Subgroups — C₆₇×#33669 #30712920

Order 17152.

650 conjugacy classes of subgroups.

The subgroup lattice has 352 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, complemented
2	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2233	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2226	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2206	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2184	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2177	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	#532×C₆₇	—	2	1	no	—
11	C₂×C₄×D₄×C₆₇	—	2	1	no	—
12	C₂×D₁₆×C₆₇	C₂²	1	4	yes	—
13	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	#486×C₆₇	—	2	1	no	—
16	#466×C₆₇	—	2	1	no	—
17	#466×C₆₇	C₂²	1	2	yes	—
18	C₄×D₈×C₆₇	—	2	1	no	—
19	C₄×D₈×C₆₇	—	2	1	no	—
20	C₄×D₈×C₆₇	—	2	1	no	—
21	C₄×D₈×C₆₇	C₂²	1	2	yes	—
22	C₄×D₈×C₆₇	C₂²	1	2	yes	—
23	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#102×C₆₇	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#367×C₆₇	C₂²	1	4	yes	complemented
28	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
29	#357×C₆₇	C₂²	1	4	yes	—
30	#357×C₆₇	C₂²	1	4	yes	complemented
31	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
32	#346×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#41×C₆₇	—	4	1	no	—
34	C₂²×D₄×C₆₇	—	2	1	no	—
35	C₂³×C₄×C₆₇	—	4	1	no	—
36	C₂×D₈×C₆₇	—	2	4	no	—
37	C₂×D₈×C₆₇	—	2	1	no	—
38	C₂×D₈×C₆₇	—	2	1	no	—
39	C₂×D₈×C₆₇	C₂³	1	2	yes	—
40	C₂×D₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₆₇	—	2	1	no	—
44	#127×C₆₇	—	2	1	no	—
45	#124×C₆₇	—	4	1	no	—
46	#121×C₆₇	—	2	2	no	—
47	#121×C₆₇	—	4	1	no	—
48	C₄×D₄×C₆₇	—	2	2	no	—
49	C₄×D₄×C₆₇	—	2	2	no	—
50	C₄×D₄×C₆₇	—	2	2	no	—
51	C₄×D₄×C₆₇	—	4	1	no	—
52	C₄×D₄×C₆₇	—	4	1	no	—
53	C₄×D₄×C₆₇	—	4	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	D₁₆×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	#102×C₆₇	—	2	2	no	—
64	#102×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	C₂⁴×C₆₇	—	4	1	no	—
72	#41×C₆₇	—	8	4	no	—
73	C₂×Q₈×C₆₇	—	4	1	no	—
74	C₂×D₄×C₆₇	—	2	4	no	—
75	C₂×D₄×C₆₇	—	2	2	no	—
76	C₂×D₄×C₆₇	—	2	2	no	—
77	C₂×D₄×C₆₇	—	4	2	no	—
78	C₂×D₄×C₆₇	—	4	2	no	—
79	C₂×D₄×C₆₇	—	4	1	no	—
80	C₂×D₄×C₆₇	—	4	1	no	—
81	C₂×D₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	—	4	2	no	—
84	C₂²×C₄×C₆₇	—	4	2	no	—
85	C₂²×C₄×C₆₇	—	4	2	no	—
86	C₂²×C₄×C₆₇	—	8	4	no	—
87	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	D₈×C₆₇	—	2	4	no	—
91	D₈×C₆₇	—	2	2	no	—
92	D₈×C₆₇	—	2	4	no	—
93	D₈×C₆₇	—	4	2	no	—
94	D₈×C₆₇	#41	1	4	yes	complemented
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	4	1	no	—
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	4	1	no	—
104	C₄⋊C₄×C₆₇	—	4	1	no	—
105	C₂²⋊C₄×C₆₇	—	4	2	no	—
106	C₂²⋊C₄×C₆₇	—	4	2	no	—
107	C₂²⋊C₄×C₆₇	—	4	2	no	—
108	C₂²⋊C₄×C₆₇	—	4	2	no	—
109	C₄²×C₆₇	D₈	1	1	yes	characteristic
110	C₄²×C₆₇	D₈	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₆₇	—	2	4	no	—
114	C₂³×C₆₇	—	4	4	no	—
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	—	4	2	no	—
117	C₂³×C₆₇	—	8	2	no	—
118	C₂³×C₆₇	—	8	2	no	—
119	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₆₇	—	8	2	no	—
