Subgroups — C₆₇×#22440 #30701691

Order 17152.

1386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#441×C₆₇	C₂	1	2	yes	maximal, complemented
3	Q₈×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#1971	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1878	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1787	C₂	1	2	yes	maximal
8	C₆₇×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₆₇	—	2	8	no	—
11	C₂²×Q₁₆×C₆₇	C₄	1	8	yes	complemented
12	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
17	#441×C₆₇	—	2	8	no	—
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	C₂×#103×C₆₇	C₂²	1	4	yes	—
20	C₂×#103×C₆₇	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₆₇	C₂²	1	2	yes	—
23	#391×C₆₇	—	2	4	no	—
24	#391×C₆₇	C₂²	1	4	yes	—
25	#380×C₆₇	—	2	2	no	—
26	#380×C₆₇	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
28	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#340×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	C₂²×Q₈×C₆₇	—	2	2	no	—
32	C₂²×Q₈×C₆₇	—	2	4	no	—
33	C₂²×Q₈×C₆₇	—	2	8	no	—
34	C₂²×Q₈×C₆₇	—	4	4	no	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
36	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₄×C₆₇	—	2	2	no	—
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×Q₁₆×C₆₇	—	2	8	no	—
41	C₂×Q₁₆×C₆₇	—	2	32	no	—
42	C₂×Q₁₆×C₆₇	—	2	4	no	—
43	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	—
44	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×Q₁₆×C₆₇	D₄	1	8	yes	—
47	C₂²×C₈×C₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₆₇	—	2	8	no	—
53	C₄×Q₈×C₆₇	—	4	4	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
58	C₂×C₄²×C₆₇	—	2	4	no	—
59	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
60	#106×C₆₇	—	2	4	no	—
61	#103×C₆₇	—	2	8	no	—
62	#103×C₆₇	—	4	4	no	—
63	C₂²⋊C₈×C₆₇	—	2	2	no	—
64	#97×C₆₇	—	2	4	no	—
65	#95×C₆₇	—	2	4	no	—
66	#95×C₆₇	—	2	4	no	—
67	#95×C₆₇	—	2	2	no	—
68	#95×C₆₇	D₄	1	2	yes	—
69	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×Q₈×C₆₇	—	2	8	no	—
71	C₂×Q₈×C₆₇	—	2	2	no	—
72	C₂×Q₈×C₆₇	—	2	2	no	—
73	C₂×Q₈×C₆₇	—	2	4	no	—
74	C₂×Q₈×C₆₇	—	2	16	no	—
75	C₂×Q₈×C₆₇	—	2	4	no	—
76	C₂×Q₈×C₆₇	—	2	4	no	—
77	C₂×Q₈×C₆₇	—	4	8	no	—
78	C₂×Q₈×C₆₇	—	4	8	no	—
79	C₂×Q₈×C₆₇	—	4	16	no	—
80	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
81	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
82	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	8	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	4	8	no	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	#41	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	Q₁₆×C₆₇	—	2	16	no	—
96	Q₁₆×C₆₇	—	2	8	no	—
97	Q₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	8	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	4	8	no	—
109	C₂²⋊C₄×C₆₇	—	2	4	no	—
110	C₂²⋊C₄×C₆₇	—	4	4	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	—	4	8	no	—
114	C₂³×C₆₇	—	2	2	no	—
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	C₄×D₄	1	2	yes	—
117	C₂³×C₆₇	#137	1	4	yes	—
118	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₆₇	—	2	4	no	—
120	Q₈×C₆₇	—	2	8	no	—
121	Q₈×C₆₇	—	2	4	no	—
122	Q₈×C₆₇	—	2	4	no	—
123	Q₈×C₆₇	—	4	16	no	—
124	Q₈×C₆₇	—	4	4	no	—
125	Q₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	8	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	4	16	no	—
137	C₂×C₄×C₆₇	—	4	4	no	—
138	C₂×C₄×C₆₇	—	4	4	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
146	C₈×C₆₇	—	2	2	no	—
147	C₈×C₆₇	—	4	2	no	—
148	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	8	no	—
152	C₂²×C₆₇	#441	1	4	yes	complemented
153	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂²×C₆₇	D₄²	1	2	yes	—
155	C₂²×C₆₇	C₂×#137	1	2	yes	—
156	C₂²×C₆₇	#578	1	2	yes	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	8	no	—
159	C₄×C₆₇	—	4	4	no	—
160	C₄×C₆₇	—	4	4	no	—
161	C₄×C₆₇	—	8	4	no	—
162	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₆₇	#518	1	1	yes	characteristic
164	C₄×C₆₇	#546	1	2	yes	—
165	#22440	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂×C₆₇	—	2	4	no	—
167	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
169	C₂×C₆₇	#2950	1	1	yes	characteristic
170	C₂×C₆₇	#3270	1	4	yes	—
171	C₂³×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
172	C₂×#441	C₂×C₆₇	1	2	yes	complemented
173	Q₈×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
174	#1971	C₂×C₆₇	1	2	yes	complemented
175	#1878	C₂×C₆₇	1	4	yes	complemented
176	#1807	C₂×C₆₇	1	1	yes	characteristic
177	#1787	C₂×C₆₇	1	2	yes	—
178	#1752	C₂×C₆₇	1	1	yes	characteristic
179	C₆₇	#22440	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
181	C₂²×Q₁₆	—	2	8	no	—
182	C₂²×Q₁₆	C₄×C₆₇	1	8	yes	complemented
183	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
184	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
