Subgroups — C₆₇×#33211 #30712462

Order 17152.

826 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2851	C₂	1	2	yes	maximal
4	C₆₇×#1991	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1944	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1943	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1809	C₂	1	2	yes	maximal
9	C₆₇×#1809	C₂	1	2	yes	maximal
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#125×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#499×C₆₇	—	2	4	no	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#106×C₆₇	C₂²	1	2	yes	—
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	C₄×M₁₆×C₆₇	—	2	4	no	—
20	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#389×C₆₇	C₂²	1	4	yes	—
25	#387×C₆₇	—	2	4	no	—
26	#385×C₆₇	—	2	4	no	—
27	#382×C₆₇	C₂²	1	2	yes	—
28	#380×C₆₇	C₂²	1	4	yes	—
29	#340×C₆₇	C₂²	1	4	yes	—
30	#340×C₆₇	C₂²	1	4	yes	—
31	C₂³×C₄×C₆₇	D₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
38	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#125×C₆₇	—	2	4	no	—
44	#125×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₄²×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#107×C₆₇	—	2	4	no	—
56	#107×C₆₇	—	2	4	no	—
57	#106×C₆₇	—	2	4	no	—
58	#106×C₆₇	—	2	4	no	—
59	#97×C₆₇	—	2	2	no	—
60	#97×C₆₇	—	2	2	no	—
61	C₄×C₈×C₆₇	—	2	2	no	—
62	C₄×C₈×C₆₇	D₄	1	4	yes	—
63	#95×C₆₇	—	2	4	no	—
64	#95×C₆₇	—	2	4	no	—
65	#95×C₆₇	—	2	4	no	—
66	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	1	no	—
68	C₂²×C₄×C₆₇	—	2	1	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	4	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₆₇	—	2	8	no	—
82	M₁₆×C₆₇	—	2	8	no	—
83	C₂×C₈×C₆₇	—	2	1	no	—
84	C₂×C₈×C₆₇	—	2	1	no	—
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₆₇	#41	1	2	yes	—
94	C₂×C₈×C₆₇	#41	1	2	yes	—
95	C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₄⋊C₄×C₆₇	—	2	4	no	—
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	4	8	no	—
100	C₄⋊C₄×C₆₇	—	4	8	no	—
101	C₂²⋊C₄×C₆₇	—	4	8	no	—
102	C₄²×C₆₇	—	2	4	no	—
103	C₄²×C₆₇	—	2	1	no	—
104	C₄²×C₆₇	—	2	1	no	—
105	C₄²×C₆₇	C₂×D₄	1	2	yes	—
106	C₄²×C₆₇	#41	1	2	yes	—
107	C₂³×C₆₇	—	2	1	no	—
108	C₂³×C₆₇	—	2	1	no	—
109	C₂³×C₆₇	—	2	4	no	—
110	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
111	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	4	8	no	—
122	C₂×C₄×C₆₇	—	4	4	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	—	4	4	no	—
125	C₂×C₄×C₆₇	—	4	4	no	—
126	C₂×C₄×C₆₇	—	4	4	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	#121	1	4	yes	—
129	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₈×C₆₇	—	2	2	no	—
137	C₈×C₆₇	—	2	2	no	—
138	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
139	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
140	C₂²×C₆₇	—	2	2	no	—
141	C₂²×C₆₇	—	2	4	no	—
142	C₂²×C₆₇	—	4	2	no	—
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₆₇	#532	1	1	yes	characteristic
147	C₂²×C₆₇	#575	1	4	yes	—
148	C₄×C₆₇	—	2	2	no	—
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	—	4	2	no	—
151	C₄×C₆₇	—	8	4	no	—
152	C₄×C₆₇	—	8	4	no	—
153	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
154	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
155	C₄×C₆₇	#533	1	1	yes	characteristic
156	C₄×C₆₇	#533	1	1	yes	characteristic
157	#33211	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂×C₆₇	—	4	2	no	—
159	C₂×C₆₇	#2857	1	1	yes	characteristic
160	C₂×C₆₇	#3054	1	2	yes	—
161	C₂×C₆₇	#3166	1	2	yes	—
162	C₂×C₆₇	#3183	1	2	yes	—
163	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
164	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
165	#2851	C₂×C₆₇	1	2	yes	—
166	#1991	C₂×C₆₇	1	2	yes	complemented
167	#1944	C₂×C₆₇	1	2	yes	complemented
168	#1943	C₂×C₆₇	1	2	yes	complemented
169	#1846	C₂×C₆₇	1	1	yes	characteristic, complemented
170	#1809	C₂×C₆₇	1	2	yes	—
171	#1809	C₂×C₆₇	1	2	yes	—
