Subgroups — C₆₇×#33638 #30712889

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2957	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2231	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2230	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#365×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2145	C₂	1	1	yes	maximal, characteristic
9	C₂×#360×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
11	#568×C₆₇	—	2	1	no	—
12	#544×C₆₇	—	2	1	no	—
13	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
15	#501×C₆₇	—	2	2	no	—
16	#499×C₆₇	C₂²	1	2	yes	complemented
17	#498×C₆₇	C₂²	1	2	yes	complemented
18	#444×C₆₇	—	2	2	no	—
19	#443×C₆₇	C₂²	1	4	yes	—
20	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
21	#428×C₆₇	—	2	1	no	—
22	#427×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
24	#421×C₆₇	—	2	1	no	—
25	#405×C₆₇	C₂²	1	1	yes	characteristic
26	#365×C₆₇	C₂²	1	4	yes	complemented
27	#365×C₆₇	C₂²	1	2	yes	complemented
28	#365×C₆₇	C₂²	1	2	yes	complemented
29	#364×C₆₇	C₂²	1	4	yes	complemented
30	#360×C₆₇	C₂²	1	4	yes	—
31	#360×C₆₇	C₂²	1	2	yes	—
32	#360×C₆₇	C₂²	1	2	yes	—
33	C₂×#41×C₆₇	—	2	1	no	—
34	#135×C₆₇	—	2	1	no	—
35	#135×C₆₇	—	2	1	no	—
36	#135×C₆₇	Q₈	1	2	yes	complemented
37	#135×C₆₇	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₆₇	Q₈	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₆₇	—	2	1	no	—
40	C₂×D₈×C₆₇	Q₈	1	1	yes	characteristic, complemented
41	#131×C₆₇	—	2	4	no	—
42	#131×C₆₇	—	4	2	no	—
43	C₂×M₁₆×C₆₇	—	2	1	no	—
44	C₂×M₁₆×C₆₇	—	4	2	no	—
45	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	—	4	1	no	—
49	#125×C₆₇	—	2	2	no	—
50	#125×C₆₇	—	4	1	no	—
51	C₂²⋊Q₈×C₆₇	—	4	2	no	—
52	#117×C₆₇	—	2	1	no	—
53	#117×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	M₃₂×C₆₇	—	2	1	no	—
56	M₃₂×C₆₇	—	2	1	no	—
57	M₃₂×C₆₇	C₂³	1	2	yes	—
58	M₃₂×C₆₇	C₂³	1	2	yes	—
59	C₂×C₁₆×C₆₇	—	2	1	no	—
60	C₂×C₁₆×C₆₇	—	2	1	no	—
61	#108×C₆₇	—	2	1	no	—
62	#108×C₆₇	C₂³	1	2	yes	—
63	#107×C₆₇	—	2	2	no	—
64	#107×C₆₇	—	2	1	no	—
65	#106×C₆₇	—	2	2	no	—
66	#106×C₆₇	—	2	1	no	—
67	#106×C₆₇	C₂³	1	1	yes	characteristic
68	#106×C₆₇	C₂³	1	2	yes	—
69	#106×C₆₇	C₂³	1	1	yes	characteristic
70	#104×C₆₇	—	2	4	no	—
71	#97×C₆₇	D₄	1	2	yes	—
72	C₄×C₈×C₆₇	C₂³	1	2	yes	—
73	#41×C₆₇	—	2	2	no	—
74	#41×C₆₇	—	2	2	no	—
75	#41×C₆₇	—	4	1	no	—
76	#41×C₆₇	—	4	1	no	—
77	C₂×Q₈×C₆₇	—	2	1	no	—
78	C₂×Q₈×C₆₇	—	4	2	no	—
79	C₂×D₄×C₆₇	—	2	1	no	—
80	C₂×D₄×C₆₇	—	4	1	no	—
81	C₂²×C₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₆₇	—	2	1	no	—
85	Q₁₆×C₆₇	C₂×Q₈	1	2	yes	—
86	SD₁₆×C₆₇	—	2	2	no	—
87	SD₁₆×C₆₇	—	2	2	no	—
88	D₈×C₆₇	—	2	1	no	—
89	D₈×C₆₇	C₂×Q₈	1	2	yes	—
90	M₁₆×C₆₇	—	2	2	no	—
91	M₁₆×C₆₇	—	4	1	no	—
92	M₁₆×C₆₇	—	4	4	no	—
93	M₁₆×C₆₇	SD₁₆	1	2	yes	—
94	M₁₆×C₆₇	SD₁₆	1	2	yes	—
95	C₂×C₈×C₆₇	—	4	1	no	—
96	C₂×C₈×C₆₇	—	4	4	no	—
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
102	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₆₇	—	2	1	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₄⋊C₄×C₆₇	—	2	1	no	—
108	C₄⋊C₄×C₆₇	—	4	2	no	—
109	C₄⋊C₄×C₆₇	—	4	2	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₂²⋊C₄×C₆₇	—	2	1	no	—
113	C₂²⋊C₄×C₆₇	—	4	1	no	—
