Subgroups — C₆₇×#33208 #30712459

Order 17152.

878 conjugacy classes of subgroups.

The subgroup lattice has 340 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	C₂²×C₄×C₈×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×#1942	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1941	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1808	C₂	1	2	yes	maximal
9	C₆₇×#1807	C₂	1	2	yes	maximal
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×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	C₂×C₄×C₈×C₆₇	—	2	4	no	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×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	C₂²×C₈×C₆₇	—	2	2	no	—
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	C₂²×C₈×C₆₇	—	2	2	no	—
36	C₂²×C₈×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
38	C₂²×C₈×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	C₄×C₈×C₆₇	—	2	2	no	—
60	C₄×C₈×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	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₆₇	—	2	1	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	8	no	—
87	C₂×C₈×C₆₇	—	2	8	no	—
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×D₄	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	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₆₇	#41	1	2	yes	—
95	C₂×C₈×C₆₇	#41	1	2	yes	—
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	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	4	8	no	—
101	C₄⋊C₄×C₆₇	—	4	8	no	—
102	C₂²⋊C₄×C₆₇	—	4	8	no	—
103	C₄²×C₆₇	—	2	4	no	—
104	C₄²×C₆₇	—	2	1	no	—
105	C₄²×C₆₇	—	2	1	no	—
106	C₄²×C₆₇	C₂×D₄	1	2	yes	—
107	C₄²×C₆₇	#41	1	2	yes	—
108	C₂³×C₆₇	—	2	1	no	—
109	C₂³×C₆₇	—	2	1	no	—
110	C₂³×C₆₇	—	2	2	no	—
111	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
112	C₂³×C₆₇	#135	1	4	yes	—
113	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	—	2	16	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	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	—	4	4	no	—
129	C₂×C₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	#121	1	4	yes	—
132	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₆₇	#135	1	4	yes	—
134	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₈×C₆₇	—	2	2	no	—
141	C₈×C₆₇	—	2	2	no	—
142	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
143	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	—	2	4	no	—
147	C₂²×C₆₇	—	2	4	no	—
148	C₂²×C₆₇	#499	1	4	yes	complemented
149	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
150	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂²×C₆₇	#532	1	1	yes	characteristic
152	C₂²×C₆₇	C₂×#135	1	2	yes	—
153	C₂²×C₆₇	C₂×#135	1	2	yes	—
154	C₄×C₆₇	—	2	2	no	—
155	C₄×C₆₇	—	2	4	no	—
156	C₄×C₆₇	—	2	4	no	—
157	C₄×C₆₇	—	8	4	no	—
158	C₄×C₆₇	—	8	4	no	—
159	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
160	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₆₇	#533	1	1	yes	characteristic
162	C₄×C₆₇	#533	1	1	yes	characteristic
163	#33208	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₆₇	—	2	4	no	—
165	C₂×C₆₇	#2857	1	1	yes	characteristic
166	C₂×C₆₇	#3050	1	2	yes	—
167	C₂×C₆₇	C₂×#499	1	1	yes	characteristic
168	C₂×C₆₇	#3160	1	1	yes	characteristic
169	C₂×C₆₇	#3168	1	2	yes	—
170	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
171	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
172	#2851	C₂×C₆₇	1	2	yes	—
173	#1991	C₂×C₆₇	1	2	yes	complemented
174	#1942	C₂×C₆₇	1	2	yes	complemented
175	#1941	C₂×C₆₇	1	2	yes	complemented
176	#1846	C₂×C₆₇	1	1	yes	characteristic, complemented
177	#1808	C₂×C₆₇	1	2	yes	—
178	#1807	C₂×C₆₇	1	2	yes	—
179	C₆₇	#33208	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
181	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
182	C₂×#125	C₂²×C₆₇	1	2	yes	—
183	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
184	#499	—	2	4	no	—
185	C₂×#107	C₂²×C₆₇	1	2	yes	—
186	C₂×#107	C₂²×C₆₇	1	2	yes	—
187	C₂×#106	C₂²×C₆₇	1	2	yes	—
188	C₂×#106	C₂²×C₆₇	1	2	yes	—
