Subgroups — C₆₇×#33588 #30712839

Order 17152.

680 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#510×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3156	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2211	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2209	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
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×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	#536×C₆₇	—	2	1	no	—
15	#519×C₆₇	—	2	1	no	—
16	#510×C₆₇	—	2	4	no	—
17	C₂×Q₃₂×C₆₇	—	2	4	no	—
18	C₂×Q₃₂×C₆₇	C₂²	1	4	yes	—
19	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
20	C₂×SD₃₂×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	—
25	#494×C₆₇	C₂²	1	2	yes	—
26	#441×C₆₇	—	2	2	no	—
27	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
28	#427×C₆₇	C₂²	1	1	yes	characteristic
29	#419×C₆₇	—	2	1	no	—
30	#405×C₆₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
32	#358×C₆₇	C₂²	1	2	yes	—
33	#358×C₆₇	C₂²	1	4	yes	—
34	#357×C₆₇	C₂²	1	2	yes	—
35	C₂×#41×C₆₇	—	2	1	no	—
36	C₂²×Q₈×C₆₇	—	2	1	no	—
37	C₂²×Q₈×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	1	no	—
42	C₂×Q₁₆×C₆₇	—	2	2	no	—
43	C₂×Q₁₆×C₆₇	—	2	2	no	—
44	C₂×Q₁₆×C₆₇	—	2	1	no	—
45	C₂×Q₁₆×C₆₇	—	2	2	no	—
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	D₄	1	4	yes	—
48	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂×SD₁₆×C₆₇	—	2	2	no	—
52	C₂×SD₁₆×C₆₇	—	2	1	no	—
53	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
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×Q₈×C₆₇	—	2	4	no	—
61	C₄×Q₈×C₆₇	—	4	2	no	—
62	#117×C₆₇	—	2	1	no	—
63	#117×C₆₇	—	4	2	no	—
64	#117×C₆₇	D₄	1	1	yes	characteristic
65	Q₃₂×C₆₇	—	2	8	no	—
66	Q₃₂×C₆₇	—	2	4	no	—
67	SD₃₂×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	#103×C₆₇	—	2	4	no	—
75	#97×C₆₇	D₄	1	2	yes	—
76	C₄×C₈×C₆₇	C₂³	1	2	yes	—
77	#41×C₆₇	—	4	2	no	—
78	#41×C₆₇	—	4	2	no	—
79	C₂×Q₈×C₆₇	—	2	4	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	—	2	1	no	—
83	C₂×Q₈×C₆₇	—	4	1	no	—
84	C₂×Q₈×C₆₇	—	4	1	no	—
85	C₂×Q₈×C₆₇	—	4	2	no	—
86	C₂×Q₈×C₆₇	—	4	2	no	—
87	C₂×Q₈×C₆₇	—	4	2	no	—
88	C₂×D₄×C₆₇	—	2	1	no	—
89	C₂×D₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	—	4	1	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₆₇	—	2	4	no	—
95	Q₁₆×C₆₇	—	2	2	no	—
96	Q₁₆×C₆₇	—	2	1	no	—
97	Q₁₆×C₆₇	—	2	2	no	—
98	Q₁₆×C₆₇	—	2	2	no	—
99	Q₁₆×C₆₇	—	2	1	no	—
100	Q₁₆×C₆₇	—	4	4	no	—
101	Q₁₆×C₆₇	—	4	2	no	—
102	Q₁₆×C₆₇	C₂×D₄	1	4	yes	—
103	SD₁₆×C₆₇	—	4	4	no	—
104	SD₁₆×C₆₇	—	4	2	no	—
105	D₈×C₆₇	—	2	1	no	—
106	D₈×C₆₇	—	2	1	no	—
107	M₁₆×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₆₇	#41	1	2	yes	—
112	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	4	4	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	D₈	1	2	yes	—
120	C₄²×C₆₇	—	4	1	no	—
121	C₄²×C₆₇	—	4	4	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	1	no	—
126	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₆₇	—	2	4	no	—
128	Q₈×C₆₇	—	4	2	no	—
129	Q₈×C₆₇	—	4	4	no	—
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	—	4	1	no	—
136	D₄×C₆₇	—	4	1	no	—
137	D₄×C₆₇	—	8	2	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	1	no	—
143	C₂×C₄×C₆₇	—	8	2	no	—
144	C₂×C₄×C₆₇	—	8	2	no	—
145	C₂×C₄×C₆₇	#121	1	2	yes	—
146	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
147	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₆₇	—	2	2	no	—
151	C₈×C₆₇	#121	1	2	yes	—
152	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	—	8	2	no	—
156	C₂²×C₆₇	—	8	1	no	—
157	C₂²×C₆₇	#459	1	2	yes	—
158	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₆₇	—	2	4	no	—
160	C₄×C₆₇	—	4	4	no	—
161	C₄×C₆₇	—	8	2	no	—
162	C₄×C₆₇	—	8	1	no	—
163	C₄×C₆₇	—	8	2	no	—
164	C₄×C₆₇	#459	1	2	yes	—
165	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
166	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	#33588	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₆₇	—	2	2	no	—
169	C₂×C₆₇	—	16	1	no	—
170	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
171	C₂×C₆₇	#3417	1	2	yes	—
172	C₂×#510	C₂×C₆₇	1	1	yes	characteristic
173	C₂²×Q₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
174	#3156	C₂×C₆₇	1	1	yes	characteristic
175	#2950	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#2211	C₂×C₆₇	1	4	yes	complemented
177	#2209	C₂×C₆₇	1	4	yes	complemented
178	#2151	C₂×C₆₇	1	1	yes	characteristic, complemented
179	#2141	C₂×C₆₇	1	1	yes	characteristic, complemented
180	#2138	C₂×C₆₇	1	1	yes	characteristic
181	C₆₇	#33588	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×#137	—	2	1	no	—
183	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
