Subgroups — C₆₇×#25913 #30705164

Order 17152.

1218 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#490×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2025	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1992	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1874	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×#125×C₆₇	—	2	8	no	—
10	C₂×#125×C₆₇	—	2	2	no	—
11	C₂×#125×C₆₇	C₂²	1	2	yes	—
12	C₂×#125×C₆₇	C₂²	1	2	yes	—
13	C₂×#121×C₆₇	—	2	2	no	—
14	C₂×#121×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#490×C₆₇	—	2	8	no	—
19	C₂×#102×C₆₇	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	#390×C₆₇	—	2	2	no	—
23	#390×C₆₇	C₂²	1	2	yes	—
24	#384×C₆₇	C₂²	1	1	yes	characteristic
25	#340×C₆₇	C₂²	1	8	yes	—
26	#336×C₆₇	C₂²	1	2	yes	—
27	C₂²×D₄×C₆₇	—	2	2	no	—
28	C₂²×D₄×C₆₇	—	2	2	no	—
29	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
30	C₂³×C₄×C₆₇	—	2	2	no	—
31	C₂³×C₄×C₆₇	D₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
34	#127×C₆₇	—	2	4	no	—
35	#125×C₆₇	—	2	16	no	—
36	#125×C₆₇	—	2	8	no	—
37	#125×C₆₇	—	2	4	no	—
38	#125×C₆₇	D₄	1	8	yes	complemented
39	#121×C₆₇	—	2	8	no	—
40	#121×C₆₇	—	4	8	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	—	2	4	no	—
52	C₂×C₄²×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#102×C₆₇	—	2	16	no	—
56	C₂²⋊C₈×C₆₇	—	2	4	no	—
57	#97×C₆₇	—	2	4	no	—
58	C₂⁴×C₆₇	—	2	2	no	—
59	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×D₄×C₆₇	—	2	4	no	—
61	C₂×D₄×C₆₇	—	2	4	no	—
62	C₂×D₄×C₆₇	—	2	4	no	—
63	C₂×D₄×C₆₇	—	4	4	no	—
64	C₂×D₄×C₆₇	—	4	4	no	—
65	C₂×D₄×C₆₇	—	4	4	no	—
66	C₂×D₄×C₆₇	—	4	8	no	—
67	C₂×D₄×C₆₇	—	4	4	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	8	no	—
71	C₂²×C₄×C₆₇	—	2	4	no	—
72	C₂²×C₄×C₆₇	—	2	4	no	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	—	4	4	no	—
77	C₂²×C₄×C₆₇	—	4	4	no	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	4	yes	—
82	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	—	2	8	no	—
87	C₄⋊C₄×C₆₇	—	2	16	no	—
88	C₄⋊C₄×C₆₇	—	2	16	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
93	C₂²⋊C₄×C₆₇	—	4	8	no	—
94	C₂²⋊C₄×C₆₇	—	4	4	no	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	—	2	2	no	—
97	C₄²×C₆₇	C₂×D₄	1	4	yes	—
98	C₂³×C₆₇	—	2	1	no	—
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	—	4	2	no	—
103	C₂³×C₆₇	—	4	2	no	—
104	C₂³×C₆₇	—	4	4	no	—
105	C₂³×C₆₇	—	4	4	no	—
106	C₂³×C₆₇	#124	1	2	yes	—
107	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
108	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	D₄×C₆₇	—	4	8	no	—
110	D₄×C₆₇	—	8	8	no	—
111	D₄×C₆₇	—	8	8	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	8	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	8	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	4	8	no	—
122	C₂×C₄×C₆₇	—	4	16	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	—	4	4	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	4	no	—
127	C₂×C₄×C₆₇	#120	1	4	yes	—
128	C₂×C₄×C₆₇	#124	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
132	C₈×C₆₇	—	4	4	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	2	4	no	—
136	C₂²×C₆₇	—	2	4	no	—
137	C₂²×C₆₇	—	8	2	no	—
138	C₂²×C₆₇	—	8	2	no	—
139	C₂²×C₆₇	—	8	2	no	—
140	C₂²×C₆₇	—	8	4	no	—
141	C₂²×C₆₇	—	8	4	no	—
142	C₂²×C₆₇	#490	1	4	yes	complemented
143	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
144	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
145	C₂²×C₆₇	#536	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
147	C₂²×C₆₇	#575	1	2	yes	—
148	C₄×C₆₇	—	2	2	no	—
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	—	4	2	no	—
151	C₄×C₆₇	—	4	8	no	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	#548	1	4	yes	—
154	#25913	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₆₇	—	2	4	no	—
156	C₂×C₆₇	—	16	2	no	—
157	C₂×C₆₇	#2436	1	1	yes	characteristic
158	C₂×C₆₇	#3006	1	2	yes	—
159	C₂×C₆₇	C₂×#490	1	1	yes	characteristic
160	C₂×C₆₇	#3142	1	1	yes	characteristic
161	C₂×C₆₇	#3234	1	2	yes	—
162	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
163	C₂×#490	C₂×C₆₇	1	2	yes	complemented
