Subgroups — C₆₇×#25919 #30705170

Order 17152.

954 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3502	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3139	C₂	1	2	yes	maximal
3	C₆₇×#2546	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2028	C₂	1	4	yes	maximal
5	C₆₇×#1986	C₂	1	2	yes	maximal
6	C₆₇×#1886	C₂	1	4	yes	maximal
7	C₄×#101×C₆₇	C₂	1	1	yes	maximal, characteristic
8	Q₈²×C₆₇	—	2	2	no	—
9	#557×C₆₇	—	2	2	no	—
10	#557×C₆₇	—	2	4	no	—
11	#544×C₆₇	—	2	2	no	—
12	#544×C₆₇	C₂²	1	2	yes	—
13	#533×C₆₇	—	2	2	no	—
14	#533×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	#519×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	#491×C₆₇	—	2	4	no	—
20	#487×C₆₇	—	2	2	no	—
21	#487×C₆₇	—	2	2	no	—
22	#419×C₆₇	C₂²	1	8	yes	—
23	C₂×#101×C₆₇	C₂²	1	2	yes	—
24	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
25	#398×C₆₇	—	2	2	no	—
26	#391×C₆₇	C₂²	1	2	yes	—
27	#389×C₆₇	—	2	2	no	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#343×C₆₇	C₂²	1	2	yes	—
30	#339×C₆₇	C₂²	1	8	yes	—
31	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
32	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₆₇	—	2	2	no	—
35	C₄⋊Q₈×C₆₇	—	2	2	no	—
36	C₄⋊Q₈×C₆₇	—	2	8	no	—
37	C₄⋊Q₈×C₆₇	—	2	4	no	—
38	C₄⋊Q₈×C₆₇	—	2	8	no	—
39	C₄⋊Q₈×C₆₇	—	2	2	no	—
40	C₄⋊Q₈×C₆₇	—	2	2	no	—
41	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
42	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
43	#125×C₆₇	—	2	4	no	—
44	#125×C₆₇	—	2	8	no	—
45	#125×C₆₇	—	2	4	no	—
46	#125×C₆₇	D₄	1	4	yes	—
47	C₂²⋊Q₈×C₆₇	—	2	2	no	—
48	C₂²⋊Q₈×C₆₇	—	2	4	no	—
49	C₂²⋊Q₈×C₆₇	—	2	2	no	—
50	C₄×Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	—	2	4	no	—
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₄×Q₈×C₆₇	—	2	4	no	—
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	—	2	2	no	—
58	#117×C₆₇	—	2	4	no	—
59	#117×C₆₇	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	D₄	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	#103×C₆₇	—	2	8	no	—
70	#103×C₆₇	—	2	8	no	—
71	#101×C₆₇	—	2	4	no	—
72	#97×C₆₇	—	2	2	no	—
73	C₄×C₈×C₆₇	—	2	2	no	—
74	#95×C₆₇	—	2	1	no	—
75	#95×C₆₇	—	2	4	no	—
76	C₂×Q₈×C₆₇	—	2	4	no	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	—	2	2	no	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	4	no	—
82	C₂×Q₈×C₆₇	—	2	4	no	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×Q₈×C₆₇	#41	1	4	yes	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₆₇	—	2	4	no	—
93	M₁₆×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₄⋊C₄×C₆₇	—	2	4	no	—
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	2	8	no	—
100	C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₄⋊C₄×C₆₇	—	2	4	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	4	1	no	—
107	C₄⋊C₄×C₆₇	—	4	2	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₄⋊C₄×C₆₇	—	4	8	no	—
110	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
111	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
112	C₂²⋊C₄×C₆₇	—	2	4	no	—
113	C₄²×C₆₇	—	2	8	no	—
114	C₄²×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	2	4	no	—
117	C₄²×C₆₇	—	2	4	no	—
118	C₄²×C₆₇	C₂×D₄	1	2	yes	—
119	C₄²×C₆₇	C₂×D₄	1	2	yes	—
120	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₆₇	—	2	8	no	—
122	Q₈×C₆₇	—	4	4	no	—
123	Q₈×C₆₇	—	4	8	no	—
124	Q₈×C₆₇	—	8	2	no	—
125	Q₈×C₆₇	—	8	4	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	4	2	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	#120	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#120	1	2	yes	—
142	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#124	1	4	yes	—
144	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
146	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
147	C₈×C₆₇	—	4	4	no	—
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
152	C₂²×C₆₇	#536	1	1	yes	characteristic
153	C₄×C₆₇	—	2	4	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	4	4	no	—
157	C₄×C₆₇	—	4	4	no	—
158	C₄×C₆₇	—	4	2	no	—
159	C₄×C₆₇	—	4	4	no	—
160	C₄×C₆₇	—	8	2	no	—
161	C₄×C₆₇	#456	1	2	yes	—
162	C₄×C₆₇	#548	1	4	yes	—
163	#25919	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₆₇	—	2	1	no	—
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	#2436	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#456	1	1	yes	characteristic
