Subgroups — C₆₇×#33692 #30712943

Order 17152.

536 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3176	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2955	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2238	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2231	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2213	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2190	C₂	1	2	yes	maximal, complemented
7	C₆₇×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2166	C₂	1	1	yes	maximal, characteristic, complemented
9	#575×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₆₇	—	2	1	no	—
11	#540×C₆₇	—	2	1	no	—
12	#509×C₆₇	C₂²	1	4	yes	—
13	#499×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	#490×C₆₇	—	2	1	no	—
16	#469×C₆₇	C₂²	1	2	yes	—
17	#466×C₆₇	—	2	1	no	—
18	#444×C₆₇	—	2	1	no	—
19	#444×C₆₇	C₂²	1	2	yes	—
20	#443×C₆₇	—	2	1	no	—
21	#443×C₆₇	C₂²	1	2	yes	—
22	#442×C₆₇	C₂²	1	2	yes	complemented
23	C₄×D₈×C₆₇	—	2	1	no	—
24	#430×C₆₇	C₂²	1	1	yes	characteristic
25	#421×C₆₇	—	2	1	no	—
26	#405×C₆₇	C₂²	1	1	yes	characteristic
27	#365×C₆₇	C₂²	1	4	yes	complemented
28	#364×C₆₇	C₂²	1	2	yes	complemented
29	#361×C₆₇	C₂²	1	4	yes	complemented
30	#360×C₆₇	C₂²	1	4	yes	—
31	#349×C₆₇	C₂²	1	2	yes	—
32	#347×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#41×C₆₇	—	2	1	no	—
34	#137×C₆₇	—	2	2	no	—
35	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
36	#135×C₆₇	—	2	1	no	—
37	#135×C₆₇	C₂³	1	2	yes	—
38	C₂×D₈×C₆₇	—	2	1	no	—
39	C₂×D₈×C₆₇	C₂³	1	2	yes	—
40	#131×C₆₇	—	2	2	no	—
41	#131×C₆₇	—	2	2	no	—
42	#131×C₆₇	—	4	1	no	—
43	#131×C₆₇	—	4	1	no	—
44	C₂×M₁₆×C₆₇	—	2	1	no	—
45	C₂×M₁₆×C₆₇	—	4	1	no	—
46	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	—	4	1	no	—
48	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
49	#127×C₆₇	—	2	1	no	—
50	#125×C₆₇	—	2	1	no	—
51	#123×C₆₇	—	4	2	no	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	—	4	2	no	—
54	C₄×D₄×C₆₇	—	2	2	no	—
55	#117×C₆₇	D₄	1	1	yes	characteristic
56	SD₃₂×C₆₇	—	2	4	no	—
57	D₁₆×C₆₇	—	2	4	no	—
58	M₃₂×C₆₇	C₂³	1	4	yes	—
59	#108×C₆₇	—	2	1	no	—
60	#108×C₆₇	C₂³	1	2	yes	—
61	#107×C₆₇	—	2	1	no	—
62	#106×C₆₇	C₂³	1	2	yes	—
63	#104×C₆₇	—	2	2	no	—
64	#104×C₆₇	—	2	2	no	—
65	#102×C₆₇	—	2	2	no	—
66	#102×C₆₇	—	2	2	no	—
67	#97×C₆₇	D₄	1	1	yes	characteristic
68	#97×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	#41×C₆₇	—	2	2	no	—
72	#41×C₆₇	—	2	2	no	—
73	#41×C₆₇	—	4	1	no	—
74	#41×C₆₇	—	4	1	no	—
75	C₂×Q₈×C₆₇	—	2	1	no	—
76	C₂×D₄×C₆₇	—	2	1	no	—
77	C₂×D₄×C₆₇	—	2	2	no	—
78	C₂×D₄×C₆₇	—	4	2	no	—
79	C₂×D₄×C₆₇	—	4	1	no	—
80	C₂×D₄×C₆₇	—	4	2	no	—
81	C₂²×C₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	—	4	2	no	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₆₇	—	2	1	no	—
85	Q₁₆×C₆₇	#41	1	2	yes	—
86	SD₁₆×C₆₇	—	2	2	no	—
87	SD₁₆×C₆₇	—	2	2	no	—
88	D₈×C₆₇	—	2	1	no	—
89	D₈×C₆₇	—	2	4	no	—
90	D₈×C₆₇	—	4	1	no	—
91	D₈×C₆₇	—	4	1	no	—
92	D₈×C₆₇	#41	1	2	yes	—
93	M₁₆×C₆₇	—	2	2	no	—
94	M₁₆×C₆₇	—	2	2	no	—
95	M₁₆×C₆₇	—	4	2	no	—
96	M₁₆×C₆₇	—	4	1	no	—
97	M₁₆×C₆₇	—	4	2	no	—
98	C₂×C₈×C₆₇	—	2	1	no	—
