Subgroups — C₆₇×#33586 #30712837

Order 17152.

716 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×#509×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×SD₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3156	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2948	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2210	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2208	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₆₇	—	2	1	no	—
11	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	#536×C₆₇	—	2	1	no	—
15	#518×C₆₇	—	2	1	no	—
16	#509×C₆₇	—	2	4	no	—
17	C₂×SD₃₂×C₆₇	—	2	4	no	—
18	C₂×SD₃₂×C₆₇	C₂²	1	4	yes	—
19	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
20	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	#497×C₆₇	—	2	2	no	—
24	#495×C₆₇	C₂²	1	2	yes	complemented
25	#494×C₆₇	C₂²	1	2	yes	complemented
26	#442×C₆₇	—	2	2	no	—
27	C₄×D₈×C₆₇	C₂²	1	4	yes	—
28	#427×C₆₇	C₂²	1	1	yes	characteristic
29	#418×C₆₇	—	2	1	no	—
30	#405×C₆₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	complemented
32	#358×C₆₇	C₂²	1	2	yes	complemented
33	#358×C₆₇	C₂²	1	4	yes	complemented
34	#357×C₆₇	C₂²	1	2	yes	complemented
35	C₂×#41×C₆₇	—	2	1	no	—
36	C₂²×Q₈×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	#137×C₆₇	—	4	4	no	—
39	#135×C₆₇	—	2	2	no	—
40	#135×C₆₇	—	2	2	no	—
41	C₂×Q₁₆×C₆₇	—	2	2	no	—
42	C₂×Q₁₆×C₆₇	—	2	2	no	—
43	C₂×Q₁₆×C₆₇	—	2	1	no	—
44	C₂×Q₁₆×C₆₇	—	2	2	no	—
45	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂×SD₁₆×C₆₇	—	2	2	no	—
48	C₂×SD₁₆×C₆₇	—	2	1	no	—
49	C₂×D₈×C₆₇	—	2	1	no	—
50	C₂×D₈×C₆₇	—	2	2	no	—
51	C₂×D₈×C₆₇	D₄	1	4	yes	—
52	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×D₈×C₆₇	C₂³	1	2	yes	—
54	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₇	—	2	2	no	—
57	#124×C₆₇	—	2	2	no	—
58	#123×C₆₇	—	4	2	no	—
59	C₂²⋊Q₈×C₆₇	—	4	2	no	—
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	4	2	no	—
62	#117×C₆₇	—	2	1	no	—
63	#117×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
65	SD₃₂×C₆₇	—	2	8	no	—
66	SD₃₂×C₆₇	—	2	4	no	—
67	D₁₆×C₆₇	—	2	4	no	—
68	M₃₂×C₆₇	—	2	2	no	—
69	C₂×C₁₆×C₆₇	—	2	2	no	—
70	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
71	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
72	#107×C₆₇	C₂³	1	2	yes	—
73	#106×C₆₇	—	2	1	no	—
74	#102×C₆₇	—	2	4	no	—
75	#97×C₆₇	D₄	1	2	yes	—
76	C₄×C₈×C₆₇	C₂³	1	2	yes	—
77	C₂⁴×C₆₇	—	4	1	no	—
78	#41×C₆₇	—	4	2	no	—
79	#41×C₆₇	—	4	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	1	no	—
82	C₂×Q₈×C₆₇	—	4	1	no	—
83	C₂×Q₈×C₆₇	—	4	2	no	—
84	C₂×Q₈×C₆₇	—	4	2	no	—
85	C₂×D₄×C₆₇	—	2	4	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	1	no	—
88	C₂×D₄×C₆₇	—	4	1	no	—
89	C₂×D₄×C₆₇	—	4	1	no	—
90	C₂×D₄×C₆₇	—	4	2	no	—
91	C₂²×C₄×C₆₇	—	4	1	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	4	no	—
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	Q₁₆×C₆₇	—	2	2	no	—
96	Q₁₆×C₆₇	—	2	1	no	—
97	Q₁₆×C₆₇	—	2	2	no	—
98	Q₁₆×C₆₇	—	2	1	no	—
99	Q₁₆×C₆₇	—	4	4	no	—
100	Q₁₆×C₆₇	—	4	2	no	—
101	SD₁₆×C₆₇	—	4	4	no	—
102	SD₁₆×C₆₇	—	4	2	no	—
103	D₈×C₆₇	—	2	4	no	—
104	D₈×C₆₇	—	2	1	no	—
105	D₈×C₆₇	—	2	2	no	—
106	D₈×C₆₇	—	2	1	no	—
107	D₈×C₆₇	C₂×D₄	1	4	yes	—
108	M₁₆×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₆₇	#41	1	2	yes	—
113	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	4	4	no	—
116	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
117	C₂²⋊C₄×C₆₇	—	4	1	no	—
118	C₂²⋊C₄×C₆₇	—	4	4	no	—
119	C₂²⋊C₄×C₆₇	—	4	4	no	—
120	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
121	C₄²×C₆₇	—	4	1	no	—
122	C₄²×C₆₇	C₂×D₄	1	2	yes	—
123	C₁₆×C₆₇	—	2	2	no	—
124	C₁₆×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	4	2	no	—
126	C₂³×C₆₇	—	4	4	no	—
127	C₂³×C₆₇	—	4	1	no	—
128	C₂³×C₆₇	—	16	2	no	—
