Subgroups — C₆₇×#25860 #30705111

Order 17152.

1970 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#490×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2851	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2041	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1961	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1786	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄×C₆₇	—	2	2	no	—
9	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
10	C₂×#127×C₆₇	—	2	8	no	—
11	C₂×#127×C₆₇	—	2	2	no	—
12	C₂×#127×C₆₇	C₂²	1	2	yes	—
13	C₂×#127×C₆₇	C₂²	1	2	yes	—
14	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#490×C₆₇	—	2	8	no	—
17	C₂×#102×C₆₇	C₂²	1	8	yes	complemented
18	C₂×#102×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
21	#389×C₆₇	C₂²	1	2	yes	complemented
22	#387×C₆₇	—	2	2	no	—
23	#385×C₆₇	—	2	2	no	—
24	#382×C₆₇	C₂²	1	1	yes	characteristic, complemented
25	#380×C₆₇	C₂²	1	2	yes	—
26	#336×C₆₇	C₂²	1	2	yes	complemented
27	C₂⁵×C₆₇	—	2	2	no	—
28	C₂²×D₄×C₆₇	—	2	16	no	—
29	C₂²×D₄×C₆₇	—	2	8	no	—
30	C₂²×D₄×C₆₇	—	2	2	no	—
31	C₂²×D₄×C₆₇	—	2	4	no	—
32	C₂²×D₄×C₆₇	—	2	4	no	—
33	C₂²×D₄×C₆₇	—	2	8	no	—
34	C₂²×D₄×C₆₇	—	4	4	no	—
35	C₂²×D₄×C₆₇	—	4	8	no	—
36	C₂²×D₄×C₆₇	—	4	8	no	—
37	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
38	C₂³×C₄×C₆₇	D₄	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
41	#127×C₆₇	—	2	16	no	—
42	#127×C₆₇	—	2	8	no	—
43	#127×C₆₇	—	2	4	no	—
44	#127×C₆₇	D₄	1	8	yes	—
45	#125×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	#102×C₆₇	—	4	8	no	—
57	C₂²⋊C₈×C₆₇	—	2	4	no	—
58	#97×C₆₇	—	2	4	no	—
59	#95×C₆₇	—	2	2	no	—
60	#95×C₆₇	—	2	2	no	—
61	#95×C₆₇	—	2	2	no	—
62	C₂⁴×C₆₇	—	2	8	no	—
63	C₂⁴×C₆₇	—	2	4	no	—
64	C₂⁴×C₆₇	—	4	4	no	—
65	C₂⁴×C₆₇	—	4	8	no	—
66	C₂⁴×C₆₇	—	4	4	no	—
67	C₂⁴×C₆₇	—	4	8	no	—
68	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×D₄×C₆₇	—	2	8	no	—
70	C₂×D₄×C₆₇	—	2	16	no	—
71	C₂×D₄×C₆₇	—	2	16	no	—
72	C₂×D₄×C₆₇	—	2	4	no	—
73	C₂×D₄×C₆₇	—	2	8	no	—
74	C₂×D₄×C₆₇	—	2	8	no	—
75	C₂×D₄×C₆₇	—	4	32	no	—
76	C₂×D₄×C₆₇	—	4	64	no	—
77	C₂×D₄×C₆₇	—	4	4	no	—
78	C₂×D₄×C₆₇	—	4	8	no	—
79	C₂×D₄×C₆₇	—	4	8	no	—
80	C₂×D₄×C₆₇	—	4	8	no	—
81	C₂×D₄×C₆₇	—	4	4	no	—
82	C₂×D₄×C₆₇	—	4	8	no	—
83	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₆₇	#41	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₂×C₈×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	—	2	4	no	—
98	C₂×C₈×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₄⋊C₄×C₆₇	—	4	4	no	—
102	C₄⋊C₄×C₆₇	—	4	4	no	—
103	C₂²⋊C₄×C₆₇	—	4	4	no	—
104	C₄²×C₆₇	—	2	4	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	C₂×D₄	1	4	yes	—
107	C₂³×C₆₇	—	2	1	no	—
108	C₂³×C₆₇	—	2	1	no	—
109	C₂³×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	—	2	8	no	—
111	C₂³×C₆₇	—	4	16	no	—
112	C₂³×C₆₇	—	4	32	no	—
113	C₂³×C₆₇	—	4	4	no	—
114	C₂³×C₆₇	—	4	4	no	—
115	C₂³×C₆₇	—	4	8	no	—
116	C₂³×C₆₇	—	4	8	no	—
117	C₂³×C₆₇	—	8	8	no	—
118	C₂³×C₆₇	—	8	16	no	—
119	C₂³×C₆₇	—	8	8	no	—
120	C₂³×C₆₇	#124	1	2	yes	—
121	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
122	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	D₄×C₆₇	—	2	16	no	—
124	D₄×C₆₇	—	4	16	no	—
125	D₄×C₆₇	—	4	32	no	—
126	D₄×C₆₇	—	4	32	no	—
127	D₄×C₆₇	—	4	8	no	—
128	C₂×C₄×C₆₇	—	2	16	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	8	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	#120	1	4	yes	—