99	C₂×C₈×C₆₇	—	2	1	no	—
100	C₂×C₈×C₆₇	—	4	2	no	—
101	C₂×C₈×C₆₇	—	4	1	no	—
102	C₂×C₈×C₆₇	—	4	2	no	—
103	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₆₇	—	2	1	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	4	2	no	—
110	C₂²⋊C₄×C₆₇	—	2	1	no	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₂²⋊C₄×C₆₇	—	4	2	no	—
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₆₇	—	2	4	no	—
116	C₂³×C₆₇	—	4	1	no	—
117	C₂³×C₆₇	—	4	2	no	—
118	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	Q₈×C₆₇	—	2	2	no	—
120	Q₈×C₆₇	—	4	1	no	—
121	D₄×C₆₇	—	2	2	no	—
122	D₄×C₆₇	—	4	1	no	—
123	D₄×C₆₇	—	4	2	no	—
124	D₄×C₆₇	—	4	2	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	4	2	no	—
127	D₄×C₆₇	—	16	2	no	—
128	D₄×C₆₇	—	16	2	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	16	2	no	—
138	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	4	2	no	—
143	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₈×C₆₇	C₂×#41	1	2	yes	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	4	1	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	—	8	2	no	—
152	C₂²×C₆₇	—	8	2	no	—
153	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	4	1	no	—
156	C₄×C₆₇	—	4	1	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	—	8	2	no	—
159	C₄×C₆₇	#525	1	1	yes	characteristic
160	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
161	#33692	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	—	4	1	no	—
164	C₂×C₆₇	—	8	2	no	—
165	C₂×C₆₇	—	16	2	no	—
166	C₂×C₆₇	#3109	1	1	yes	characteristic
167	#3176	C₂×C₆₇	1	1	yes	characteristic, complemented
168	#2955	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#2238	C₂×C₆₇	1	2	yes	complemented
170	#2231	C₂×C₆₇	1	2	yes	complemented
171	#2213	C₂×C₆₇	1	2	yes	complemented
172	#2190	C₂×C₆₇	1	2	yes	complemented
173	AGL(1,ℤ/16)	C₂×C₆₇	1	4	yes	complemented
174	#2166	C₂×C₆₇	1	1	yes	characteristic, complemented
175	C₆₇	#33692	1	1	yes	characteristic, direct product factor, Sylow
176	#575	C₂²×C₆₇	1	1	yes	characteristic
177	C₂×#131	—	2	1	no	—
178	#540	—	2	1	no	—
179	#509	C₂²×C₆₇	1	4	yes	—
180	#499	C₂²×C₆₇	1	1	yes	characteristic, complemented
181	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
182	#490	—	2	1	no	—
183	#469	C₂²×C₆₇	1	2	yes	—
184	#466	—	2	1	no	—
185	#444	—	2	1	no	—
186	#444	C₂²×C₆₇	1	2	yes	—
187	#443	—	2	1	no	—
188	#443	C₂²×C₆₇	1	2	yes	—
189	#442	C₂²×C₆₇	1	2	yes	complemented
190	C₄×D₈	—	2	1	no	—
191	#430	C₂²×C₆₇	1	1	yes	characteristic
192	#421	—	2	1	no	—
193	#405	C₂²×C₆₇	1	1	yes	characteristic
194	#365	C₂²×C₆₇	1	4	yes	complemented
195	#364	C₂²×C₆₇	1	2	yes	complemented
196	#361	C₂²×C₆₇	1	4	yes	complemented
197	#360	C₂²×C₆₇	1	4	yes	—
198	#349	C₂²×C₆₇	1	2	yes	—
199	#347	C₂²×C₆₇	1	2	yes	complemented
200	C₂×#41	—	2	1	no	—
201	#137	—	2	2	no	—
202	AGL(1,ℤ/8)	—	2	2	no	—
203	#135	—	2	1	no	—
204	#135	C₂³×C₆₇	1	2	yes	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	C₂³×C₆₇	1	2	yes	—
207	#131	—	2	2	no	—
208	#131	—	2	2	no	—
209	#131	—	4	1	no	—
210	#131	—	4	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	4	1	no	—
