Subgroups — C₆₇×#33059 #30712310

Order 17152.

870 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	C₂×#491×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2851	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1992	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1933	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1894	C₂	1	4	yes	maximal
9	C₆₇×#1836	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#491×C₆₇	—	2	4	no	—
16	C₂×#103×C₆₇	C₂²	1	4	yes	—
17	C₂×#103×C₆₇	C₂²	1	4	yes	—
18	C₄×M₁₆×C₆₇	—	2	4	no	—
19	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#391×C₆₇	C₂²	1	2	yes	—
23	#389×C₆₇	—	2	2	no	—
24	#389×C₆₇	C₂²	1	2	yes	—
25	#387×C₆₇	—	2	2	no	—
26	#385×C₆₇	—	2	2	no	—
27	#384×C₆₇	C₂²	1	1	yes	characteristic
28	#382×C₆₇	C₂²	1	1	yes	characteristic
29	#380×C₆₇	—	2	2	no	—
30	#380×C₆₇	C₂²	1	2	yes	—
31	#340×C₆₇	C₂²	1	4	yes	—
32	#340×C₆₇	C₂²	1	4	yes	—
33	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₆₇	D₄	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₆₇	—	2	4	no	—
37	C₂×M₁₆×C₆₇	—	2	4	no	—
38	C₂×M₁₆×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₆₇	—	2	2	no	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	#125×C₆₇	—	2	2	no	—
44	#125×C₆₇	—	2	2	no	—
45	C₂²⋊Q₈×C₆₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	D₄	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#103×C₆₇	—	2	8	no	—
62	#103×C₆₇	—	2	8	no	—
63	#97×C₆₇	—	2	2	no	—
64	#97×C₆₇	D₄	1	4	yes	—
65	C₄×C₈×C₆₇	—	2	4	no	—
66	#95×C₆₇	—	2	4	no	—
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	—	2	2	no	—
69	#95×C₆₇	—	2	2	no	—
70	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₆₇	—	2	2	no	—
72	C₂×Q₈×C₆₇	—	2	4	no	—
73	C₂×Q₈×C₆₇	—	2	4	no	—
74	C₂×Q₈×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	4	2	no	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	#41	1	4	yes	—
89	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₆₇	—	2	16	no	—
91	C₂×C₈×C₆₇	—	2	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	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₆₇	#41	1	4	yes	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	4	4	no	—
102	C₄⋊C₄×C₆₇	—	4	4	no	—
103	C₄⋊C₄×C₆₇	—	4	4	no	—
104	C₄⋊C₄×C₆₇	—	4	4	no	—
105	C₄⋊C₄×C₆₇	—	4	4	no	—
106	C₂²⋊C₄×C₆₇	—	4	4	no	—
107	C₂²⋊C₄×C₆₇	—	4	4	no	—
108	C₄²×C₆₇	—	2	4	no	—
109	C₄²×C₆₇	—	2	1	no	—
110	C₄²×C₆₇	—	2	1	no	—
111	C₄²×C₆₇	C₂×D₄	1	2	yes	—
112	C₄²×C₆₇	#41	1	2	yes	—
113	C₂³×C₆₇	—	2	1	no	—
114	C₂³×C₆₇	—	2	1	no	—
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	#124	1	2	yes	—
118	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₆₇	—	4	4	no	—
120	Q₈×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	4	8	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	#121	1	4	yes	—
142	C₂×C₄×C₆₇	#124	1	2	yes	—
143	C₂×C₄×C₆₇	#137	1	4	yes	—
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
146	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
147	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₈×C₆₇	—	2	4	no	—
149	C₈×C₆₇	—	2	4	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	4	2	no	—
154	C₂²×C₆₇	—	4	2	no	—
155	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
156	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
157	C₂²×C₆₇	#532	1	1	yes	characteristic
158	C₂²×C₆₇	C₂×#137	1	2	yes	—
159	C₂²×C₆₇	#575	1	2	yes	—
160	C₄×C₆₇	—	2	2	no	—
161	C₄×C₆₇	—	2	4	no	—
162	C₄×C₆₇	—	2	4	no	—
163	C₄×C₆₇	—	8	2	no	—
164	C₄×C₆₇	—	8	2	no	—
165	C₄×C₆₇	—	8	2	no	—
166	C₄×C₆₇	—	8	2	no	—
167	C₄×C₆₇	#525	1	2	yes	—
168	C₄×C₆₇	#529	1	2	yes	—
169	#33059	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₆₇	—	4	2	no	—
171	C₂×C₆₇	#2850	1	1	yes	characteristic
172	C₂×C₆₇	#3056	1	2	yes	—
173	C₂×C₆₇	#3139	1	1	yes	characteristic
174	C₂×C₆₇	#3142	1	1	yes	characteristic
175	C₂×C₆₇	#3172	1	2	yes	—
176	C₂×#491	C₂×C₆₇	1	1	yes	characteristic, complemented
177	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