189	C₂×C₄×C₈	—	2	4	no	—
190	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
194	#389	C₂²×C₆₇	1	4	yes	—
195	#387	—	2	4	no	—
196	#385	—	2	4	no	—
197	#382	C₂²×C₆₇	1	2	yes	—
198	#380	C₂²×C₆₇	1	4	yes	—
199	#340	C₂²×C₆₇	1	4	yes	—
200	#340	C₂²×C₆₇	1	4	yes	—
201	C₂³×C₄	D₄×C₆₇	1	2	yes	—
202	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
208	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
209	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
213	#125	—	2	4	no	—
214	#125	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
218	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
219	C₂×C₂²⋊C₄	—	2	4	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	D₄×C₆₇	1	2	yes	—
223	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
225	#107	—	2	4	no	—
226	#107	—	2	4	no	—
227	#106	—	2	4	no	—
228	#106	—	2	4	no	—
229	C₄×C₈	—	2	2	no	—
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	D₄×C₆₇	1	4	yes	—
233	#95	—	2	4	no	—
234	#95	—	2	4	no	—
235	#95	—	2	4	no	—
236	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
249	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
250	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
251	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	8	no	—
257	C₂×C₈	—	2	8	no	—
258	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
259	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
260	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
261	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
262	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
263	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
264	C₂×C₈	#41×C₆₇	1	2	yes	—
265	C₂×C₈	#41×C₆₇	1	2	yes	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	8	no	—
271	C₄⋊C₄	—	4	8	no	—
272	C₂²⋊C₄	—	4	8	no	—
273	C₄²	—	2	4	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	C₂×D₄×C₆₇	1	2	yes	—
277	C₄²	#41×C₆₇	1	2	yes	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	2	no	—
281	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
282	C₂³	#135×C₆₇	1	4	yes	—
283	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
284	C₂×C₄	—	2	16	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	#121×C₆₇	1	4	yes	—
302	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
303	C₂×C₄	#135×C₆₇	1	4	yes	—
304	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	2	no	—
312	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
313	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	4	no	—
318	C₂²	#499×C₆₇	1	4	yes	complemented
319	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
320	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
321	C₂²	#532×C₆₇	1	1	yes	characteristic
322	C₂²	C₂×#135×C₆₇	1	2	yes	—
323	C₂²	C₂×#135×C₆₇	1	2	yes	—
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	2	2	no	—
327	C₄	—	8	4	no	—
328	C₄	—	8	4	no	—
329	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
330	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
331	C₄	#533×C₆₇	1	1	yes	characteristic
332	C₄	#533×C₆₇	1	1	yes	characteristic
333	C₂	—	2	4	no	—
334	C₂	C₆₇×#2857	1	1	yes	characteristic
335	C₂	C₆₇×#3050	1	2	yes	—
336	C₂	C₂×#499×C₆₇	1	1	yes	characteristic
337	C₂	C₆₇×#3160	1	1	yes	characteristic
338	C₂	C₆₇×#3168	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×#1942	C₂	semidirect
6. C₆₇×#1941	C₂	semidirect
7. C₆₇×#1846	C₂	semidirect
148. C₂²×C₆₇	#499	semidirect
163. #33208	C₆₇	direct (N × H)
170. C₂×#499	C₂×C₆₇	semidirect
173. #1991	C₂×C₆₇	semidirect
174. #1942	C₂×C₆₇	semidirect
175. #1941	C₂×C₆₇	semidirect
176. #1846	C₂×C₆₇	semidirect
179. C₆₇	#33208	direct (N × H)
318. C₂²	#499×C₆₇	semidirect

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
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2851	2
C₆₇×#1991	2
C₆₇×#1942	2
C₆₇×#1941	2
C₆₇×#1846	2
C₆₇×#1808	2
C₆₇×#1807	2
#33208	67

Subgroups — C67×#33208 #30712459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33208 #30712459