213	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
214	C₂²×C₈	—	4	1	no	—
215	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
216	#127	—	2	1	no	—
217	#125	—	2	1	no	—
218	#123	—	4	2	no	—
219	#121	—	2	2	no	—
220	#121	—	4	2	no	—
221	C₄×D₄	—	2	2	no	—
222	#117	D₄×C₆₇	1	1	yes	characteristic
223	SD₃₂	—	2	4	no	—
224	D₁₆	—	2	4	no	—
225	M₃₂	C₂³×C₆₇	1	4	yes	—
226	#108	—	2	1	no	—
227	#108	C₂³×C₆₇	1	2	yes	—
228	#107	—	2	1	no	—
229	#106	C₂³×C₆₇	1	2	yes	—
230	#104	—	2	2	no	—
231	#104	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#97	D₄×C₆₇	1	1	yes	characteristic
235	#97	C₂³×C₆₇	1	1	yes	characteristic
236	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
238	#41	—	2	2	no	—
239	#41	—	2	2	no	—
240	#41	—	4	1	no	—
241	#41	—	4	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
251	Q₁₆	—	2	1	no	—
252	Q₁₆	#41×C₆₇	1	2	yes	—
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	2	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	4	no	—
257	D₈	—	4	1	no	—
258	D₈	—	4	1	no	—
259	D₈	#41×C₆₇	1	2	yes	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	2	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
273	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₁₆	—	2	4	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	16	2	no	—
295	D₄	—	16	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	4	2	no	—
310	C₈	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
311	C₈	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
312	C₈	C₂×#41×C₆₇	1	2	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#525×C₆₇	1	1	yes	characteristic
327	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	4	1	no	—
330	C₂	—	8	2	no	—
331	C₂	—	16	2	no	—
332	C₂	C₆₇×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×C₄×C₆₇ ◁ #405×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3176	C₂	semidirect
2. C₆₇×#2955	C₂	semidirect
3. C₆₇×#2238	C₂	semidirect
4. C₆₇×#2231	C₂	semidirect
5. C₆₇×#2213	C₂	semidirect
6. C₆₇×#2190	C₂	semidirect
7. C₆₇×AGL(1,ℤ/16)	C₂	semidirect
8. C₆₇×#2166	C₂	semidirect
13. #499×C₆₇	C₂²	semidirect
14. #493×C₆₇	C₂²	semidirect
22. #442×C₆₇	C₂²	semidirect
27. #365×C₆₇	C₂²	semidirect
28. #364×C₆₇	C₂²	semidirect
29. #361×C₆₇	C₂²	semidirect
32. #347×C₆₇	C₂²	semidirect
161. #33692	C₆₇	direct (N × H)
167. #3176	C₂×C₆₇	semidirect
168. #2955	C₂×C₆₇	semidirect
169. #2238	C₂×C₆₇	semidirect
170. #2231	C₂×C₆₇	semidirect
171. #2213	C₂×C₆₇	semidirect
172. #2190	C₂×C₆₇	semidirect
173. AGL(1,ℤ/16)	C₂×C₆₇	semidirect
174. #2166	C₂×C₆₇	semidirect
175. C₆₇	#33692	direct (N × H)
180. #499	C₂²×C₆₇	semidirect
181. #493	C₂²×C₆₇	semidirect
189. #442	C₂²×C₆₇	semidirect
194. #365	C₂²×C₆₇	semidirect
195. #364	C₂²×C₆₇	semidirect
196. #361	C₂²×C₆₇	semidirect
199. #347	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3176	2
C₆₇×#2955	2
C₆₇×#2238	2
C₆₇×#2231	2
C₆₇×#2213	2
C₆₇×#2190	2
C₆₇×AGL(1,ℤ/16)	2
C₆₇×#2166	2
#33692	67

Subgroups — C67×#33692 #30712943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33692 #30712943