185	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
186	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
187	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
188	#441	—	2	8	no	—
189	C₂×#106	C₂²×C₆₇	1	2	yes	—
190	C₂×#103	C₂²×C₆₇	1	4	yes	—
191	C₂×#103	C₂²×C₆₇	1	4	yes	—
192	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
193	C₂×#97	C₂²×C₆₇	1	2	yes	—
194	#391	—	2	4	no	—
195	#391	C₂²×C₆₇	1	4	yes	—
196	#380	—	2	2	no	—
197	#380	C₂²×C₆₇	1	2	yes	—
198	C₄×C₂²⋊C₄	—	2	2	no	—
199	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
200	#340	C₂²×C₆₇	1	2	yes	—
201	#336	C₂²×C₆₇	1	2	yes	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂²×Q₈	—	2	4	no	—
204	C₂²×Q₈	—	2	8	no	—
205	C₂²×Q₈	—	4	4	no	—
206	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
207	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
208	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
209	C₂³×C₄	—	2	2	no	—
210	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
211	C₂×Q₁₆	—	2	8	no	—
212	C₂×Q₁₆	—	2	32	no	—
213	C₂×Q₁₆	—	2	4	no	—
214	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	—
215	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
216	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	—
217	C₂×Q₁₆	D₄×C₆₇	1	8	yes	—
218	C₂²×C₈	—	2	4	no	—
219	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
220	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
221	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
223	C₄×Q₈	—	2	8	no	—
224	C₄×Q₈	—	4	4	no	—
225	C₂×C₄⋊C₄	—	2	4	no	—
226	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
229	C₂×C₄²	—	2	4	no	—
230	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
231	#106	—	2	4	no	—
232	#103	—	2	8	no	—
233	#103	—	4	4	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	#97	—	2	4	no	—
236	#95	—	2	4	no	—
237	#95	—	2	4	no	—
238	#95	—	2	2	no	—
239	#95	D₄×C₆₇	1	2	yes	—
240	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	16	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	4	8	no	—
249	C₂×Q₈	—	4	8	no	—
250	C₂×Q₈	—	4	16	no	—
251	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
252	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
253	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	8	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	4	8	no	—
261	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
263	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₄	#41×C₆₇	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
266	Q₁₆	—	2	16	no	—
267	Q₁₆	—	2	8	no	—
268	Q₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	8	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
275	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
276	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	4	8	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	4	8	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	C₄×D₄×C₆₇	1	2	yes	—
288	C₂³	#137×C₆₇	1	4	yes	—
289	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
290	Q₈	—	2	4	no	—
291	Q₈	—	2	8	no	—
292	Q₈	—	2	4	no	—
293	Q₈	—	2	4	no	—
294	Q₈	—	4	16	no	—
295	Q₈	—	4	4	no	—
296	Q₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	16	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
313	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
315	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
316	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
320	C₂²	—	2	8	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	#441×C₆₇	1	4	yes	complemented
324	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
325	C₂²	D₄²×C₆₇	1	2	yes	—
326	C₂²	C₂×#137×C₆₇	1	2	yes	—
327	C₂²	#578×C₆₇	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	8	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	4	no	—
332	C₄	—	8	4	no	—
333	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
334	C₄	#518×C₆₇	1	1	yes	characteristic
335	C₄	#546×C₆₇	1	2	yes	—
336	C₂	—	2	4	no	—
337	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
338	C₂	C₂×#441×C₆₇	1	1	yes	characteristic
339	C₂	C₆₇×#2950	1	1	yes	characteristic
340	C₂	C₆₇×#3270	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#441×C₆₇	C₂	semidirect
4. C₆₇×#1971	C₂	semidirect
5. C₆₇×#1878	C₂	semidirect
11. C₂²×Q₁₆×C₆₇	C₄	semidirect
97. Q₁₆×C₆₇	C₂²⋊C₄	semidirect
152. C₂²×C₆₇	#441	semidirect
165. #22440	C₆₇	direct (N × H)
172. C₂×#441	C₂×C₆₇	semidirect
174. #1971	C₂×C₆₇	semidirect
175. #1878	C₂×C₆₇	semidirect
179. C₆₇	#22440	direct (N × H)
182. C₂²×Q₁₆	C₄×C₆₇	semidirect
268. Q₁₆	C₂²⋊C₄×C₆₇	semidirect
323. C₂²	#441×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₆₇	2
C₂×#441×C₆₇	2
Q₈×C₂²⋊C₄×C₆₇	2
C₆₇×#1971	2
C₆₇×#1878	2
C₆₇×#1807	2
C₆₇×#1787	2
C₆₇×#1752	2
#22440	67

Subgroups — C67×#22440 #30701691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22440 #30701691