Subgroups — C₆₇×#33065 #30712316

Order 17152.

746 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#491×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3080	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3039	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2875	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1998	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1966	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1930	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1908	C₂	1	4	yes	maximal
9	C₆₇×#1838	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₆₇	C₂²	1	2	yes	—
11	C₂×#131×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	#491×C₆₇	—	2	1	no	—
16	#491×C₆₇	—	2	1	no	—
17	#491×C₆₇	—	2	2	no	—
18	#479×C₆₇	—	2	2	no	—
19	#478×C₆₇	—	2	2	no	—
20	#462×C₆₇	—	2	2	no	—
21	#461×C₆₇	—	2	2	no	—
22	#428×C₆₇	C₂²	1	2	yes	—
23	#423×C₆₇	C₂²	1	1	yes	characteristic
24	#423×C₆₇	C₂²	1	1	yes	characteristic
25	#419×C₆₇	C₂²	1	2	yes	—
26	#417×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	4	yes	—
28	C₂×#101×C₆₇	C₂²	1	4	yes	—
29	#405×C₆₇	—	2	2	no	—
30	#405×C₆₇	C₂²	1	2	yes	—
31	C₄×M₁₆×C₆₇	—	2	1	no	—
32	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#97×C₆₇	—	2	1	no	—
34	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
35	#343×C₆₇	C₂²	1	4	yes	—
36	#341×C₆₇	C₂²	1	4	yes	—
37	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
39	#137×C₆₇	—	2	8	no	—
40	C₂×Q₁₆×C₆₇	—	2	2	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	#131×C₆₇	—	2	4	no	—
43	#131×C₆₇	—	2	4	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
46	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	—	2	1	no	—
52	C₄⋊Q₈×C₆₇	—	2	1	no	—
53	#125×C₆₇	—	2	2	no	—
54	#125×C₆₇	—	2	1	no	—
55	#125×C₆₇	—	2	1	no	—
56	C₄×Q₈×C₆₇	D₄	1	2	yes	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	D₄	1	2	yes	—
59	#117×C₆₇	—	2	1	no	—
60	#117×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	—	2	1	no	—
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	#107×C₆₇	—	2	2	no	—
67	#106×C₆₇	—	2	2	no	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	#103×C₆₇	—	2	2	no	—
71	#103×C₆₇	—	2	4	no	—
72	#103×C₆₇	—	2	4	no	—
73	#103×C₆₇	—	2	4	no	—
74	#102×C₆₇	—	2	2	no	—
75	#101×C₆₇	—	2	8	no	—
76	#97×C₆₇	—	2	2	no	—
77	#97×C₆₇	—	2	1	no	—
78	#97×C₆₇	D₄	1	1	yes	characteristic
79	#97×C₆₇	D₄	1	1	yes	characteristic
80	#97×C₆₇	D₄	1	2	yes	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	—	2	1	no	—
83	#41×C₆₇	—	2	2	no	—
84	#41×C₆₇	#41	1	4	yes	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	—	2	4	no	—
89	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	—	2	1	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	#41	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
98	Q₁₆×C₆₇	—	4	4	no	—
99	SD₁₆×C₆₇	—	4	4	no	—
100	M₁₆×C₆₇	—	2	8	no	—
101	M₁₆×C₆₇	—	2	2	no	—
102	M₁₆×C₆₇	—	2	4	no	—
103	M₁₆×C₆₇	—	2	4	no	—
104	M₁₆×C₆₇	#41	1	4	yes	—
105	C₂×C₈×C₆₇	—	2	8	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	4	2	no	—
117	C₄⋊C₄×C₆₇	—	4	2	no	—
118	C₄⋊C₄×C₆₇	—	4	2	no	—
119	C₄⋊C₄×C₆₇	—	4	2	no	—
120	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²⋊C₄×C₆₇	—	2	1	no	—
122	C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	1	no	—
125	C₄²×C₆₇	C₂×D₄	1	2	yes	—
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₆₇	—	2	1	no	—
129	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
130	Q₈×C₆₇	—	2	1	no	—
131	Q₈×C₆₇	—	4	4	no	—
132	Q₈×C₆₇	—	4	2	no	—
133	Q₈×C₆₇	—	4	2	no	—
134	Q₈×C₆₇	#124	1	2	yes	—
135	D₄×C₆₇	—	2	4	no	—
136	D₄×C₆₇	—	2	1	no	—
137	D₄×C₆₇	#124	1	2	yes	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	4	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	8	2	no	—
146	C₂×C₄×C₆₇	—	8	2	no	—
147	C₂×C₄×C₆₇	#121	1	2	yes	—
148	C₂×C₄×C₆₇	#121	1	2	yes	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
153	C₈×C₆₇	—	2	2	no	—
154	C₈×C₆₇	—	2	4	no	—
155	C₈×C₆₇	—	2	2	no	—
156	C₈×C₆₇	—	4	2	no	—
157	C₈×C₆₇	—	4	2	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	4	2	no	—
161	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
162	C₂²×C₆₇	#529	1	2	yes	—
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	4	2	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	#525	1	2	yes	—
170	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
171	C₄×C₆₇	#532	1	1	yes	characteristic
172	#33065	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₆₇	—	2	2	no	—
174	C₂×C₆₇	—	4	2	no	—
175	C₂×C₆₇	#2850	1	1	yes	characteristic
176	C₂×C₆₇	#3069	1	2	yes	—
177	C₂×#491	C₂×C₆₇	1	1	yes	characteristic, complemented
178	#3080	C₂×C₆₇	1	1	yes	characteristic