184	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
185	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
186	#536	—	2	1	no	—
187	#519	—	2	1	no	—
188	#510	—	2	4	no	—
189	C₂×Q₃₂	—	2	4	no	—
190	C₂×Q₃₂	C₂²×C₆₇	1	4	yes	—
191	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
192	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
193	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
194	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
195	#497	—	2	2	no	—
196	#495	C₂²×C₆₇	1	2	yes	—
197	#494	C₂²×C₆₇	1	2	yes	—
198	#441	—	2	2	no	—
199	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	—
200	#427	C₂²×C₆₇	1	1	yes	characteristic
201	#419	—	2	1	no	—
202	#405	C₂²×C₆₇	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	—
204	#358	C₂²×C₆₇	1	2	yes	—
205	#358	C₂²×C₆₇	1	4	yes	—
206	#357	C₂²×C₆₇	1	2	yes	—
207	C₂×#41	—	2	1	no	—
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×Q₈	—	2	1	no	—
210	#137	—	4	4	no	—
211	#135	—	2	2	no	—
212	#135	—	2	2	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×Q₁₆	—	2	2	no	—
218	C₂×Q₁₆	—	2	2	no	—
219	C₂×Q₁₆	D₄×C₆₇	1	4	yes	—
220	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
221	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
222	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
223	C₂×SD₁₆	—	2	1	no	—
224	C₂×SD₁₆	—	2	2	no	—
225	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
226	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
228	C₄⋊Q₈	—	2	2	no	—
229	#124	—	2	2	no	—
230	#123	—	4	2	no	—
231	C₂²⋊Q₈	—	4	2	no	—
232	C₄×Q₈	—	2	4	no	—
233	C₄×Q₈	—	4	2	no	—
234	#117	—	2	1	no	—
235	#117	—	4	2	no	—
236	#117	D₄×C₆₇	1	1	yes	characteristic
237	Q₃₂	—	2	8	no	—
238	Q₃₂	—	2	4	no	—
239	SD₃₂	—	2	4	no	—
240	M₃₂	—	2	2	no	—
241	C₂×C₁₆	—	2	2	no	—
242	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
243	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
244	#107	C₂³×C₆₇	1	2	yes	—
245	#106	—	2	1	no	—
246	#103	—	2	4	no	—
247	#97	D₄×C₆₇	1	2	yes	—
248	C₄×C₈	C₂³×C₆₇	1	2	yes	—
249	#41	—	4	2	no	—
250	#41	—	4	2	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×Q₈	—	4	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
266	Q₁₆	—	2	4	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	4	4	no	—
273	Q₁₆	—	4	2	no	—
274	Q₁₆	C₂×D₄×C₆₇	1	4	yes	—
275	SD₁₆	—	4	4	no	—
276	SD₁₆	—	4	2	no	—
277	D₈	—	2	1	no	—
278	D₈	—	2	1	no	—
279	M₁₆	—	2	2	no	—
280	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
283	C₂×C₈	#41×C₆₇	1	2	yes	—
284	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₄⋊C₄	—	4	4	no	—
288	C₄⋊C₄	D₈×C₆₇	1	2	yes	—
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	4	no	—
294	C₄²	C₂×D₄×C₆₇	1	2	yes	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	2	2	no	—
297	C₂³	—	4	1	no	—
298	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
299	Q₈	—	2	4	no	—
300	Q₈	—	4	4	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	1	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	8	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	#121×C₆₇	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
319	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
320	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
321	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	#121×C₆₇	1	2	yes	—
324	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
325	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	#459×C₆₇	1	2	yes	—
330	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
331	C₄	—	2	4	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	#459×C₆₇	1	2	yes	—
337	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
338	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	16	1	no	—
341	C₂	C₂×#459×C₆₇	1	1	yes	characteristic
342	C₂	C₆₇×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#135×C₆₇	C₂²
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×Q₃₂×C₆₇	C₂	semidirect
4. C₆₇×#2950	C₂	semidirect
5. C₆₇×#2211	C₂	semidirect
6. C₆₇×#2209	C₂	semidirect
7. C₆₇×#2151	C₂	semidirect
8. C₆₇×#2141	C₂	semidirect
167. #33588	C₆₇	direct (N × H)
173. C₂²×Q₃₂	C₂×C₆₇	semidirect
175. #2950	C₂×C₆₇	semidirect
176. #2211	C₂×C₆₇	semidirect
177. #2209	C₂×C₆₇	semidirect
178. #2151	C₂×C₆₇	semidirect
179. #2141	C₂×C₆₇	semidirect
181. C₆₇	#33588	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×#510×C₆₇	2
C₂²×Q₃₂×C₆₇	2
C₆₇×#3156	2
C₆₇×#2950	2
C₆₇×#2211	2
C₆₇×#2209	2
C₆₇×#2151	2
C₆₇×#2141	2
C₆₇×#2138	2
#33588	67

Subgroups — C67×#33588 #30712839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33588 #30712839