168	C₂×C₆₇	#3022	1	1	yes	characteristic
169	#3502	C₂×C₆₇	1	1	yes	characteristic
170	#3139	C₂×C₆₇	1	2	yes	—
171	#2546	C₂×C₆₇	1	1	yes	characteristic
172	#2028	C₂×C₆₇	1	4	yes	—
173	#1986	C₂×C₆₇	1	2	yes	—
174	#1886	C₂×C₆₇	1	4	yes	—
175	C₄×#101	C₂×C₆₇	1	1	yes	characteristic
176	C₆₇	#25919	1	1	yes	characteristic, direct product factor, Sylow
177	Q₈²	—	2	2	no	—
178	#557	—	2	2	no	—
179	#557	—	2	4	no	—
180	#544	—	2	2	no	—
181	#544	C₂²×C₆₇	1	2	yes	—
182	#533	—	2	2	no	—
183	#533	C₂²×C₆₇	1	2	yes	—
184	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
185	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
186	#519	C₂²×C₆₇	1	2	yes	—
187	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
188	#491	—	2	4	no	—
189	#487	—	2	2	no	—
190	#487	—	2	2	no	—
191	#419	C₂²×C₆₇	1	8	yes	—
192	C₂×#101	C₂²×C₆₇	1	2	yes	—
193	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
194	#398	—	2	2	no	—
195	#391	C₂²×C₆₇	1	2	yes	—
196	#389	—	2	2	no	—
197	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
198	#343	C₂²×C₆₇	1	2	yes	—
199	#339	C₂²×C₆₇	1	8	yes	—
200	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
201	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
202	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	8	no	—
206	C₄⋊Q₈	—	2	4	no	—
207	C₄⋊Q₈	—	2	8	no	—
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	—	2	2	no	—
210	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
211	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
212	#125	—	2	4	no	—
213	#125	—	2	8	no	—
214	#125	—	2	4	no	—
215	#125	D₄×C₆₇	1	4	yes	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	2	4	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	4	no	—
220	C₄×Q₈	—	2	4	no	—
221	C₄×Q₈	—	2	4	no	—
222	C₄×Q₈	—	2	4	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	#117	—	2	4	no	—
228	#117	C₂³×C₆₇	1	4	yes	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	4	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
234	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
235	C₂×C₄²	D₄×C₆₇	1	2	yes	—
236	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
237	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
238	#103	—	2	8	no	—
239	#103	—	2	8	no	—
240	#101	—	2	4	no	—
241	#97	—	2	2	no	—
242	C₄×C₈	—	2	2	no	—
243	#95	—	2	1	no	—
244	#95	—	2	4	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	#41×C₆₇	1	4	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
260	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	8	no	—
279	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
280	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
281	C₂²⋊C₄	—	2	4	no	—
282	C₄²	—	2	8	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	4	no	—
287	C₄²	C₂×D₄×C₆₇	1	2	yes	—
288	C₄²	C₂×D₄×C₆₇	1	2	yes	—
289	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
290	Q₈	—	2	8	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	8	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	8	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
310	C₂×C₄	#120×C₆₇	1	2	yes	—
311	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
312	C₂×C₄	#124×C₆₇	1	4	yes	—
313	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
314	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
315	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
316	C₈	—	4	4	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
320	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
321	C₂²	#536×C₆₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	4	no	—
329	C₄	—	8	2	no	—
330	C₄	#456×C₆₇	1	2	yes	—
331	C₄	#548×C₆₇	1	4	yes	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	C₆₇×#2436	1	1	yes	characteristic
335	C₂	C₂×#456×C₆₇	1	1	yes	characteristic
336	C₂	C₆₇×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂³×C₆₇	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
163. #25919	C₆₇	direct (N × H)
176. C₆₇	#25919	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3502	2
C₆₇×#3139	2
C₆₇×#2546	2
C₆₇×#2028	2
C₆₇×#1986	2
C₆₇×#1886	2
C₄×#101×C₆₇	2
#25919	67

Subgroups — C67×#25919 #30705170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25919 #30705170