129	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
130	Q₈×C₆₇	—	4	1	no	—
131	Q₈×C₆₇	—	4	2	no	—
132	Q₈×C₆₇	—	4	2	no	—
133	Q₈×C₆₇	—	4	1	no	—
134	Q₈×C₆₇	—	8	2	no	—
135	D₄×C₆₇	—	2	4	no	—
136	D₄×C₆₇	—	4	1	no	—
137	D₄×C₆₇	—	4	2	no	—
138	D₄×C₆₇	—	4	1	no	—
139	D₄×C₆₇	—	4	4	no	—
140	D₄×C₆₇	—	8	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	1	no	—
144	C₂×C₄×C₆₇	—	8	2	no	—
145	C₂×C₄×C₆₇	—	8	2	no	—
146	C₂×C₄×C₆₇	—	16	4	no	—
147	C₂×C₄×C₆₇	#121	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
149	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₆₇	—	2	2	no	—
153	C₈×C₆₇	#121	1	2	yes	—
154	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	4	4	no	—
158	C₂²×C₆₇	—	8	4	no	—
159	C₂²×C₆₇	—	8	2	no	—
160	C₂²×C₆₇	—	8	1	no	—
161	C₂²×C₆₇	—	16	4	no	—
162	C₂²×C₆₇	#459	1	2	yes	—
163	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₆₇	—	2	4	no	—
165	C₄×C₆₇	—	8	2	no	—
166	C₄×C₆₇	—	8	1	no	—
167	C₄×C₆₇	—	8	2	no	—
168	C₄×C₆₇	#459	1	2	yes	—
169	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
170	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	#33586	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₆₇	—	2	2	no	—
173	C₂×C₆₇	—	8	4	no	—
174	C₂×C₆₇	—	16	1	no	—
175	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
176	C₂×C₆₇	#3416	1	2	yes	—
177	C₂×#509	C₂×C₆₇	1	1	yes	characteristic
178	C₂²×SD₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
179	#3156	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#2948	C₂×C₆₇	1	1	yes	characteristic, complemented
181	#2210	C₂×C₆₇	1	4	yes	complemented
182	#2208	C₂×C₆₇	1	4	yes	complemented
183	#2151	C₂×C₆₇	1	1	yes	characteristic, complemented
184	#2141	C₂×C₆₇	1	1	yes	characteristic, complemented
185	#2138	C₂×C₆₇	1	1	yes	characteristic, complemented
186	C₆₇	#33586	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×#137	—	2	1	no	—
188	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
189	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
190	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
191	#536	—	2	1	no	—
192	#518	—	2	1	no	—
193	#509	—	2	4	no	—
194	C₂×SD₃₂	—	2	4	no	—
195	C₂×SD₃₂	C₂²×C₆₇	1	4	yes	—
196	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
197	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
198	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
199	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
200	#497	—	2	2	no	—
201	#495	C₂²×C₆₇	1	2	yes	complemented
202	#494	C₂²×C₆₇	1	2	yes	complemented
203	#442	—	2	2	no	—
204	C₄×D₈	C₂²×C₆₇	1	4	yes	—
205	#427	C₂²×C₆₇	1	1	yes	characteristic
206	#418	—	2	1	no	—
207	#405	C₂²×C₆₇	1	1	yes	characteristic
208	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	complemented
209	#358	C₂²×C₆₇	1	2	yes	complemented
210	#358	C₂²×C₆₇	1	4	yes	complemented
211	#357	C₂²×C₆₇	1	2	yes	complemented
212	C₂×#41	—	2	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	C₂²×D₄	—	2	1	no	—
215	#137	—	4	4	no	—
216	#135	—	2	2	no	—
217	#135	—	2	2	no	—
218	C₂×Q₁₆	—	2	2	no	—
219	C₂×Q₁₆	—	2	1	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×Q₁₆	—	2	2	no	—
222	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
223	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
224	C₂×SD₁₆	—	2	1	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	—	2	1	no	—
227	C₂×D₈	—	2	2	no	—
228	C₂×D₈	D₄×C₆₇	1	4	yes	—
229	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
230	C₂×D₈	C₂³×C₆₇	1	2	yes	—
231	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
233	C₄⋊Q₈	—	2	2	no	—
234	#124	—	2	2	no	—
235	#123	—	4	2	no	—
236	C₂²⋊Q₈	—	4	2	no	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	—	4	2	no	—
239	#117	—	2	1	no	—
240	#117	D₄×C₆₇	1	1	yes	characteristic
241	C₂×C₂²⋊C₄	—	4	2	no	—
242	SD₃₂	—	2	8	no	—
