Subgroups — C₆₇×#33672 #30712923

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2232	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2227	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2207	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2185	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2175	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	#532×C₆₇	—	2	1	no	—
11	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
12	C₂×SD₃₂×C₆₇	C₂²	1	4	yes	—
13	#500×C₆₇	C₂²	1	1	yes	characteristic
14	#493×C₆₇	C₂²	1	1	yes	characteristic
15	#486×C₆₇	—	2	1	no	—
16	#466×C₆₇	—	2	1	no	—
17	#466×C₆₇	C₂²	1	2	yes	—
18	C₄×Q₁₆×C₆₇	—	2	1	no	—
19	C₄×Q₁₆×C₆₇	—	2	1	no	—
20	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
22	C₄×D₈×C₆₇	—	2	1	no	—
23	C₄×D₈×C₆₇	C₂²	1	2	yes	—
24	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₆₇	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#366×C₆₇	C₂²	1	4	yes	—
28	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
29	#358×C₆₇	C₂²	1	4	yes	—
30	#357×C₆₇	C₂²	1	4	yes	—
31	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
32	#346×C₆₇	C₂²	1	2	yes	—
33	C₂×#41×C₆₇	—	4	1	no	—
34	C₂²×Q₈×C₆₇	—	2	1	no	—
35	C₂×Q₁₆×C₆₇	—	2	4	no	—
36	C₂×Q₁₆×C₆₇	—	2	1	no	—
37	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
38	C₂×D₈×C₆₇	—	2	1	no	—
39	C₂×D₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₆₇	—	2	1	no	—
43	#127×C₆₇	—	2	1	no	—
44	#124×C₆₇	—	4	1	no	—
45	#121×C₆₇	—	2	2	no	—
46	#121×C₆₇	—	4	1	no	—
47	C₄×Q₈×C₆₇	—	2	2	no	—
48	C₄×Q₈×C₆₇	—	2	2	no	—
49	C₄×Q₈×C₆₇	—	4	1	no	—
50	C₄×Q₈×C₆₇	—	4	1	no	—
51	C₄×D₄×C₆₇	—	2	2	no	—
52	C₄×D₄×C₆₇	—	4	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
55	C₂×C₄²×C₆₇	—	4	1	no	—
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	SD₃₂×C₆₇	—	2	8	no	—
58	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
59	#107×C₆₇	—	2	1	no	—
60	#107×C₆₇	—	2	1	no	—
61	#107×C₆₇	C₂³	1	2	yes	—
62	#107×C₆₇	C₂³	1	2	yes	—
63	#103×C₆₇	—	2	2	no	—
64	#103×C₆₇	—	2	2	no	—
65	#102×C₆₇	—	2	2	no	—
66	#102×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	#41×C₆₇	—	8	4	no	—
72	C₂×Q₈×C₆₇	—	2	4	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×Q₈×C₆₇	—	4	2	no	—
75	C₂×Q₈×C₆₇	—	4	1	no	—
76	C₂×Q₈×C₆₇	—	4	1	no	—
77	C₂×D₄×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	—	4	2	no	—
79	C₂×D₄×C₆₇	—	4	1	no	—
80	C₂×D₄×C₆₇	—	4	1	no	—
81	C₂²×C₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	—	4	2	no	—
84	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₆₇	—	2	4	no	—
88	Q₁₆×C₆₇	—	2	2	no	—
89	Q₁₆×C₆₇	#41	1	4	yes	complemented
90	D₈×C₆₇	—	2	4	no	—
91	D₈×C₆₇	—	4	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	4	1	no	—
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	4	1	no	—
101	C₄⋊C₄×C₆₇	—	4	2	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₄⋊C₄×C₆₇	—	4	1	no	—
104	C₂²⋊C₄×C₆₇	—	4	2	no	—
105	C₂²⋊C₄×C₆₇	—	4	2	no	—
106	C₄²×C₆₇	—	4	2	no	—
107	C₄²×C₆₇	—	4	2	no	—
108	C₄²×C₆₇	D₈	1	1	yes	characteristic
109	C₄²×C₆₇	D₈	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₆₇	—	2	4	no	—
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
115	Q₈×C₆₇	—	2	4	no	—
116	Q₈×C₆₇	—	4	4	no	—
117	Q₈×C₆₇	—	4	2	no	—
118	Q₈×C₆₇	—	8	2	no	—
119	D₄×C₆₇	—	4	2	no	—
120	D₄×C₆₇	—	4	2	no	—
121	D₄×C₆₇	—	8	2	no	—
122	D₄×C₆₇	—	8	4	no	—
123	D₄×C₆₇	—	8	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	4	4	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	—	8	2	no	—
134	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	C₂×#41	1	2	yes	—
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	#509	1	2	yes	—
150	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	4	4	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	—	8	2	no	—
158	C₄×C₆₇	#510	1	2	yes	—
159	C₄×C₆₇	#525	1	1	yes	characteristic
160	C₄×C₆₇	#575	1	1	yes	characteristic
161	#33672	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₆₇	—	2	2	no	—
163	C₂×C₆₇	—	16	2	no	—
164	C₂×C₆₇	#3100	1	1	yes	characteristic
