Subgroups — C₆₇×#33212 #30712463

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2857	C₂	1	2	yes	maximal
4	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1944	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1809	C₂	1	4	yes	maximal
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	#500×C₆₇	—	2	4	no	—
13	C₂×#107×C₆₇	C₂²	1	4	yes	—
14	C₂×#106×C₆₇	C₂²	1	4	yes	—
15	C₄×M₁₆×C₆₇	—	2	4	no	—
16	C₂×#97×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#391×C₆₇	—	2	2	no	—
20	#389×C₆₇	—	2	2	no	—
21	#389×C₆₇	C₂²	1	4	yes	—
22	#384×C₆₇	C₂²	1	2	yes	—
23	#380×C₆₇	—	2	2	no	—
24	#380×C₆₇	C₂²	1	4	yes	—
25	#340×C₆₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₆₇	—	2	2	no	—
27	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₆₇	—	2	4	no	—
31	C₂×M₁₆×C₆₇	—	2	4	no	—
32	C₂×M₁₆×C₆₇	Q₈	1	8	yes	complemented
33	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₄⋊Q₈×C₆₇	—	2	4	no	—
36	C₄⋊Q₈×C₆₇	—	2	4	no	—
37	C₂²⋊Q₈×C₆₇	—	4	8	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
44	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#107×C₆₇	—	2	8	no	—
52	#106×C₆₇	—	2	8	no	—
53	#97×C₆₇	—	2	4	no	—
54	C₄×C₈×C₆₇	—	2	2	no	—
55	C₄×C₈×C₆₇	D₄	1	4	yes	—
56	#95×C₆₇	—	2	4	no	—
57	#95×C₆₇	—	2	4	no	—
58	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₆₇	—	4	4	no	—
60	C₂×Q₈×C₆₇	—	4	4	no	—
61	C₂×Q₈×C₆₇	—	4	4	no	—
62	C₂²×C₄×C₆₇	—	2	1	no	—
63	C₂²×C₄×C₆₇	—	2	1	no	—
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
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	4	no	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₆₇	—	2	16	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
83	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₆₇	#41	1	4	yes	—
85	C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	—	2	8	no	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	4	8	no	—
89	C₄⋊C₄×C₆₇	—	4	4	no	—
90	C₄⋊C₄×C₆₇	—	4	4	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	8	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	2	1	no	—
97	C₄²×C₆₇	C₂×D₄	1	2	yes	—
98	C₄²×C₆₇	#41	1	2	yes	—
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	—	2	2	no	—
102	C₂³×C₆₇	—	2	2	no	—
103	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
104	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₆₇	—	8	8	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	4	4	no	—
115	C₂×C₄×C₆₇	—	4	4	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	4	4	no	—
120	C₂×C₄×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	—	4	4	no	—
122	C₂×C₄×C₆₇	#121	1	2	yes	—
123	C₂×C₄×C₆₇	#121	1	2	yes	—
124	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
126	C₂×C₄×C₆₇	#137	1	4	yes	—
127	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₈×C₆₇	—	2	4	no	—
132	C₈×C₆₇	C₂²⋊Q₈	1	8	yes	complemented
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	4	no	—
135	C₂²×C₆₇	—	4	2	no	—
136	C₂²×C₆₇	—	4	1	no	—
137	C₂²×C₆₇	—	4	1	no	—
138	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
139	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₂²×C₆₇	#532	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
142	C₂²×C₆₇	C₂×#137	1	2	yes	—
143	C₄×C₆₇	—	2	2	no	—
144	C₄×C₆₇	—	4	1	no	—
145	C₄×C₆₇	—	4	1	no	—
146	C₄×C₆₇	—	8	4	no	—
147	C₄×C₆₇	—	8	4	no	—
148	C₄×C₆₇	#469	1	4	yes	—
149	C₄×C₆₇	#470	1	4	yes	—
150	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
151	C₄×C₆₇	#533	1	2	yes	—
152	#33212	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂×C₆₇	—	4	2	no	—
154	C₂×C₆₇	#2857	1	1	yes	characteristic
155	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
156	C₂×C₆₇	C₂×#470	1	1	yes	characteristic
157	C₂×C₆₇	#3165	1	2	yes	—
158	C₂×C₆₇	#3182	1	2	yes	—
159	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
160	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
161	#2857	C₂×C₆₇	1	2	yes	—
162	#1990	C₂×C₆₇	1	2	yes	complemented