143	C₂×C₄×C₆₇	#124	1	2	yes	—
144	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
145	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
147	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
148	C₈×C₆₇	—	4	4	no	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	4	no	—
152	C₂²×C₆₇	—	2	4	no	—
153	C₂²×C₆₇	—	4	8	no	—
154	C₂²×C₆₇	—	4	16	no	—
155	C₂²×C₆₇	—	8	32	no	—
156	C₂²×C₆₇	—	8	4	no	—
157	C₂²×C₆₇	—	8	4	no	—
158	C₂²×C₆₇	—	8	8	no	—
159	C₂²×C₆₇	#490	1	4	yes	complemented
160	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
161	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
162	C₂²×C₆₇	#535	1	1	yes	characteristic
163	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
164	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	2	4	no	—
168	C₄×C₆₇	—	8	2	no	—
169	C₄×C₆₇	—	8	2	no	—
170	C₄×C₆₇	#546	1	4	yes	—
171	#25860	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₆₇	—	2	4	no	—
173	C₂×C₆₇	—	8	8	no	—
174	C₂×C₆₇	#2435	1	1	yes	characteristic
175	C₂×C₆₇	#3003	1	2	yes	—
176	C₂×C₆₇	C₂×#490	1	1	yes	characteristic
177	C₂×C₆₇	#3138	1	1	yes	characteristic
178	C₂×C₆₇	#3217	1	2	yes	—
179	C₂²×#127	C₂×C₆₇	1	1	yes	characteristic
180	C₂×#490	C₂×C₆₇	1	2	yes	complemented
181	#2851	C₂×C₆₇	1	1	yes	characteristic, complemented
182	#2041	C₂×C₆₇	1	4	yes	complemented
183	#1961	C₂×C₆₇	1	2	yes	complemented
184	#1786	C₂×C₆₇	1	4	yes	complemented
185	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
186	C₆₇	#25860	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³×D₄	—	2	2	no	—
188	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
189	C₂×#127	—	2	8	no	—
190	C₂×#127	—	2	2	no	—
191	C₂×#127	C₂²×C₆₇	1	2	yes	—
192	C₂×#127	C₂²×C₆₇	1	2	yes	—
193	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
194	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
195	#490	—	2	8	no	—
196	C₂×#102	C₂²×C₆₇	1	8	yes	complemented
197	C₂×#102	C₂²×C₆₇	1	8	yes	complemented
198	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
199	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
200	#389	C₂²×C₆₇	1	2	yes	complemented
201	#387	—	2	2	no	—
202	#385	—	2	2	no	—
203	#382	C₂²×C₆₇	1	1	yes	characteristic, complemented
204	#380	C₂²×C₆₇	1	2	yes	—
205	#336	C₂²×C₆₇	1	2	yes	complemented
206	C₂⁵	—	2	2	no	—
207	C₂²×D₄	—	2	16	no	—
208	C₂²×D₄	—	2	8	no	—
209	C₂²×D₄	—	2	2	no	—
210	C₂²×D₄	—	2	4	no	—
211	C₂²×D₄	—	2	4	no	—
212	C₂²×D₄	—	2	8	no	—
213	C₂²×D₄	—	4	4	no	—
214	C₂²×D₄	—	4	8	no	—
215	C₂²×D₄	—	4	8	no	—
216	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
217	C₂³×C₄	D₄×C₆₇	1	2	yes	—
218	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
220	#127	—	2	16	no	—
221	#127	—	2	8	no	—
222	#127	—	2	4	no	—
223	#127	D₄×C₆₇	1	8	yes	—
224	#125	—	2	4	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₄²	—	2	4	no	—
231	C₂×C₄²	D₄×C₆₇	1	2	yes	—
232	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
234	#102	—	2	16	no	—
235	#102	—	4	8	no	—
236	C₂²⋊C₈	—	2	4	no	—
237	#97	—	2	4	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	C₂⁴	—	2	8	no	—
242	C₂⁴	—	2	4	no	—
243	C₂⁴	—	4	4	no	—
244	C₂⁴	—	4	8	no	—
245	C₂⁴	—	4	4	no	—
246	C₂⁴	—	4	8	no	—
247	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
248	C₂×D₄	—	2	8	no	—
249	C₂×D₄	—	2	16	no	—
250	C₂×D₄	—	2	16	no	—
251	C₂×D₄	—	2	4	no	—
252	C₂×D₄	—	2	8	no	—
253	C₂×D₄	—	2	8	no	—
254	C₂×D₄	—	4	32	no	—
255	C₂×D₄	—	4	64	no	—