164	#2856	C₂×C₆₇	1	1	yes	characteristic
165	#2025	C₂×C₆₇	1	4	yes	complemented
166	#1992	C₂×C₆₇	1	2	yes	complemented
167	#1874	C₂×C₆₇	1	4	yes	complemented
168	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
169	C₆₇	#25913	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
171	C₂×#125	—	2	8	no	—
172	C₂×#125	—	2	2	no	—
173	C₂×#125	C₂²×C₆₇	1	2	yes	—
174	C₂×#125	C₂²×C₆₇	1	2	yes	—
175	C₂×#121	—	2	2	no	—
176	C₂×#121	C₂²×C₆₇	1	2	yes	—
177	C₂²×C₄⋊C₄	—	2	2	no	—
178	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
179	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
180	#490	—	2	8	no	—
181	C₂×#102	C₂²×C₆₇	1	8	yes	—
182	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
183	C₂×#97	C₂²×C₆₇	1	2	yes	—
184	#390	—	2	2	no	—
185	#390	C₂²×C₆₇	1	2	yes	—
186	#384	C₂²×C₆₇	1	1	yes	characteristic
187	#340	C₂²×C₆₇	1	8	yes	—
188	#336	C₂²×C₆₇	1	2	yes	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	D₄×C₆₇	1	2	yes	—
194	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
196	#127	—	2	4	no	—
197	#125	—	2	16	no	—
198	#125	—	2	8	no	—
199	#125	—	2	4	no	—
200	#125	D₄×C₆₇	1	8	yes	complemented
201	#121	—	2	8	no	—
202	#121	—	4	8	no	—
203	C₂×C₄⋊C₄	—	2	16	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	8	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	4	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	4	no	—
214	C₂×C₄²	D₄×C₆₇	1	2	yes	—
215	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
217	#102	—	2	16	no	—
218	C₂²⋊C₈	—	2	4	no	—
219	#97	—	2	4	no	—
220	C₂⁴	—	2	2	no	—
221	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	8	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
243	C₂²×C₄	#41×C₆₇	1	4	yes	—
244	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	16	no	—
250	C₄⋊C₄	—	2	16	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
255	C₂²⋊C₄	—	4	8	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	C₂×D₄×C₆₇	1	4	yes	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	4	4	no	—
268	C₂³	#124×C₆₇	1	2	yes	—
269	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
270	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
271	D₄	—	4	8	no	—
272	D₄	—	8	8	no	—
273	D₄	—	8	8	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	16	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	#120×C₆₇	1	4	yes	—
290	C₂×C₄	#124×C₆₇	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
292	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
293	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
294	C₈	—	4	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	4	no	—
303	C₂²	—	8	4	no	—
304	C₂²	#490×C₆₇	1	4	yes	complemented
305	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
306	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
307	C₂²	#536×C₆₇	1	1	yes	characteristic
308	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
309	C₂²	#575×C₆₇	1	2	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	8	no	—
314	C₄	—	8	2	no	—
315	C₄	#548×C₆₇	1	4	yes	—
316	C₂	—	2	4	no	—
317	C₂	—	16	2	no	—
318	C₂	C₆₇×#2436	1	1	yes	characteristic
319	C₂	C₆₇×#3006	1	2	yes	—
320	C₂	C₂×#490×C₆₇	1	1	yes	characteristic
321	C₂	C₆₇×#3142	1	1	yes	characteristic
322	C₂	C₆₇×#3234	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#125×C₆₇	C₂	semidirect
2. C₂×#490×C₆₇	C₂	semidirect
4. C₆₇×#2025	C₂	semidirect
5. C₆₇×#1992	C₂	semidirect
6. C₆₇×#1874	C₂	semidirect
7. C₆₇×#1749	C₂	semidirect
38. #125×C₆₇	D₄	semidirect
142. C₂²×C₆₇	#490	semidirect
154. #25913	C₆₇	direct (N × H)
162. C₂²×#125	C₂×C₆₇	semidirect
163. C₂×#490	C₂×C₆₇	semidirect
165. #2025	C₂×C₆₇	semidirect
166. #1992	C₂×C₆₇	semidirect
167. #1874	C₂×C₆₇	semidirect
168. #1749	C₂×C₆₇	semidirect
169. C₆₇	#25913	direct (N × H)
200. #125	D₄×C₆₇	semidirect
304. C₂²	#490×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#125×C₆₇	2
C₂×#490×C₆₇	2
C₆₇×#2856	2
C₆₇×#2025	2
C₆₇×#1992	2
C₆₇×#1874	2
C₆₇×#1749	2
#25913	67

Subgroups — C67×#25913 #30705164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25913 #30705164