163	#1944	C₂×C₆₇	1	4	yes	complemented
164	#1844	C₂×C₆₇	1	1	yes	characteristic, complemented
165	#1809	C₂×C₆₇	1	4	yes	—
166	C₆₇	#33212	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
168	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
169	C₂×C₂²⋊Q₈	—	2	2	no	—
170	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
171	#500	—	2	4	no	—
172	C₂×#107	C₂²×C₆₇	1	4	yes	—
173	C₂×#106	C₂²×C₆₇	1	4	yes	—
174	C₄×M₁₆	—	2	4	no	—
175	C₂×#97	C₂²×C₆₇	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
178	#391	—	2	2	no	—
179	#389	—	2	2	no	—
180	#389	C₂²×C₆₇	1	4	yes	—
181	#384	C₂²×C₆₇	1	2	yes	—
182	#380	—	2	2	no	—
183	#380	C₂²×C₆₇	1	4	yes	—
184	#340	C₂²×C₆₇	1	8	yes	—
185	C₂²×Q₈	—	2	2	no	—
186	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
187	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	Q₈×C₆₇	1	8	yes	complemented
192	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
193	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	—	2	4	no	—
196	C₂²⋊Q₈	—	4	8	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
202	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
207	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
210	#107	—	2	8	no	—
211	#106	—	2	8	no	—
212	#97	—	2	4	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	D₄×C₆₇	1	4	yes	—
215	#95	—	2	4	no	—
216	#95	—	2	4	no	—
217	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	4	4	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
237	M₁₆	—	2	16	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
241	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
242	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
243	C₂×C₈	#41×C₆₇	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	4	8	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	—	4	8	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂×D₄×C₆₇	1	2	yes	—
257	C₄²	#41×C₆₇	1	2	yes	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
263	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
264	Q₈	—	8	8	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	#121×C₆₇	1	2	yes	—
282	C₂×C₄	#121×C₆₇	1	2	yes	—
283	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
284	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
285	C₂×C₄	#137×C₆₇	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
288	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
289	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
290	C₈	—	2	4	no	—
291	C₈	C₂²⋊Q₈×C₆₇	1	8	yes	complemented
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
298	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
299	C₂²	#532×C₆₇	1	1	yes	characteristic
300	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
301	C₂²	C₂×#137×C₆₇	1	2	yes	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	8	4	no	—
306	C₄	—	8	4	no	—
307	C₄	#469×C₆₇	1	4	yes	—
308	C₄	#470×C₆₇	1	4	yes	—
309	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
310	C₄	#533×C₆₇	1	2	yes	—
311	C₂	—	4	2	no	—
312	C₂	C₆₇×#2857	1	1	yes	characteristic
313	C₂	C₂×#469×C₆₇	1	1	yes	characteristic
314	C₂	C₂×#470×C₆₇	1	1	yes	characteristic
315	C₂	C₆₇×#3165	1	2	yes	—
316	C₂	C₆₇×#3182	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×#500×C₆₇	C₂	semidirect
4. C₆₇×#1990	C₂	semidirect
5. C₆₇×#1944	C₂	semidirect
6. C₆₇×#1844	C₂	semidirect
32. C₂×M₁₆×C₆₇	Q₈	semidirect
132. C₈×C₆₇	C₂²⋊Q₈	semidirect
152. #33212	C₆₇	direct (N × H)
159. C₂×#500	C₂×C₆₇	semidirect
162. #1990	C₂×C₆₇	semidirect
163. #1944	C₂×C₆₇	semidirect
164. #1844	C₂×C₆₇	semidirect
166. C₆₇	#33212	direct (N × H)
191. C₂×M₁₆	Q₈×C₆₇	semidirect
291. C₈	C₂²⋊Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2857	2
C₆₇×#1990	2
C₆₇×#1944	2
C₆₇×#1844	2
C₆₇×#1809	2
#33212	67

Subgroups — C67×#33212 #30712463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33212 #30712463