Subgroups — C₂×C₆₇×#2568 #30714195

Order 17152.

2614 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2568	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	4	yes	maximal
5	C₂×#391×C₆₇	C₂	1	2	yes	maximal
6	C₂×#391×C₆₇	C₂	1	4	yes	maximal
7	C₂×#382×C₆₇	C₂	1	2	yes	maximal
8	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₆₇	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	8	yes	—
13	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#400×C₆₇	C₂²	1	32	yes	—
24	#391×C₆₇	C₂²	1	32	yes	—
25	#391×C₆₇	C₂²	1	16	yes	—
26	#382×C₆₇	C₂²	1	16	yes	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
28	C₂×#95×C₆₇	C₂²	1	2	yes	—
29	C₂×#95×C₆₇	C₂²	1	8	yes	—
30	C₂×#95×C₆₇	C₂²	1	8	yes	—
31	C₂²×Q₈×C₆₇	—	2	4	no	—
32	C₂²×Q₈×C₆₇	—	2	8	no	—
33	C₂²×Q₈×C₆₇	—	2	4	no	—
34	C₂²×Q₈×C₆₇	—	2	16	no	—
35	C₂²×Q₈×C₆₇	Q₈	1	8	yes	complemented
36	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
37	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊Q₈×C₆₇	—	2	32	no	—
46	C₄×Q₈×C₆₇	—	2	32	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
57	C₂×C₄²×C₆₇	—	2	8	no	—
58	C₂×C₄²×C₆₇	—	2	8	no	—
59	C₂×C₄²×C₆₇	D₄	1	8	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
63	#95×C₆₇	C₂³	1	8	yes	—
64	#95×C₆₇	C₂³	1	32	yes	—
65	#95×C₆₇	C₂³	1	32	yes	—
66	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₂×Q₈×C₆₇	—	2	32	no	—
68	C₂×Q₈×C₆₇	—	2	8	no	—
69	C₂×Q₈×C₆₇	—	2	16	no	—
70	C₂×Q₈×C₆₇	—	2	8	no	—
71	C₂×Q₈×C₆₇	—	2	32	no	—
72	C₂×Q₈×C₆₇	—	4	32	no	—
73	C₂×Q₈×C₆₇	C₂×Q₈	1	16	yes	complemented
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	8	no	—
79	C₂²×C₄×C₆₇	—	2	8	no	—
80	C₂²×C₄×C₆₇	—	2	8	no	—
81	C₂²×C₄×C₆₇	—	2	16	no	—
82	C₂²×C₄×C₆₇	—	2	8	no	—
83	C₂²×C₄×C₆₇	—	2	8	no	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
88	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
89	C₂²×C₄×C₆₇	#41	1	4	yes	—
90	C₂²×C₄×C₆₇	#41	1	8	yes	—
91	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
93	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
94	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
98	C₄⋊C₄×C₆₇	—	2	16	no	—
99	C₄⋊C₄×C₆₇	—	2	8	no	—
100	C₄⋊C₄×C₆₇	—	2	16	no	—
101	C₄⋊C₄×C₆₇	—	2	16	no	—
102	C₄⋊C₄×C₆₇	—	2	16	no	—
103	C₄²×C₆₇	—	2	16	no	—
104	C₄²×C₆₇	—	2	16	no	—
105	C₄²×C₆₇	C₂×D₄	1	16	yes	—
106	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
107	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
108	C₂³×C₆₇	C₂×#41	1	2	yes	—
109	C₂³×C₆₇	C₂×#41	1	4	yes	—
110	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
111	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
112	Q₈×C₆₇	—	2	32	no	—
113	Q₈×C₆₇	—	4	32	no	—
114	C₂×C₄×C₆₇	—	2	16	no	—
115	C₂×C₄×C₆₇	—	2	8	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	4	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	8	no	—
121	C₂×C₄×C₆₇	—	2	8	no	—
122	C₂×C₄×C₆₇	—	2	8	no	—
123	C₂×C₄×C₆₇	—	2	16	no	—
124	C₂×C₄×C₆₇	—	2	8	no	—
125	C₂×C₄×C₆₇	—	2	8	no	—
126	C₂×C₄×C₆₇	—	4	16	no	—
127	C₂×C₄×C₆₇	—	4	8	no	—
128	C₂×C₄×C₆₇	—	4	16	no	—
129	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	—
130	C₂×C₄×C₆₇	#124	1	16	yes	—
131	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
132	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
133	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
134	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
135	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
136	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
137	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
138	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
139	C₂²×C₆₇	C₂×#124	1	4	yes	—
140	C₂²×C₆₇	#536	1	4	yes	—
141	C₂²×C₆₇	#556	1	8	yes	—
142	C₂²×C₆₇	#557	1	8	yes	—
143	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
144	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
145	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
146	C₂²×C₆₇	C₂²×#41	1	2	yes	—
147	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
148	C₄×C₆₇	—	2	8	no	—
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	—	4	8	no	—
151	C₄×C₆₇	—	4	4	no	—
152	C₄×C₆₇	—	4	8	no	—
153	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	—
154	C₄×C₆₇	C₂×#124	1	8	yes	—
155	C₂×#2568	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₆₇	#2568	1	8	yes	direct product factor
157	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
159	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
160	C₂×C₆₇	C₂×#556	1	2	yes	—
161	C₂×C₆₇	C₂×#557	1	2	yes	—
162	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
163	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
164	#2568	C₂×C₆₇	1	16	yes	direct product factor
165	C₂×#400	C₂×C₆₇	1	4	yes	—
166	C₂×#391	C₂×C₆₇	1	2	yes	—
167	C₂×#391	C₂×C₆₇	1	4	yes	—
168	C₂×#382	C₂×C₆₇	1	2	yes	—
169	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
170	C₆₇	C₂×#2568	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
172	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
173	C₂×C₄⋊Q₈	—	2	8	no	—
174	C₂×C₄⋊Q₈	C₂²×C₆₇	1	8	yes	—
175	C₂×C₄×Q₈	—	2	8	no	—
176	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	—
177	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
178	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
179	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
181	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
182	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
183	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
184	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