172	C₆₇	#33211	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
174	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×#125	C₂²×C₆₇	1	2	yes	—
176	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
177	#499	—	2	4	no	—
178	C₂×#107	C₂²×C₆₇	1	2	yes	—
179	C₂×#107	C₂²×C₆₇	1	2	yes	—
180	C₂×#106	C₂²×C₆₇	1	2	yes	—
181	C₂×#106	C₂²×C₆₇	1	2	yes	—
182	C₄×M₁₆	—	2	4	no	—
183	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
184	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
187	#389	C₂²×C₆₇	1	4	yes	—
188	#387	—	2	4	no	—
189	#385	—	2	4	no	—
190	#382	C₂²×C₆₇	1	2	yes	—
191	#380	C₂²×C₆₇	1	4	yes	—
192	#340	C₂²×C₆₇	1	4	yes	—
193	#340	C₂²×C₆₇	1	4	yes	—
194	C₂³×C₄	D₄×C₆₇	1	2	yes	—
195	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
201	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
202	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
206	#125	—	2	4	no	—
207	#125	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
211	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	4	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	D₄×C₆₇	1	2	yes	—
216	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
218	#107	—	2	4	no	—
219	#107	—	2	4	no	—
220	#106	—	2	4	no	—
221	#106	—	2	4	no	—
222	#97	—	2	2	no	—
223	#97	—	2	2	no	—
224	C₄×C₈	—	2	2	no	—
225	C₄×C₈	D₄×C₆₇	1	4	yes	—
226	#95	—	2	4	no	—
227	#95	—	2	4	no	—
228	#95	—	2	4	no	—
229	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
242	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
243	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
244	M₁₆	—	2	8	no	—
245	M₁₆	—	2	8	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
252	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
253	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
254	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
255	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
256	C₂×C₈	#41×C₆₇	1	2	yes	—
257	C₂×C₈	#41×C₆₇	1	2	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	8	no	—
263	C₄⋊C₄	—	4	8	no	—
264	C₂²⋊C₄	—	4	8	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	C₂×D₄×C₆₇	1	2	yes	—
269	C₄²	#41×C₆₇	1	2	yes	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	4	no	—
273	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
274	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	#121×C₆₇	1	4	yes	—
292	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
298	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	2	2	no	—
301	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
302	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
308	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
309	C₂²	#532×C₆₇	1	1	yes	characteristic
310	C₂²	#575×C₆₇	1	4	yes	—
311	C₄	—	2	4	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	8	4	no	—
315	C₄	—	8	4	no	—
316	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
317	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
318	C₄	#533×C₆₇	1	1	yes	characteristic
319	C₄	#533×C₆₇	1	1	yes	characteristic
320	C₂	—	4	2	no	—
321	C₂	C₆₇×#2857	1	1	yes	characteristic
322	C₂	C₆₇×#3054	1	2	yes	—
323	C₂	C₆₇×#3166	1	2	yes	—
324	C₂	C₆₇×#3183	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×#499×C₆₇	C₂	semidirect
4. C₆₇×#1991	C₂	semidirect
5. C₆₇×#1944	C₂	semidirect
6. C₆₇×#1943	C₂	semidirect
7. C₆₇×#1846	C₂	semidirect
157. #33211	C₆₇	direct (N × H)
163. C₂×#499	C₂×C₆₇	semidirect
166. #1991	C₂×C₆₇	semidirect
167. #1944	C₂×C₆₇	semidirect
168. #1943	C₂×C₆₇	semidirect
169. #1846	C₂×C₆₇	semidirect
172. C₆₇	#33211	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×#499×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2851	2
C₆₇×#1991	2
C₆₇×#1944	2
C₆₇×#1943	2
C₆₇×#1846	2
C₆₇×#1809	2
#33211	67

Subgroups — C67×#33211 #30712462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33211 #30712462