Subgroups — C₆₇×#25920 #30705171

Order 17152.

1022 conjugacy classes of subgroups.

The subgroup lattice has 364 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, complemented
2	C₆₇×#3138	C₂	1	2	yes	maximal
3	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2027	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1986	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1888	C₂	1	4	yes	maximal, complemented
7	C₄×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
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×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	2	yes	—
17	#519×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	#490×C₆₇	—	2	4	no	—
20	#486×C₆₇	—	2	2	no	—
21	#486×C₆₇	—	2	2	no	—
22	#418×C₆₇	C₂²	1	8	yes	—
23	C₂×#101×C₆₇	C₂²	1	2	yes	—
24	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
25	#394×C₆₇	—	2	2	no	—
26	#390×C₆₇	—	2	2	no	—
27	#390×C₆₇	C₂²	1	2	yes	—
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	1	yes	characteristic
32	C₂²×D₄×C₆₇	—	2	1	no	—
33	C₂²×D₄×C₆₇	—	2	1	no	—
34	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
35	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
36	C₄⋊Q₈×C₆₇	—	2	2	no	—
37	C₄⋊Q₈×C₆₇	—	2	2	no	—
38	C₄⋊Q₈×C₆₇	—	2	8	no	—
39	C₄⋊Q₈×C₆₇	—	2	4	no	—
40	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
41	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
42	#127×C₆₇	—	2	2	no	—
43	#127×C₆₇	—	2	2	no	—
44	#125×C₆₇	—	2	4	no	—
45	#125×C₆₇	—	2	8	no	—
46	#125×C₆₇	—	2	4	no	—
47	#125×C₆₇	D₄	1	4	yes	—
48	#124×C₆₇	—	2	8	no	—
49	C₂²⋊Q₈×C₆₇	—	2	2	no	—
50	C₂²⋊Q₈×C₆₇	—	2	4	no	—
51	C₂²⋊Q₈×C₆₇	—	2	2	no	—
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₄×Q₈×C₆₇	—	2	4	no	—
54	C₄×Q₈×C₆₇	—	2	4	no	—
55	C₄×Q₈×C₆₇	—	2	4	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	—	2	2	no	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	#117×C₆₇	—	2	4	no	—
61	#117×C₆₇	C₂³	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
67	C₂×C₄²×C₆₇	D₄	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	#102×C₆₇	—	2	8	no	—
71	#102×C₆₇	—	2	8	no	—
72	#101×C₆₇	—	2	4	no	—
73	#97×C₆₇	—	2	2	no	—
74	C₄×C₈×C₆₇	—	2	2	no	—
75	#95×C₆₇	—	2	1	no	—
76	C₂⁴×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	2	4	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	2	no	—
82	C₂×Q₈×C₆₇	#41	1	4	yes	—
83	C₂×D₄×C₆₇	—	2	4	no	—
84	C₂×D₄×C₆₇	—	2	4	no	—
85	C₂×D₄×C₆₇	—	2	4	no	—
86	C₂×D₄×C₆₇	—	4	4	no	—
87	C₂×D₄×C₆₇	—	4	1	no	—
88	C₂×D₄×C₆₇	—	4	2	no	—
89	C₂×D₄×C₆₇	—	4	1	no	—
90	C₂×D₄×C₆₇	—	8	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₆₇	—	2	4	no	—
98	M₁₆×C₆₇	—	2	4	no	—
99	C₂×C₈×C₆₇	—	2	4	no	—
100	C₄⋊C₄×C₆₇	—	2	4	no	—
101	C₄⋊C₄×C₆₇	—	2	4	no	—
102	C₄⋊C₄×C₆₇	—	2	4	no	—
103	C₄⋊C₄×C₆₇	—	2	8	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
112	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
113	C₂²⋊C₄×C₆₇	—	2	4	no	—
114	C₂²⋊C₄×C₆₇	—	4	8	no	—
115	C₂²⋊C₄×C₆₇	—	8	4	no	—
116	C₄²×C₆₇	—	2	8	no	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	4	no	—
120	C₄²×C₆₇	—	2	4	no	—
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₄²×C₆₇	C₂×D₄	1	2	yes	—
123	C₂³×C₆₇	—	4	4	no	—
124	C₂³×C₆₇	—	4	4	no	—
125	C₂³×C₆₇	—	4	2	no	—
126	C₂³×C₆₇	—	4	2	no	—