256	C₂×D₄	—	4	4	no	—
257	C₂×D₄	—	4	8	no	—
258	C₂×D₄	—	4	8	no	—
259	C₂×D₄	—	4	8	no	—
260	C₂×D₄	—	4	4	no	—
261	C₂×D₄	—	4	8	no	—
262	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
271	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
273	C₂²×C₄	#41×C₆₇	1	4	yes	—
274	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	4	4	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	C₂×D₄×C₆₇	1	4	yes	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	8	no	—
290	C₂³	—	4	16	no	—
291	C₂³	—	4	32	no	—
292	C₂³	—	4	4	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	4	8	no	—
295	C₂³	—	4	8	no	—
296	C₂³	—	8	8	no	—
297	C₂³	—	8	16	no	—
298	C₂³	—	8	8	no	—
299	C₂³	#124×C₆₇	1	2	yes	—
300	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
301	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
302	D₄	—	2	16	no	—
303	D₄	—	4	16	no	—
304	D₄	—	4	32	no	—
305	D₄	—	4	32	no	—
306	D₄	—	4	8	no	—
307	C₂×C₄	—	2	16	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	#120×C₆₇	1	4	yes	—
322	C₂×C₄	#124×C₆₇	1	2	yes	—
323	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
324	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
325	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
326	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
327	C₈	—	4	4	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	4	8	no	—
333	C₂²	—	4	16	no	—
334	C₂²	—	8	32	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	4	no	—
337	C₂²	—	8	8	no	—
338	C₂²	#490×C₆₇	1	4	yes	complemented
339	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
340	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
341	C₂²	#535×C₆₇	1	1	yes	characteristic
342	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
343	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
344	C₄	—	2	2	no	—
345	C₄	—	2	4	no	—
346	C₄	—	2	4	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	#546×C₆₇	1	4	yes	—
350	C₂	—	2	4	no	—
351	C₂	—	8	8	no	—
352	C₂	C₆₇×#2435	1	1	yes	characteristic
353	C₂	C₆₇×#3003	1	2	yes	—
354	C₂	C₂×#490×C₆₇	1	1	yes	characteristic
355	C₂	C₆₇×#3138	1	1	yes	characteristic
356	C₂	C₆₇×#3217	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#127×C₆₇	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
2. C₂×#490×C₆₇	C₂	semidirect
3. C₆₇×#2851	C₂	semidirect
4. C₆₇×#2041	C₂	semidirect
5. C₆₇×#1961	C₂	semidirect
6. C₆₇×#1786	C₂	semidirect
7. C₆₇×#1749	C₂	semidirect
14. C₂×#125×C₆₇	C₂²	semidirect
17. C₂×#102×C₆₇	C₂²	semidirect
18. C₂×#102×C₆₇	C₂²	semidirect
20. C₂×#97×C₆₇	C₂²	semidirect
21. #389×C₆₇	C₂²	semidirect
24. #382×C₆₇	C₂²	semidirect
26. #336×C₆₇	C₂²	semidirect
159. C₂²×C₆₇	#490	semidirect
171. #25860	C₆₇	direct (N × H)
180. C₂×#490	C₂×C₆₇	semidirect
181. #2851	C₂×C₆₇	semidirect
182. #2041	C₂×C₆₇	semidirect
183. #1961	C₂×C₆₇	semidirect
184. #1786	C₂×C₆₇	semidirect
185. #1749	C₂×C₆₇	semidirect
186. C₆₇	#25860	direct (N × H)
193. C₂×#125	C₂²×C₆₇	semidirect
196. C₂×#102	C₂²×C₆₇	semidirect
197. C₂×#102	C₂²×C₆₇	semidirect
199. C₂×#97	C₂²×C₆₇	semidirect
200. #389	C₂²×C₆₇	semidirect
203. #382	C₂²×C₆₇	semidirect
205. #336	C₂²×C₆₇	semidirect
338. 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×#127×C₆₇	2
C₂×#490×C₆₇	2
C₆₇×#2851	2
C₆₇×#2041	2
C₆₇×#1961	2
C₆₇×#1786	2
C₆₇×#1749	2
#25860	67

Subgroups — C67×#25860 #30705111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25860 #30705111