185	#400	C₂²×C₆₇	1	32	yes	—
186	#391	C₂²×C₆₇	1	32	yes	—
187	#391	C₂²×C₆₇	1	16	yes	—
188	#382	C₂²×C₆₇	1	16	yes	—
189	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
190	C₂×#95	C₂²×C₆₇	1	2	yes	—
191	C₂×#95	C₂²×C₆₇	1	8	yes	—
192	C₂×#95	C₂²×C₆₇	1	8	yes	—
193	C₂²×Q₈	—	2	4	no	—
194	C₂²×Q₈	—	2	8	no	—
195	C₂²×Q₈	—	2	4	no	—
196	C₂²×Q₈	—	2	16	no	—
197	C₂²×Q₈	Q₈×C₆₇	1	8	yes	complemented
198	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
199	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
200	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
201	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
202	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
203	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
205	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
206	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
207	C₄⋊Q₈	—	2	32	no	—
208	C₄×Q₈	—	2	32	no	—
209	C₂×C₄⋊C₄	—	2	8	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	8	no	—
212	C₂×C₄⋊C₄	—	2	8	no	—
213	C₂×C₄⋊C₄	—	2	8	no	—
214	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
215	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
216	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
217	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
218	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
219	C₂×C₄²	—	2	8	no	—
220	C₂×C₄²	—	2	8	no	—
221	C₂×C₄²	D₄×C₆₇	1	8	yes	—
222	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
223	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
224	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
225	#95	C₂³×C₆₇	1	8	yes	—
226	#95	C₂³×C₆₇	1	32	yes	—
227	#95	C₂³×C₆₇	1	32	yes	—
228	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
229	C₂×Q₈	—	2	32	no	—
230	C₂×Q₈	—	2	8	no	—
231	C₂×Q₈	—	2	16	no	—
232	C₂×Q₈	—	2	8	no	—
233	C₂×Q₈	—	2	32	no	—
234	C₂×Q₈	—	4	32	no	—
235	C₂×Q₈	C₂×Q₈×C₆₇	1	16	yes	complemented
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	16	no	—
244	C₂²×C₄	—	2	8	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
247	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
248	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
249	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
250	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
251	C₂²×C₄	#41×C₆₇	1	4	yes	—
252	C₂²×C₄	#41×C₆₇	1	8	yes	—
253	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
254	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
255	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
256	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
258	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
259	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
260	C₄⋊C₄	—	2	16	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₄⋊C₄	—	2	16	no	—
263	C₄⋊C₄	—	2	16	no	—
264	C₄⋊C₄	—	2	16	no	—
265	C₄²	—	2	16	no	—
266	C₄²	—	2	16	no	—
267	C₄²	C₂×D₄×C₆₇	1	16	yes	—
268	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
269	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
270	C₂³	C₂×#41×C₆₇	1	2	yes	—
271	C₂³	C₂×#41×C₆₇	1	4	yes	—
272	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
273	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
274	Q₈	—	2	32	no	—
275	Q₈	—	4	32	no	—
276	C₂×C₄	—	2	16	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	16	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	4	16	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	—
292	C₂×C₄	#124×C₆₇	1	16	yes	—
293	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
294	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
295	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
296	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
297	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
298	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
299	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
300	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
301	C₂²	C₂×#124×C₆₇	1	4	yes	—
302	C₂²	#536×C₆₇	1	4	yes	—
303	C₂²	#556×C₆₇	1	8	yes	—
304	C₂²	#557×C₆₇	1	8	yes	—
305	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
306	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
307	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
308	C₂²	C₂²×#41×C₆₇	1	2	yes	—
309	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
310	C₄	—	2	8	no	—
311	C₄	—	2	4	no	—
312	C₄	—	4	8	no	—
313	C₄	—	4	4	no	—
314	C₄	—	4	8	no	—
315	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	—
316	C₄	C₂×#124×C₆₇	1	8	yes	—
317	C₂	C₆₇×#2568	1	8	yes	direct product factor
318	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
319	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
320	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
321	C₂	C₂×#556×C₆₇	1	2	yes	—
322	C₂	C₂×#557×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2568	C₂	direct (N × H)
35. C₂²×Q₈×C₆₇	Q₈	semidirect
73. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
155. C₂×#2568	C₆₇	direct (N × H)
156. C₂×C₆₇	#2568	direct (N × H)
164. #2568	C₂×C₆₇	direct (N × H)
170. C₆₇	C₂×#2568	direct (N × H)
197. C₂²×Q₈	Q₈×C₆₇	semidirect
235. C₂×Q₈	C₂×Q₈×C₆₇	semidirect
317. C₂	C₆₇×#2568	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₆₇×#2568	2
C₂×#400×C₆₇	2
C₂×#391×C₆₇	2
C₂×#382×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2568	67

Subgroups — C2×C67×#2568 #30714195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2568 #30714195