178	#2857	C₂×C₆₇	1	1	yes	characteristic
179	#2851	C₂×C₆₇	1	1	yes	characteristic
180	#1992	C₂×C₆₇	1	1	yes	characteristic, complemented
181	#1960	C₂×C₆₇	1	1	yes	characteristic, complemented
182	#1933	C₂×C₆₇	1	4	yes	complemented
183	#1894	C₂×C₆₇	1	4	yes	—
184	#1836	C₂×C₆₇	1	1	yes	characteristic, complemented
185	C₆₇	#33059	1	1	yes	characteristic, direct product factor, Sylow
186	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
187	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
188	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
189	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
190	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
191	#491	—	2	4	no	—
192	C₂×#103	C₂²×C₆₇	1	4	yes	—
193	C₂×#103	C₂²×C₆₇	1	4	yes	—
194	C₄×M₁₆	—	2	4	no	—
195	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
198	#391	C₂²×C₆₇	1	2	yes	—
199	#389	—	2	2	no	—
200	#389	C₂²×C₆₇	1	2	yes	—
201	#387	—	2	2	no	—
202	#385	—	2	2	no	—
203	#384	C₂²×C₆₇	1	1	yes	characteristic
204	#382	C₂²×C₆₇	1	1	yes	characteristic
205	#380	—	2	2	no	—
206	#380	C₂²×C₆₇	1	2	yes	—
207	#340	C₂²×C₆₇	1	4	yes	—
208	#340	C₂²×C₆₇	1	4	yes	—
209	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
210	C₂³×C₄	D₄×C₆₇	1	2	yes	—
211	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
212	C₂×M₁₆	—	2	4	no	—
213	C₂×M₁₆	—	2	4	no	—
214	C₂×M₁₆	—	2	4	no	—
215	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
216	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	—	2	2	no	—
219	#125	—	2	2	no	—
220	#125	—	2	2	no	—
221	C₂²⋊Q₈	—	2	8	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	D₄×C₆₇	1	2	yes	—
235	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
237	#103	—	2	8	no	—
238	#103	—	2	8	no	—
239	#97	—	2	2	no	—
240	#97	D₄×C₆₇	1	4	yes	—
241	C₄×C₈	—	2	4	no	—
242	#95	—	2	4	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
264	C₂²×C₄	#41×C₆₇	1	4	yes	—
265	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
266	M₁₆	—	2	16	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
272	C₂×C₈	#41×C₆₇	1	4	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂×D₄×C₆₇	1	2	yes	—
288	C₄²	#41×C₆₇	1	2	yes	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	#124×C₆₇	1	2	yes	—
294	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
295	Q₈	—	4	4	no	—
296	Q₈	—	4	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	#121×C₆₇	1	4	yes	—
318	C₂×C₄	#124×C₆₇	1	2	yes	—
319	C₂×C₄	#137×C₆₇	1	4	yes	—
320	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
321	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
322	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
323	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
332	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
333	C₂²	#532×C₆₇	1	1	yes	characteristic
334	C₂²	C₂×#137×C₆₇	1	2	yes	—
335	C₂²	#575×C₆₇	1	2	yes	—
336	C₄	—	2	4	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	#525×C₆₇	1	2	yes	—
344	C₄	#529×C₆₇	1	2	yes	—
345	C₂	—	4	2	no	—
346	C₂	C₆₇×#2850	1	1	yes	characteristic
347	C₂	C₆₇×#3056	1	2	yes	—
348	C₂	C₆₇×#3139	1	1	yes	characteristic
349	C₂	C₆₇×#3142	1	1	yes	characteristic
350	C₂	C₆₇×#3172	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×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×#1992	C₂	semidirect
6. C₆₇×#1960	C₂	semidirect
7. C₆₇×#1933	C₂	semidirect
9. C₆₇×#1836	C₂	semidirect
169. #33059	C₆₇	direct (N × H)
176. C₂×#491	C₂×C₆₇	semidirect
180. #1992	C₂×C₆₇	semidirect
181. #1960	C₂×C₆₇	semidirect
182. #1933	C₂×C₆₇	semidirect
184. #1836	C₂×C₆₇	semidirect
185. C₆₇	#33059	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
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2857	2
C₆₇×#2851	2
C₆₇×#1992	2
C₆₇×#1960	2
C₆₇×#1933	2
C₆₇×#1894	2
C₆₇×#1836	2
#33059	67

Subgroups — C67×#33059 #30712310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33059 #30712310