165	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
166	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
167	#3169	C₂×C₆₇	1	1	yes	characteristic
168	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#2232	C₂×C₆₇	1	2	yes	complemented
170	#2227	C₂×C₆₇	1	2	yes	complemented
171	#2207	C₂×C₆₇	1	2	yes	complemented
172	#2185	C₂×C₆₇	1	2	yes	complemented
173	#2175	C₂×C₆₇	1	4	yes	complemented
174	#2163	C₂×C₆₇	1	1	yes	characteristic, complemented
175	C₆₇	#33672	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
177	#532	—	2	1	no	—
178	C₂×C₄×Q₈	—	2	1	no	—
179	C₂×SD₃₂	C₂²×C₆₇	1	4	yes	—
180	#500	C₂²×C₆₇	1	1	yes	characteristic
181	#493	C₂²×C₆₇	1	1	yes	characteristic
182	#486	—	2	1	no	—
183	#466	—	2	1	no	—
184	#466	C₂²×C₆₇	1	2	yes	—
185	C₄×Q₁₆	—	2	1	no	—
186	C₄×Q₁₆	—	2	1	no	—
187	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
188	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
189	C₄×D₈	—	2	1	no	—
190	C₄×D₈	C₂²×C₆₇	1	2	yes	—
191	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
192	C₂×#103	—	2	1	no	—
193	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
194	#366	C₂²×C₆₇	1	4	yes	—
195	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	—
196	#358	C₂²×C₆₇	1	4	yes	—
197	#357	C₂²×C₆₇	1	4	yes	—
198	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
199	#346	C₂²×C₆₇	1	2	yes	—
200	C₂×#41	—	4	1	no	—
201	C₂²×Q₈	—	2	1	no	—
202	C₂×Q₁₆	—	2	4	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	C₂³×C₆₇	1	2	yes	—
207	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	1	no	—
210	#127	—	2	1	no	—
211	#124	—	4	1	no	—
212	#121	—	2	2	no	—
213	#121	—	4	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	4	1	no	—
217	C₄×Q₈	—	4	1	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	4	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	4	1	no	—
222	C₂×C₄²	—	4	1	no	—
223	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
224	SD₃₂	—	2	8	no	—
225	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	C₂³×C₆₇	1	2	yes	—
229	#107	C₂³×C₆₇	1	2	yes	—
230	#103	—	2	2	no	—
231	#103	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
235	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
238	#41	—	8	4	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
254	Q₁₆	—	2	4	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	#41×C₆₇	1	4	yes	complemented
257	D₈	—	2	4	no	—
258	D₈	—	4	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
264	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	D₈×C₆₇	1	1	yes	characteristic
276	C₄²	D₈×C₆₇	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₁₆	—	2	4	no	—
280	C₂³	—	4	2	no	—
281	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	8	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	C₂×#41×C₆₇	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	#509×C₆₇	1	2	yes	—
317	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	#510×C₆₇	1	2	yes	—
326	C₄	#525×C₆₇	1	1	yes	characteristic
327	C₄	#575×C₆₇	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	16	2	no	—
330	C₂	C₆₇×#3100	1	1	yes	characteristic
331	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
332	C₂	C₂×#510×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
3. C₆₇×#2232	C₂	semidirect
4. C₆₇×#2227	C₂	semidirect
5. C₆₇×#2207	C₂	semidirect
6. C₆₇×#2185	C₂	semidirect
7. C₆₇×#2175	C₂	semidirect
8. C₆₇×#2163	C₂	semidirect
89. Q₁₆×C₆₇	#41	semidirect
161. #33672	C₆₇	direct (N × H)
168. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
169. #2232	C₂×C₆₇	semidirect
170. #2227	C₂×C₆₇	semidirect
171. #2207	C₂×C₆₇	semidirect
172. #2185	C₂×C₆₇	semidirect
173. #2175	C₂×C₆₇	semidirect
174. #2163	C₂×C₆₇	semidirect
175. C₆₇	#33672	direct (N × H)
256. Q₁₆	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3169	2
C₂×C₄×Q₁₆×C₆₇	2
C₆₇×#2232	2
C₆₇×#2227	2
C₆₇×#2207	2
C₆₇×#2185	2
C₆₇×#2175	2
C₆₇×#2163	2
#33672	67

Subgroups — C67×#33672 #30712923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33672 #30712923