243	SD₃₂	—	2	4	no	—
244	D₁₆	—	2	4	no	—
245	M₃₂	—	2	2	no	—
246	C₂×C₁₆	—	2	2	no	—
247	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
248	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
249	#107	C₂³×C₆₇	1	2	yes	—
250	#106	—	2	1	no	—
251	#102	—	2	4	no	—
252	#97	D₄×C₆₇	1	2	yes	—
253	C₄×C₈	C₂³×C₆₇	1	2	yes	—
254	C₂⁴	—	4	1	no	—
255	#41	—	4	2	no	—
256	#41	—	4	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×D₄	—	2	4	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	4	no	—
271	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
272	Q₁₆	—	2	2	no	—
273	Q₁₆	—	2	1	no	—
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	4	4	no	—
277	Q₁₆	—	4	2	no	—
278	SD₁₆	—	4	4	no	—
279	SD₁₆	—	4	2	no	—
280	D₈	—	2	4	no	—
281	D₈	—	2	1	no	—
282	D₈	—	2	2	no	—
283	D₈	—	2	1	no	—
284	D₈	C₂×D₄×C₆₇	1	4	yes	—
285	M₁₆	—	2	2	no	—
286	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
287	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
289	C₂×C₈	#41×C₆₇	1	2	yes	—
290	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	4	no	—
293	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	4	no	—
296	C₂²⋊C₄	—	4	4	no	—
297	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
298	C₄²	—	4	1	no	—
299	C₄²	C₂×D₄×C₆₇	1	2	yes	—
300	C₁₆	—	2	2	no	—
301	C₁₆	—	2	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	4	no	—
304	C₂³	—	4	1	no	—
305	C₂³	—	16	2	no	—
306	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
307	Q₈	—	4	1	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	1	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	1	no	—
317	D₄	—	8	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	16	4	no	—
324	C₂×C₄	#121×C₆₇	1	2	yes	—
325	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
326	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
327	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
328	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	#121×C₆₇	1	2	yes	—
331	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
332	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	16	4	no	—
339	C₂²	#459×C₆₇	1	2	yes	—
340	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	#459×C₆₇	1	2	yes	—
346	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
347	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
348	C₂	—	2	2	no	—
349	C₂	—	8	4	no	—
350	C₂	—	16	1	no	—
351	C₂	C₂×#459×C₆₇	1	1	yes	characteristic
352	C₂	C₆₇×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#509×C₆₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #117×C₆₇ ◁ #405×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×SD₃₂×C₆₇	C₂	semidirect
3. C₆₇×#3156	C₂	semidirect
4. C₆₇×#2948	C₂	semidirect
5. C₆₇×#2210	C₂	semidirect
6. C₆₇×#2208	C₂	semidirect
7. C₆₇×#2151	C₂	semidirect
8. C₆₇×#2141	C₂	semidirect
9. C₆₇×#2138	C₂	semidirect
24. #495×C₆₇	C₂²	semidirect
25. #494×C₆₇	C₂²	semidirect
31. C₄⋊C₁₆×C₆₇	C₂²	semidirect
32. #358×C₆₇	C₂²	semidirect
33. #358×C₆₇	C₂²	semidirect
34. #357×C₆₇	C₂²	semidirect
171. #33586	C₆₇	direct (N × H)
178. C₂²×SD₃₂	C₂×C₆₇	semidirect
179. #3156	C₂×C₆₇	semidirect
180. #2948	C₂×C₆₇	semidirect
181. #2210	C₂×C₆₇	semidirect
182. #2208	C₂×C₆₇	semidirect
183. #2151	C₂×C₆₇	semidirect
184. #2141	C₂×C₆₇	semidirect
185. #2138	C₂×C₆₇	semidirect
186. C₆₇	#33586	direct (N × H)
201. #495	C₂²×C₆₇	semidirect
202. #494	C₂²×C₆₇	semidirect
208. C₄⋊C₁₆	C₂²×C₆₇	semidirect
209. #358	C₂²×C₆₇	semidirect
210. #358	C₂²×C₆₇	semidirect
211. #357	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#509×C₆₇	2
C₂²×SD₃₂×C₆₇	2
C₆₇×#3156	2
C₆₇×#2948	2
C₆₇×#2210	2
C₆₇×#2208	2
C₆₇×#2151	2
C₆₇×#2141	2
C₆₇×#2138	2
#33586	67

Subgroups — C67×#33586 #30712837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33586 #30712837