114	C₄²×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	C₂×D₄	1	2	yes	—
116	C₁₆×C₆₇	—	2	2	no	—
117	C₁₆×C₆₇	—	2	2	no	—
118	C₂³×C₆₇	—	4	1	no	—
119	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₆₇	—	2	2	no	—
121	Q₈×C₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	16	2	no	—
123	D₄×C₆₇	—	2	2	no	—
124	D₄×C₆₇	—	4	2	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	16	1	no	—
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×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₆₇	—	4	2	no	—
142	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₆₇	C₂×SD₁₆	1	4	yes	—
145	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
146	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₂²×C₆₇	—	4	1	no	—
148	C₂²×C₆₇	—	4	2	no	—
149	C₂²×C₆₇	—	16	1	no	—
150	C₂²×C₆₇	#476	1	1	yes	characteristic
151	C₂²×C₆₇	#476	1	1	yes	characteristic
152	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	—	4	2	no	—
155	C₄×C₆₇	—	8	1	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	—	8	1	no	—
158	C₄×C₆₇	#476	1	1	yes	characteristic
159	C₄×C₆₇	#476	1	1	yes	characteristic
160	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
162	#33638	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₆₇	—	2	1	no	—
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	—	8	2	no	—
167	C₂×C₆₇	C₂×#476	1	1	yes	characteristic
168	#3163	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#2957	C₂×C₆₇	1	1	yes	characteristic
170	#2231	C₂×C₆₇	1	4	yes	complemented
171	#2230	C₂×C₆₇	1	4	yes	complemented
172	#2155	C₂×C₆₇	1	1	yes	characteristic, complemented
173	C₂×#365	C₂×C₆₇	1	1	yes	characteristic, complemented
174	#2153	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#2145	C₂×C₆₇	1	1	yes	characteristic
176	C₂×#360	C₂×C₆₇	1	1	yes	characteristic
177	C₆₇	#33638	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
179	#568	—	2	1	no	—
180	#544	—	2	1	no	—
181	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
183	#501	—	2	2	no	—
184	#499	C₂²×C₆₇	1	2	yes	complemented
185	#498	C₂²×C₆₇	1	2	yes	complemented
186	#444	—	2	2	no	—
187	#443	C₂²×C₆₇	1	4	yes	—
188	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
189	#428	—	2	1	no	—
190	#427	C₂²×C₆₇	1	1	yes	characteristic
191	C₂×#106	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	#365	C₂²×C₆₇	1	2	yes	complemented
196	#365	C₂²×C₆₇	1	2	yes	complemented
197	#364	C₂²×C₆₇	1	4	yes	complemented
198	#360	C₂²×C₆₇	1	4	yes	—
199	#360	C₂²×C₆₇	1	2	yes	—
200	#360	C₂²×C₆₇	1	2	yes	—
201	C₂×#41	—	2	1	no	—
202	#135	—	2	1	no	—
203	#135	—	2	1	no	—
204	#135	Q₈×C₆₇	1	2	yes	complemented
205	#135	C₂³×C₆₇	1	2	yes	—
206	C₂×Q₁₆	Q₈×C₆₇	1	1	yes	characteristic, complemented
207	C₂×SD₁₆	—	2	1	no	—
208	C₂×D₈	Q₈×C₆₇	1	1	yes	characteristic, complemented
209	#131	—	2	4	no	—
210	#131	—	4	2	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	4	2	no	—
213	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	4	1	no	—
217	#125	—	2	2	no	—
218	#125	—	4	1	no	—
219	C₂²⋊Q₈	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	D₄×C₆₇	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	2	1	no	—
225	M₃₂	C₂³×C₆₇	1	2	yes	—
