Subgroups — C₆₇×#9854 #30689105

Order 17152.

460 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3167	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1962	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1492	C₂	1	1	yes	maximal, characteristic
8	#533×C₆₇	C₂²	1	1	yes	characteristic
9	#532×C₆₇	C₂²	1	1	yes	characteristic
10	#498×C₆₇	C₄	1	1	yes	characteristic, complemented
11	#492×C₆₇	C₄	1	1	yes	characteristic, complemented
12	#487×C₆₇	C₄	1	1	yes	characteristic, complemented
13	#486×C₆₇	C₄	1	1	yes	characteristic, complemented
14	#465×C₆₇	—	2	2	no	—
15	C₄×SD₁₆×C₆₇	—	2	2	no	—
16	#423×C₆₇	C₂²	1	1	yes	characteristic
17	#423×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#104×C₆₇	—	2	2	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	#339×C₆₇	—	2	2	no	—
23	#336×C₆₇	—	2	2	no	—
24	#335×C₆₇	—	2	2	no	—
25	Q₈⋊C₈×C₆₇	—	2	2	no	—
26	D₄⋊C₈×C₆₇	—	2	2	no	—
27	C₂×#41×C₆₇	—	2	1	no	—
28	C₂×SD₁₆×C₆₇	—	2	2	no	—
29	C₂×M₁₆×C₆₇	—	2	1	no	—
30	C₂×M₁₆×C₆₇	—	2	1	no	—
31	C₂²×C₈×C₆₇	—	2	1	no	—
32	C₂²×C₈×C₆₇	D₄	1	2	yes	—
33	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
37	#125×C₆₇	—	2	1	no	—
38	#124×C₆₇	—	2	1	no	—
39	C₂²⋊Q₈×C₆₇	—	2	1	no	—
40	C₂²⋊Q₈×C₆₇	—	2	1	no	—
41	#121×C₆₇	—	2	1	no	—
42	#121×C₆₇	—	2	1	no	—
43	C₄×Q₈×C₆₇	—	2	1	no	—
44	C₄×Q₈×C₆₇	—	2	1	no	—
45	C₄×D₄×C₆₇	—	2	1	no	—
46	C₄×D₄×C₆₇	—	2	1	no	—
47	#117×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	—	2	1	no	—
49	C₂×C₄²×C₆₇	—	2	1	no	—
50	C₂×C₄²×C₆₇	D₄	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
52	#106×C₆₇	—	2	2	no	—
53	C₄⋊C₈×C₆₇	—	2	1	no	—
54	C₄⋊C₈×C₆₇	—	2	1	no	—
55	C₄⋊C₈×C₆₇	—	2	1	no	—
56	C₄⋊C₈×C₆₇	—	2	1	no	—
57	#104×C₆₇	—	4	4	no	—
58	#103×C₆₇	—	2	2	no	—
59	#102×C₆₇	—	2	2	no	—
60	C₄×C₈×C₆₇	—	2	1	no	—
61	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₆₇	D₄	1	2	yes	—
64	C₄×C₈×C₆₇	D₄	1	2	yes	—
65	#41×C₆₇	—	4	2	no	—
66	#41×C₆₇	—	4	2	no	—
67	C₂×Q₈×C₆₇	—	2	1	no	—
68	C₂×Q₈×C₆₇	—	2	1	no	—
69	C₂×Q₈×C₆₇	—	2	1	no	—
70	C₂×D₄×C₆₇	—	2	1	no	—
71	C₂×D₄×C₆₇	—	2	1	no	—
72	C₂×D₄×C₆₇	—	2	1	no	—
73	C₂×D₄×C₆₇	—	4	1	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	4	1	no	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
79	SD₁₆×C₆₇	—	4	4	no	—
80	M₁₆×C₆₇	—	4	2	no	—
81	M₁₆×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	4	2	no	—
84	C₂×C₈×C₆₇	—	4	2	no	—
85	C₂×C₈×C₆₇	—	4	1	no	—
86	C₂×C₈×C₆₇	—	4	1	no	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₄⋊C₄×C₆₇	—	4	1	no	—
96	C₄⋊C₄×C₆₇	—	4	1	no	—
97	C₂²⋊C₄×C₆₇	—	4	1	no	—
98	C₂²⋊C₄×C₆₇	—	4	1	no	—
99	C₂²⋊C₄×C₆₇	—	4	1	no	—
100	C₄²×C₆₇	—	2	4	no	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	4	2	no	—
104	C₄²×C₆₇	—	4	1	no	—
105	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₆₇	#41	1	1	yes	characteristic
109	C₂³×C₆₇	—	4	1	no	—
110	C₂³×C₆₇	#121	1	1	yes	characteristic
111	Q₈×C₆₇	—	4	2	no	—
112	Q₈×C₆₇	—	4	2	no	—
113	Q₈×C₆₇	—	4	2	no	—
114	D₄×C₆₇	—	4	2	no	—
115	D₄×C₆₇	—	4	2	no	—
116	D₄×C₆₇	—	4	2	no	—
117	D₄×C₆₇	—	8	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	8	1	no	—
127	C₂×C₄×C₆₇	—	8	1	no	—
128	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
133	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
134	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₆₇	#137	1	2	yes	—
136	C₈×C₆₇	—	2	2	no	—
137	C₈×C₆₇	—	2	2	no	—
138	C₈×C₆₇	—	4	2	no	—
139	C₈×C₆₇	—	8	1	no	—
140	C₈×C₆₇	—	8	1	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	2	1	no	—
143	C₂²×C₆₇	—	8	1	no	—
144	C₂²×C₆₇	—	8	1	no	—
145	C₂²×C₆₇	—	8	1	no	—
146	C₂²×C₆₇	#390	1	1	yes	characteristic
147	C₂²×C₆₇	#468	1	2	yes	—
148	C₄×C₆₇	—	2	1	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	4	4	no	—
151	C₄×C₆₇	—	8	1	no	—
152	C₄×C₆₇	—	8	1	no	—
153	C₄×C₆₇	—	8	1	no	—
154	C₄×C₆₇	#421	1	2	yes	—
155	C₄×C₆₇	#440	1	2	yes	—
156	C₄×C₆₇	#443	1	2	yes	—
157	C₄×C₆₇	#496	1	1	yes	characteristic
158	C₄×C₆₇	#497	1	1	yes	characteristic
159	#9854	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂×C₆₇	—	2	2	no	—
161	C₂×C₆₇	—	16	1	no	—
162	C₂×C₆₇	#1963	1	1	yes	characteristic
163	C₂×C₆₇	#1973	1	1	yes	characteristic
164	C₂×C₆₇	#1974	1	1	yes	characteristic
165	#3167	C₂×C₆₇	1	1	yes	characteristic
166	#1994	C₂×C₆₇	1	1	yes	characteristic, complemented
167	#1962	C₂×C₆₇	1	1	yes	characteristic