121	D₄×C₆₇	—	2	4	no	—
122	D₄×C₆₇	—	4	4	no	—
123	D₄×C₆₇	—	4	2	no	—
124	D₄×C₆₇	—	4	2	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	8	2	no	—
127	D₄×C₆₇	—	8	4	no	—
128	D₄×C₆₇	—	8	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	8	4	no	—
136	C₂×C₄×C₆₇	—	8	4	no	—
137	C₂×C₄×C₆₇	—	8	2	no	—
138	C₂×C₄×C₆₇	—	8	4	no	—
139	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₆₇	—	2	1	no	—
146	C₈×C₆₇	—	2	1	no	—
147	C₈×C₆₇	—	2	1	no	—
148	C₈×C₆₇	C₂×#41	1	2	yes	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	4	4	no	—
152	C₂²×C₆₇	—	8	8	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	—	8	2	no	—
156	C₂²×C₆₇	—	8	2	no	—
157	C₂²×C₆₇	—	8	2	no	—
158	C₂²×C₆₇	#509	1	2	yes	—
159	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₆₇	—	2	1	no	—
161	C₄×C₆₇	—	2	1	no	—
162	C₄×C₆₇	—	2	1	no	—
163	C₄×C₆₇	—	2	1	no	—
164	C₄×C₆₇	—	8	2	no	—
165	C₄×C₆₇	—	8	2	no	—
166	C₄×C₆₇	#509	1	2	yes	—
167	C₄×C₆₇	#525	1	1	yes	characteristic
168	C₄×C₆₇	#575	1	1	yes	characteristic
169	#33669	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₆₇	—	2	2	no	—
171	C₂×C₆₇	—	8	4	no	—
172	C₂×C₆₇	—	16	2	no	—
173	C₂×C₆₇	#3100	1	1	yes	characteristic
174	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
175	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
176	#3169	C₂×C₆₇	1	1	yes	characteristic, complemented
177	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
178	#2233	C₂×C₆₇	1	2	yes	complemented
179	#2226	C₂×C₆₇	1	2	yes	complemented
180	#2206	C₂×C₆₇	1	2	yes	complemented
181	#2184	C₂×C₆₇	1	2	yes	complemented
182	#2177	C₂×C₆₇	1	4	yes	complemented
183	#2163	C₂×C₆₇	1	1	yes	characteristic, complemented
184	C₆₇	#33669	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
186	#532	—	2	1	no	—
187	C₂×C₄×D₄	—	2	1	no	—
188	C₂×D₁₆	C₂²×C₆₇	1	4	yes	—
189	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
190	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
191	#486	—	2	1	no	—
192	#466	—	2	1	no	—
193	#466	C₂²×C₆₇	1	2	yes	—
194	C₄×D₈	—	2	1	no	—
195	C₄×D₈	—	2	1	no	—
196	C₄×D₈	—	2	1	no	—
197	C₄×D₈	C₂²×C₆₇	1	2	yes	—
198	C₄×D₈	C₂²×C₆₇	1	2	yes	—
199	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
200	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
201	C₂×#102	—	2	1	no	—
202	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
203	#367	C₂²×C₆₇	1	4	yes	complemented
204	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
205	#357	C₂²×C₆₇	1	4	yes	—
206	#357	C₂²×C₆₇	1	4	yes	complemented
207	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
208	#346	C₂²×C₆₇	1	2	yes	complemented
209	C₂×#41	—	4	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂³×C₄	—	4	1	no	—
212	C₂×D₈	—	2	4	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂×D₈	C₂³×C₆₇	1	2	yes	—
216	C₂×D₈	C₂³×C₆₇	1	2	yes	—
217	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
219	C₄⋊Q₈	—	2	1	no	—
220	#127	—	2	1	no	—
221	#124	—	4	1	no	—
222	#121	—	2	2	no	—
223	#121	—	4	1	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	4	1	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	4	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₂²⋊C₄	—	4	1	no	—
