	C₂	C₆₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#3169	C₂
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×#3228	C₂	semidirect
2. C₆₇×#3210	C₂	semidirect
3. C₆₇×#3170	C₂	semidirect
5. C₆₇×#1723	C₂	semidirect
6. C₆₇×#1722	C₂	semidirect
7. C₆₇×#1720	C₂	semidirect
8. C₆₇×#1717	C₂	semidirect
9. C₆₇×#1452	C₂	semidirect
159. #19712	C₆₇	direct (N × H)
165. #3228	C₂×C₆₇	semidirect
166. #3210	C₂×C₆₇	semidirect
167. #3170	C₂×C₆₇	semidirect
169. #1723	C₂×C₆₇	semidirect
170. #1722	C₂×C₆₇	semidirect
171. #1720	C₂×C₆₇	semidirect
172. #1717	C₂×C₆₇	semidirect
173. #1452	C₂×C₆₇	semidirect
174. C₆₇	#19712	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×#3228	2
C₆₇×#3210	2
C₆₇×#3170	2
C₆₇×#3169	2
C₆₇×#1723	2
C₆₇×#1722	2
C₆₇×#1720	2
C₆₇×#1717	2
C₆₇×#1452	2
#19712	67

Subgroups — C67×#19712 #30698963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19712 #30698963