168	#1744	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#1567	C₂×C₆₇	1	1	yes	characteristic, complemented
170	#1493	C₂×C₆₇	1	1	yes	characteristic, complemented
171	#1492	C₂×C₆₇	1	1	yes	characteristic
172	C₆₇	#9854	1	1	yes	characteristic, direct product factor, Sylow
173	#533	C₂²×C₆₇	1	1	yes	characteristic
174	#532	C₂²×C₆₇	1	1	yes	characteristic
175	#498	C₄×C₆₇	1	1	yes	characteristic, complemented
176	#492	C₄×C₆₇	1	1	yes	characteristic, complemented
177	#487	C₄×C₆₇	1	1	yes	characteristic, complemented
178	#486	C₄×C₆₇	1	1	yes	characteristic, complemented
179	#465	—	2	2	no	—
180	C₄×SD₁₆	—	2	2	no	—
181	#423	C₂²×C₆₇	1	1	yes	characteristic
182	#423	C₂²×C₆₇	1	1	yes	characteristic
183	C₂×#104	—	2	2	no	—
184	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
186	C₄³	C₂²×C₆₇	1	1	yes	characteristic
187	#339	—	2	2	no	—
188	#336	—	2	2	no	—
189	#335	—	2	2	no	—
190	Q₈⋊C₈	—	2	2	no	—
191	D₄⋊C₈	—	2	2	no	—
192	C₂×#41	—	2	1	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	D₄×C₆₇	1	2	yes	—
198	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
200	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
201	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
202	#125	—	2	1	no	—
203	#124	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	C₂²⋊Q₈	—	2	1	no	—
206	#121	—	2	1	no	—
207	#121	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	1	no	—
212	#117	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	D₄×C₆₇	1	2	yes	—
216	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
217	#106	—	2	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	2	1	no	—
222	#104	—	4	4	no	—
223	#103	—	2	2	no	—
224	#102	—	2	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
227	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
228	C₄×C₈	D₄×C₆₇	1	2	yes	—
229	C₄×C₈	D₄×C₆₇	1	2	yes	—
230	#41	—	4	2	no	—
231	#41	—	4	2	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
243	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
244	SD₁₆	—	4	4	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
253	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
273	C₄²	#41×C₆₇	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	C₂³	#121×C₆₇	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	8	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
294	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
296	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
297	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
298	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
299	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
300	C₂×C₄	#137×C₆₇	1	2	yes	—
301	C₈	—	2	2	no	—
302	C₈	—	2	2	no	—
303	C₈	—	4	2	no	—
304	C₈	—	8	1	no	—
305	C₈	—	8	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	8	1	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	#390×C₆₇	1	1	yes	characteristic
312	C₂²	#468×C₆₇	1	2	yes	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	4	no	—
316	C₄	—	8	1	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	1	no	—
319	C₄	#421×C₆₇	1	2	yes	—
320	C₄	#440×C₆₇	1	2	yes	—
321	C₄	#443×C₆₇	1	2	yes	—
322	C₄	#496×C₆₇	1	1	yes	characteristic
323	C₄	#497×C₆₇	1	1	yes	characteristic
324	C₂	—	2	2	no	—
325	C₂	—	16	1	no	—
326	C₂	C₆₇×#1963	1	1	yes	characteristic
327	C₂	C₆₇×#1973	1	1	yes	characteristic
328	C₂	C₆₇×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#127
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	#390
prime radical	#532×C₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1994	C₂	semidirect
4. C₆₇×#1744	C₂	semidirect
5. C₆₇×#1567	C₂	semidirect
6. C₆₇×#1493	C₂	semidirect
10. #498×C₆₇	C₄	semidirect
11. #492×C₆₇	C₄	semidirect
12. #487×C₆₇	C₄	semidirect
13. #486×C₆₇	C₄	semidirect
159. #9854	C₆₇	direct (N × H)
166. #1994	C₂×C₆₇	semidirect
168. #1744	C₂×C₆₇	semidirect
169. #1567	C₂×C₆₇	semidirect
170. #1493	C₂×C₆₇	semidirect
172. C₆₇	#9854	direct (N × H)
175. #498	C₄×C₆₇	semidirect
176. #492	C₄×C₆₇	semidirect
177. #487	C₄×C₆₇	semidirect
178. #486	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3167	2
C₆₇×#1994	2
C₆₇×#1962	2
C₆₇×#1744	2
C₆₇×#1567	2
C₆₇×#1493	2
C₆₇×#1492	2
#9854	67

Subgroups — C67×#9854 #30689105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9854 #30689105