Subgroups — C₆₇×#22744 #30701995

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2355	C₂	1	2	yes	maximal
3	C₆₇×#1949	C₂	1	2	yes	maximal
4	C₆₇×#1949	C₂	1	2	yes	maximal
5	C₆₇×#1947	C₂	1	2	yes	maximal
6	C₆₇×#1836	C₂	1	2	yes	maximal
7	C₆₇×#1835	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1776	C₂	1	2	yes	maximal
9	C₄×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	#446×C₆₇	C₄	1	8	yes	complemented
12	C₂×#107×C₆₇	C₂²	1	2	yes	—
13	C₂×#107×C₆₇	C₂²	1	2	yes	—
14	C₂×#106×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#391×C₆₇	—	2	2	no	—
20	#387×C₆₇	—	2	2	no	—
21	#387×C₆₇	—	2	2	no	—
22	#384×C₆₇	C₂²	1	2	yes	—
23	#382×C₆₇	C₂²	1	2	yes	—
24	#382×C₆₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	#340×C₆₇	C₂²	1	1	yes	characteristic
30	#340×C₆₇	C₂²	1	1	yes	characteristic
31	#340×C₆₇	C₂²	1	2	yes	—
32	#340×C₆₇	C₂²	1	1	yes	characteristic
33	#340×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	2	yes	—
35	#340×C₆₇	C₂²	1	2	yes	—
36	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
55	#107×C₆₇	—	2	4	no	—
56	#107×C₆₇	—	2	4	no	—
57	#106×C₆₇	—	2	4	no	—
58	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
60	#97×C₆₇	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
62	#95×C₆₇	—	2	2	no	—
63	#95×C₆₇	—	2	2	no	—
64	#95×C₆₇	—	2	2	no	—
65	#95×C₆₇	—	2	2	no	—
66	#95×C₆₇	—	2	2	no	—
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	—	2	2	no	—
69	C₂×Q₈×C₆₇	—	4	2	no	—
70	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₆₇	—	2	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₄⋊C₄×C₆₇	—	4	4	no	—
95	C₄⋊C₄×C₆₇	—	8	2	no	—
96	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
97	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
98	C₄⋊C₄×C₆₇	#41	1	4	yes	—
99	C₄²×C₆₇	—	4	4	no	—
100	C₄²×C₆₇	—	8	2	no	—
101	C₄²×C₆₇	—	8	2	no	—
102	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
103	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
104	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
105	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
106	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	2	no	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
135	C₈×C₆₇	—	2	4	no	—
136	C₈×C₆₇	—	2	2	no	—
137	C₈×C₆₇	—	2	2	no	—
138	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₆₇	#548	1	2	yes	—
140	C₂²×C₆₇	#575	1	2	yes	—
141	C₂²×C₆₇	#576	1	1	yes	characteristic
142	C₂²×C₆₇	#578	1	1	yes	characteristic
143	C₄×C₆₇	—	2	4	no	—
144	C₄×C₆₇	—	2	2	no	—
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	4	no	—
147	C₄×C₆₇	—	4	4	no	—
148	C₄×C₆₇	—	8	2	no	—
149	C₄×C₆₇	—	8	2	no	—
150	C₄×C₆₇	—	8	2	no	—
151	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
152	C₄×C₆₇	#519	1	1	yes	characteristic
153	C₄×C₆₇	#557	1	2	yes	—
154	#22744	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂×C₆₇	#2351	1	1	yes	characteristic
156	C₂×C₆₇	#2911	1	1	yes	characteristic
157	C₂×C₆₇	#2913	1	1	yes	characteristic
158	C₂×C₆₇	#3370	1	2	yes	—
159	C₂×C₆₇	#3377	1	2	yes	—
160	C₂×#446	C₂×C₆₇	1	1	yes	characteristic
161	#2355	C₂×C₆₇	1	2	yes	—
162	#1949	C₂×C₆₇	1	2	yes	—
163	#1949	C₂×C₆₇	1	2	yes	—
164	#1947	C₂×C₆₇	1	2	yes	—
165	#1836	C₂×C₆₇	1	2	yes	—
166	#1835	C₂×C₆₇	1	1	yes	characteristic
167	#1776	C₂×C₆₇	1	2	yes	—
168	C₄×#107	C₂×C₆₇	1	1	yes	characteristic
169	C₆₇	#22744	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
171	#446	C₄×C₆₇	1	8	yes	complemented
172	C₂×#107	C₂²×C₆₇	1	2	yes	—
173	C₂×#107	C₂²×C₆₇	1	2	yes	—
174	C₂×#106	C₂²×C₆₇	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
177	C₂×#97	C₂²×C₆₇	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
179	#391	—	2	2	no	—
180	#387	—	2	2	no	—
181	#387	—	2	2	no	—
182	#384	C₂²×C₆₇	1	2	yes	—
183	#382	C₂²×C₆₇	1	2	yes	—
184	#382	C₂²×C₆₇	1	2	yes	—
185	C₄×C₄⋊C₄	—	2	2	no	—
186	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
187	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
188	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
189	#340	C₂²×C₆₇	1	2	yes	—
190	#340	C₂²×C₆₇	1	1	yes	characteristic
191	#340	C₂²×C₆₇	1	1	yes	characteristic
192	#340	C₂²×C₆₇	1	1	yes	characteristic
193	#340	C₂²×C₆₇	1	2	yes	—
194	#340	C₂²×C₆₇	1	2	yes	—
195	#340	C₂²×C₆₇	1	1	yes	characteristic
196	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
200	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
201	C₄×Q₈	Q₈×C₆₇	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
205	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
206	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
215	#107	—	2	4	no	—
216	#107	—	2	4	no	—
217	#106	—	2	4	no	—
218	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
219	C₄⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
220	#97	C₂×C₄×C₆₇	1	8	yes	—
221	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
245	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
246	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	8	2	no	—
256	C₄⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
257	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
258	C₄⋊C₄	#41×C₆₇	1	4	yes	—
259	C₄²	—	4	4	no	—
260	C₄²	—	8	2	no	—
261	C₄²	—	8	2	no	—
262	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
263	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
264	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
265	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
266	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
289	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
291	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
292	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
294	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
295	C₈	—	2	4	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
299	C₂²	#548×C₆₇	1	2	yes	—
300	C₂²	#575×C₆₇	1	2	yes	—
301	C₂²	#576×C₆₇	1	1	yes	characteristic
302	C₂²	#578×C₆₇	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
312	C₄	#519×C₆₇	1	1	yes	characteristic
313	C₄	#557×C₆₇	1	2	yes	—
314	C₂	C₆₇×#2351	1	1	yes	characteristic
315	C₂	C₆₇×#2911	1	1	yes	characteristic
316	C₂	C₆₇×#2913	1	1	yes	characteristic
317	C₂	C₆₇×#3370	1	2	yes	—
318	C₂	C₆₇×#3377	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. #446×C₆₇	C₄	semidirect
154. #22744	C₆₇	direct (N × H)
169. C₆₇	#22744	direct (N × H)
171. #446	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₆₇	2
C₆₇×#2355	2
C₆₇×#1949	2
C₆₇×#1947	2
C₆₇×#1836	2
C₆₇×#1835	2
C₆₇×#1776	2
C₄×#107×C₆₇	2
#22744	67

Subgroups — C67×#22744 #30701995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22744 #30701995