127	C₂³×C₆₇	—	4	2	no	—
128	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₆₇	—	2	8	no	—
130	Q₈×C₆₇	—	4	4	no	—
131	Q₈×C₆₇	—	8	2	no	—
132	Q₈×C₆₇	—	8	4	no	—
133	D₄×C₆₇	—	4	8	no	—
134	D₄×C₆₇	—	8	2	no	—
135	D₄×C₆₇	—	8	4	no	—
136	D₄×C₆₇	—	8	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	4	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	4	1	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#120	1	2	yes	—
150	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#124	1	4	yes	—
152	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
153	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
154	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
155	C₈×C₆₇	—	4	4	no	—
156	C₂²×C₆₇	—	2	1	no	—
157	C₂²×C₆₇	—	2	1	no	—
158	C₂²×C₆₇	—	8	4	no	—
159	C₂²×C₆₇	—	8	4	no	—
160	C₂²×C₆₇	—	8	2	no	—
161	C₂²×C₆₇	—	8	2	no	—
162	C₂²×C₆₇	—	8	2	no	—
163	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
165	C₂²×C₆₇	#536	1	1	yes	characteristic
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	2	no	—
169	C₄×C₆₇	—	4	4	no	—
170	C₄×C₆₇	—	4	2	no	—
171	C₄×C₆₇	—	4	4	no	—
172	C₄×C₆₇	—	8	2	no	—
173	C₄×C₆₇	#454	1	2	yes	—
174	C₄×C₆₇	#548	1	4	yes	—
175	#25920	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	2	1	no	—
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	—	16	2	no	—
179	C₂×C₆₇	#2436	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#454	1	1	yes	characteristic
181	C₂×C₆₇	#3021	1	1	yes	characteristic
182	#3502	C₂×C₆₇	1	1	yes	characteristic, complemented
183	#3138	C₂×C₆₇	1	2	yes	—
184	#2548	C₂×C₆₇	1	1	yes	characteristic
185	#2027	C₂×C₆₇	1	4	yes	complemented
186	#1986	C₂×C₆₇	1	2	yes	complemented
187	#1888	C₂×C₆₇	1	4	yes	complemented
188	C₄×#101	C₂×C₆₇	1	1	yes	characteristic, complemented
189	C₆₇	#25920	1	1	yes	characteristic, direct product factor, Sylow
190	Q₈²	—	2	2	no	—
191	#557	—	2	2	no	—
192	#557	—	2	4	no	—
193	#544	—	2	2	no	—
194	#544	C₂²×C₆₇	1	2	yes	—
195	#533	—	2	2	no	—
196	#533	C₂²×C₆₇	1	2	yes	—
197	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
198	C₂×#124	C₂²×C₆₇	1	2	yes	—
199	#519	C₂²×C₆₇	1	2	yes	—
200	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
201	#490	—	2	4	no	—
202	#486	—	2	2	no	—
203	#486	—	2	2	no	—
204	#418	C₂²×C₆₇	1	8	yes	—
205	C₂×#101	C₂²×C₆₇	1	2	yes	—
206	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
207	#394	—	2	2	no	—
208	#390	—	2	2	no	—
209	#390	C₂²×C₆₇	1	2	yes	—
210	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
211	#343	C₂²×C₆₇	1	2	yes	—
212	#339	C₂²×C₆₇	1	8	yes	—
213	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
214	C₂²×D₄	—	2	1	no	—
215	C₂²×D₄	—	2	1	no	—
216	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
217	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	—	2	8	no	—
221	C₄⋊Q₈	—	2	4	no	—
222	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
223	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
224	#127	—	2	2	no	—
225	#127	—	2	2	no	—
226	#125	—	2	4	no	—
227	#125	—	2	8	no	—
228	#125	—	2	4	no	—
229	#125	D₄×C₆₇	1	4	yes	—
230	#124	—	2	8	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₂²⋊Q₈	—	2	4	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₄×Q₈	—	2	4	no	—
235	C₄×Q₈	—	2	4	no	—