179	#3039	C₂×C₆₇	1	1	yes	characteristic
180	#2875	C₂×C₆₇	1	1	yes	characteristic
181	#1998	C₂×C₆₇	1	1	yes	characteristic, complemented
182	#1966	C₂×C₆₇	1	1	yes	characteristic, complemented
183	#1930	C₂×C₆₇	1	4	yes	complemented
184	#1908	C₂×C₆₇	1	4	yes	—
185	#1838	C₂×C₆₇	1	1	yes	characteristic, complemented
186	C₆₇	#33065	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×#137	C₂²×C₆₇	1	2	yes	—
188	C₂×#131	C₂²×C₆₇	1	2	yes	—
189	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
190	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
191	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
192	#491	—	2	1	no	—
193	#491	—	2	1	no	—
194	#491	—	2	2	no	—
195	#479	—	2	2	no	—
196	#478	—	2	2	no	—
197	#462	—	2	2	no	—
198	#461	—	2	2	no	—
199	#428	C₂²×C₆₇	1	2	yes	—
200	#423	C₂²×C₆₇	1	1	yes	characteristic
201	#423	C₂²×C₆₇	1	1	yes	characteristic
202	#419	C₂²×C₆₇	1	2	yes	—
203	#417	C₂²×C₆₇	1	2	yes	—
204	C₂×#103	C₂²×C₆₇	1	4	yes	—
205	C₂×#101	C₂²×C₆₇	1	4	yes	—
206	#405	—	2	2	no	—
207	#405	C₂²×C₆₇	1	2	yes	—
208	C₄×M₁₆	—	2	1	no	—
209	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×#97	—	2	1	no	—
211	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
212	#343	C₂²×C₆₇	1	4	yes	—
213	#341	C₂²×C₆₇	1	4	yes	—
214	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
215	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
216	#137	—	2	8	no	—
217	C₂×Q₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	2	no	—
219	#131	—	2	4	no	—
220	#131	—	2	4	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
223	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
224	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	#125	—	2	2	no	—
231	#125	—	2	1	no	—
232	#125	—	2	1	no	—
233	C₄×Q₈	D₄×C₆₇	1	2	yes	—
234	C₄×D₄	—	2	2	no	—
235	C₄×D₄	D₄×C₆₇	1	2	yes	—
236	#117	—	2	1	no	—
237	#117	C₂³×C₆₇	1	1	yes	characteristic
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
243	#107	—	2	2	no	—
244	#106	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#103	—	2	2	no	—
248	#103	—	2	4	no	—
249	#103	—	2	4	no	—
250	#103	—	2	4	no	—
251	#102	—	2	2	no	—
252	#101	—	2	8	no	—
253	#97	—	2	2	no	—
254	#97	—	2	1	no	—
255	#97	D₄×C₆₇	1	1	yes	characteristic
256	#97	D₄×C₆₇	1	1	yes	characteristic
257	#97	D₄×C₆₇	1	2	yes	—
258	C₄×C₈	—	2	2	no	—
259	C₄×C₈	—	2	1	no	—
260	#41	—	2	2	no	—
261	#41	#41×C₆₇	1	4	yes	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	—	2	4	no	—
266	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	#41×C₆₇	1	2	yes	—
274	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
275	Q₁₆	—	4	4	no	—
276	SD₁₆	—	4	4	no	—
277	M₁₆	—	2	8	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	#41×C₆₇	1	4	yes	—
282	C₂×C₈	—	2	8	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	—	4	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
298	C₂²⋊C₄	—	2	1	no	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	1	no	—
302	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
304	C₄²	C₂×D₄×C₆₇	1	2	yes	—
305	C₂³	—	2	1	no	—
306	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
307	Q₈	—	2	1	no	—
308	Q₈	—	4	4	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	#124×C₆₇	1	2	yes	—
312	D₄	—	2	4	no	—
313	D₄	—	2	1	no	—
314	D₄	#124×C₆₇	1	2	yes	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	#121×C₆₇	1	2	yes	—
325	C₂×C₄	#121×C₆₇	1	2	yes	—
326	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
329	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
330	C₈	—	2	2	no	—
331	C₈	—	2	4	no	—
332	C₈	—	2	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
339	C₂²	#529×C₆₇	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	#525×C₆₇	1	2	yes	—
347	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
348	C₄	#532×C₆₇	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	2	no	—
351	C₂	C₆₇×#2850	1	1	yes	characteristic
352	C₂	C₆₇×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#41×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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. C₂×#491×C₆₇	C₂	semidirect
5. C₆₇×#1998	C₂	semidirect
6. C₆₇×#1966	C₂	semidirect
7. C₆₇×#1930	C₂	semidirect
9. C₆₇×#1838	C₂	semidirect
172. #33065	C₆₇	direct (N × H)
177. C₂×#491	C₂×C₆₇	semidirect
181. #1998	C₂×C₆₇	semidirect
182. #1966	C₂×C₆₇	semidirect
183. #1930	C₂×C₆₇	semidirect
185. #1838	C₂×C₆₇	semidirect
186. C₆₇	#33065	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#491×C₆₇	2
C₆₇×#3080	2
C₆₇×#3039	2
C₆₇×#2875	2
C₆₇×#1998	2
C₆₇×#1966	2
C₆₇×#1930	2
C₆₇×#1908	2
C₆₇×#1838	2
#33065	67

Subgroups — C67×#33065 #30712316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33065 #30712316