232	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
233	D₁₆	—	2	8	no	—
234	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
235	#107	—	2	1	no	—
236	#107	—	2	1	no	—
237	#107	C₂³×C₆₇	1	2	yes	—
238	#107	C₂³×C₆₇	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
244	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
247	C₂⁴	—	4	1	no	—
248	#41	—	8	4	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	8	4	no	—
263	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
264	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
266	D₈	—	2	4	no	—
267	D₈	—	2	2	no	—
268	D₈	—	2	4	no	—
269	D₈	—	4	2	no	—
270	D₈	#41×C₆₇	1	4	yes	complemented
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₄²	D₈×C₆₇	1	1	yes	characteristic
286	C₄²	D₈×C₆₇	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
289	C₁₆	—	2	4	no	—
290	C₂³	—	4	4	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
296	Q₈	—	8	2	no	—
297	D₄	—	2	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	4	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
316	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
317	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
318	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
319	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	C₂×#41×C₆₇	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	8	8	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	#509×C₆₇	1	2	yes	—
335	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	#509×C₆₇	1	2	yes	—
343	C₄	#525×C₆₇	1	1	yes	characteristic
344	C₄	#575×C₆₇	1	1	yes	characteristic
345	C₂	—	2	2	no	—
346	C₂	—	8	4	no	—
347	C₂	—	16	2	no	—
348	C₂	C₆₇×#3100	1	1	yes	characteristic
349	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
350	C₂	C₂×#509×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
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
1. C₆₇×#3169	C₂	semidirect
2. C₂×C₄×D₈×C₆₇	C₂	semidirect
3. C₆₇×#2233	C₂	semidirect
4. C₆₇×#2226	C₂	semidirect
5. C₆₇×#2206	C₂	semidirect
6. C₆₇×#2184	C₂	semidirect
7. C₆₇×#2177	C₂	semidirect
8. C₆₇×#2163	C₂	semidirect
13. #500×C₆₇	C₂²	semidirect
14. #493×C₆₇	C₂²	semidirect
23. C₄×D₈×C₆₇	C₂²	semidirect
27. #367×C₆₇	C₂²	semidirect
28. C₄⋊C₁₆×C₆₇	C₂²	semidirect
30. #357×C₆₇	C₂²	semidirect
32. #346×C₆₇	C₂²	semidirect
94. D₈×C₆₇	#41	semidirect
169. #33669	C₆₇	direct (N × H)
176. #3169	C₂×C₆₇	semidirect
177. C₂×C₄×D₈	C₂×C₆₇	semidirect
178. #2233	C₂×C₆₇	semidirect
179. #2226	C₂×C₆₇	semidirect
180. #2206	C₂×C₆₇	semidirect
181. #2184	C₂×C₆₇	semidirect
182. #2177	C₂×C₆₇	semidirect
183. #2163	C₂×C₆₇	semidirect
184. C₆₇	#33669	direct (N × H)
189. #500	C₂²×C₆₇	semidirect
190. #493	C₂²×C₆₇	semidirect
199. C₄×D₈	C₂²×C₆₇	semidirect
203. #367	C₂²×C₆₇	semidirect
204. C₄⋊C₁₆	C₂²×C₆₇	semidirect
206. #357	C₂²×C₆₇	semidirect
208. #346	C₂²×C₆₇	semidirect
270. D₈	#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×D₈×C₆₇	2
C₆₇×#2233	2
C₆₇×#2226	2
C₆₇×#2206	2
C₆₇×#2184	2
C₆₇×#2177	2
C₆₇×#2163	2
#33669	67

Subgroups — C67×#33669 #30712920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33669 #30712920