236	C₄×Q₈	—	2	4	no	—
237	C₄×Q₈	—	2	4	no	—
238	C₄×Q₈	—	2	2	no	—
239	C₄×Q₈	—	2	2	no	—
240	C₄×Q₈	—	2	2	no	—
241	C₄×Q₈	—	2	2	no	—
242	#117	—	2	4	no	—
243	#117	C₂³×C₆₇	1	4	yes	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
246	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
247	C₂×C₂²⋊C₄	—	2	4	no	—
248	C₂×C₂²⋊C₄	—	2	4	no	—
249	C₂×C₄²	D₄×C₆₇	1	2	yes	—
250	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
251	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
252	#102	—	2	8	no	—
253	#102	—	2	8	no	—
254	#101	—	2	4	no	—
255	#97	—	2	2	no	—
256	C₄×C₈	—	2	2	no	—
257	#95	—	2	1	no	—
258	C₂⁴	—	2	2	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	#41×C₆₇	1	4	yes	—
265	C₂×D₄	—	2	4	no	—
266	C₂×D₄	—	2	4	no	—
267	C₂×D₄	—	2	4	no	—
268	C₂×D₄	—	4	4	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	8	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
277	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	C₂×C₈	—	2	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
294	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
295	C₂²⋊C₄	—	2	4	no	—
296	C₂²⋊C₄	—	4	8	no	—
297	C₂²⋊C₄	—	8	4	no	—
298	C₄²	—	2	8	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	4	no	—
302	C₄²	—	2	4	no	—
303	C₄²	C₂×D₄×C₆₇	1	2	yes	—
304	C₄²	C₂×D₄×C₆₇	1	2	yes	—
305	C₂³	—	4	4	no	—
306	C₂³	—	4	4	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	2	no	—
310	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
311	Q₈	—	2	8	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	8	2	no	—
314	Q₈	—	8	4	no	—
315	D₄	—	4	8	no	—
316	D₄	—	8	2	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
331	C₂×C₄	#120×C₆₇	1	2	yes	—
332	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
333	C₂×C₄	#124×C₆₇	1	4	yes	—
334	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
335	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
336	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
337	C₈	—	4	4	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	8	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
346	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
347	C₂²	#536×C₆₇	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	4	no	—
351	C₄	—	4	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	4	no	—
354	C₄	—	8	2	no	—
355	C₄	#454×C₆₇	1	2	yes	—
356	C₄	#548×C₆₇	1	4	yes	—
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	16	2	no	—
360	C₂	C₆₇×#2436	1	1	yes	characteristic
361	C₂	C₂×#454×C₆₇	1	1	yes	characteristic
362	C₂	C₆₇×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
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
1. C₆₇×#3502	C₂	semidirect
4. C₆₇×#2027	C₂	semidirect
5. C₆₇×#1986	C₂	semidirect
6. C₆₇×#1888	C₂	semidirect
7. C₄×#101×C₆₇	C₂	semidirect
175. #25920	C₆₇	direct (N × H)
182. #3502	C₂×C₆₇	semidirect
185. #2027	C₂×C₆₇	semidirect
186. #1986	C₂×C₆₇	semidirect
187. #1888	C₂×C₆₇	semidirect
188. C₄×#101	C₂×C₆₇	semidirect
189. C₆₇	#25920	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×#3138	2
C₆₇×#2548	2
C₆₇×#2027	2
C₆₇×#1986	2
C₆₇×#1888	2
C₄×#101×C₆₇	2
#25920	67

Subgroups — C67×#25920 #30705171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25920 #30705171