226	M₃₂	C₂³×C₆₇	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	2	1	no	—
229	#108	—	2	1	no	—
230	#108	C₂³×C₆₇	1	2	yes	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#106	—	2	1	no	—
234	#106	—	2	2	no	—
235	#106	C₂³×C₆₇	1	2	yes	—
236	#106	C₂³×C₆₇	1	1	yes	characteristic
237	#106	C₂³×C₆₇	1	1	yes	characteristic
238	#104	—	2	4	no	—
239	#97	D₄×C₆₇	1	2	yes	—
240	C₄×C₈	C₂³×C₆₇	1	2	yes	—
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
252	Q₁₆	—	2	1	no	—
253	Q₁₆	C₂×Q₈×C₆₇	1	2	yes	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	2	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Q₈×C₆₇	1	2	yes	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	4	no	—
261	M₁₆	SD₁₆×C₆₇	1	2	yes	—
262	M₁₆	SD₁₆×C₆₇	1	2	yes	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
267	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
269	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
270	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
271	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	C₂×D₄×C₆₇	1	2	yes	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	16	1	no	—
303	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
307	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
311	C₈	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
312	C₈	C₂×SD₁₆×C₆₇	1	4	yes	—
313	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
314	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	#476×C₆₇	1	1	yes	characteristic
319	C₂²	#476×C₆₇	1	1	yes	characteristic
320	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	#476×C₆₇	1	1	yes	characteristic
327	C₄	#476×C₆₇	1	1	yes	characteristic
328	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
329	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#476×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	C₂×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ #405×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3163	C₂	semidirect
3. C₆₇×#2231	C₂	semidirect
4. C₆₇×#2230	C₂	semidirect
5. C₆₇×#2155	C₂	semidirect
6. C₂×#365×C₆₇	C₂	semidirect
7. C₆₇×#2153	C₂	semidirect
16. #499×C₆₇	C₂²	semidirect
17. #498×C₆₇	C₂²	semidirect
26. #365×C₆₇	C₂²	semidirect
27. #365×C₆₇	C₂²	semidirect
28. #365×C₆₇	C₂²	semidirect
29. #364×C₆₇	C₂²	semidirect
36. #135×C₆₇	Q₈	semidirect
38. C₂×Q₁₆×C₆₇	Q₈	semidirect
40. C₂×D₈×C₆₇	Q₈	semidirect
162. #33638	C₆₇	direct (N × H)
168. #3163	C₂×C₆₇	semidirect
170. #2231	C₂×C₆₇	semidirect
171. #2230	C₂×C₆₇	semidirect
172. #2155	C₂×C₆₇	semidirect
173. C₂×#365	C₂×C₆₇	semidirect
174. #2153	C₂×C₆₇	semidirect
177. C₆₇	#33638	direct (N × H)
184. #499	C₂²×C₆₇	semidirect
185. #498	C₂²×C₆₇	semidirect
194. #365	C₂²×C₆₇	semidirect
195. #365	C₂²×C₆₇	semidirect
196. #365	C₂²×C₆₇	semidirect
197. #364	C₂²×C₆₇	semidirect
204. #135	Q₈×C₆₇	semidirect
206. C₂×Q₁₆	Q₈×C₆₇	semidirect
208. C₂×D₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3163	2
C₆₇×#2957	2
C₆₇×#2231	2
C₆₇×#2230	2
C₆₇×#2155	2
C₂×#365×C₆₇	2
C₆₇×#2153	2
C₆₇×#2145	2
C₂×#360×C₆₇	2
#33638	67

Subgroups — C67×#33638 #30712889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33638 #30712889