Subgroups — C₂×C₄×C₂₇×C₆₄ #72191997

Order 13824.

648 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₇×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₇×C₆₄	C₂	1	2	yes	maximal
3	C₄×C₂₇×C₆₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₉×C₆₄	C₃	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₆×C₂₇	C₄	1	1	yes	characteristic
6	C₂²×C₂₇×C₃₂	C₄	1	1	yes	characteristic
7	C₂²×C₂₇×C₃₂	C₂²	1	1	yes	characteristic
8	C₄×C₂₇×C₃₂	C₄	1	2	yes	—
9	C₄×C₂₇×C₃₂	C₂²	1	2	yes	—
10	C₂×C₂₇×C₆₄	C₄	1	8	yes	direct product factor
11	C₂×C₂₇×C₆₄	C₂²	1	4	yes	—
12	C₂×C₄×C₉×C₃₂	C₂×C₃	1	1	yes	characteristic
13	C₂²×C₉×C₆₄	C₂×C₃	1	2	yes	—
14	C₄×C₉×C₆₄	C₂×C₃	1	4	yes	—
15	C₂×C₃×C₄×C₆₄	C₉	1	1	yes	characteristic
16	C₂²×C₁₆×C₂₇	C₈	1	1	yes	characteristic
17	C₂²×C₁₆×C₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₇	C₈	1	1	yes	characteristic
19	C₂×C₂₇×C₃₂	C₈	1	4	yes	—
20	C₂×C₂₇×C₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂×C₂₇×C₃₂	C₂×C₄	1	2	yes	—
22	C₂×C₂₇×C₃₂	C₂×C₄	1	4	yes	—
23	C₂×C₂₇×C₃₂	C₂³	1	1	yes	characteristic
24	C₄×C₁₆×C₂₇	C₈	1	2	yes	—
25	C₄×C₁₆×C₂₇	C₂×C₄	1	2	yes	—
26	C₂₇×C₆₄	C₂×C₄	1	8	yes	direct product factor
27	C₂²×C₉×C₃₂	C₃×C₄	1	1	yes	characteristic
28	C₂²×C₉×C₃₂	C₂²×C₃	1	1	yes	characteristic
29	C₂×C₄×C₉×C₁₆	C₃×C₄	1	1	yes	characteristic
30	C₂×C₉×C₆₄	C₃×C₄	1	8	yes	—
31	C₂×C₉×C₆₄	C₂²×C₃	1	4	yes	—
32	C₄×C₉×C₃₂	C₃×C₄	1	2	yes	—
33	C₄×C₉×C₃₂	C₂²×C₃	1	2	yes	—
34	C₂²×C₈×C₂₇	C₁₆	1	1	yes	characteristic
35	C₂²×C₈×C₂₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄²×C₂₇	C₁₆	1	1	yes	characteristic
37	C₂×C₁₆×C₂₇	C₁₆	1	4	yes	—
38	C₂×C₁₆×C₂₇	C₄²	1	2	yes	—
39	C₂×C₁₆×C₂₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₁₆×C₂₇	C₂×C₈	1	2	yes	—
41	C₂×C₁₆×C₂₇	C₂×C₈	1	2	yes	—
42	C₂×C₁₆×C₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₈×C₂₇	C₁₆	1	2	yes	—
44	C₄×C₈×C₂₇	C₂×C₈	1	2	yes	—
45	C₂₇×C₃₂	C₄²	1	2	yes	—
46	C₂₇×C₃₂	C₂×C₈	1	4	yes	—
47	C₂₇×C₃₂	C₂²×C₄	1	2	yes	—
48	C₂²×C₃×C₆₄	C₂×C₉	1	2	yes	—
49	C₂×C₃×C₄×C₃₂	C₂×C₉	1	1	yes	characteristic
50	C₃×C₄×C₆₄	C₂×C₉	1	4	yes	—
51	C₂²×C₉×C₁₆	C₃×C₈	1	1	yes	characteristic
52	C₂²×C₉×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈×C₉	C₃×C₈	1	1	yes	characteristic
54	C₂×C₉×C₃₂	C₃×C₈	1	4	yes	—
55	C₂×C₉×C₃₂	C₂×C₃×C₄	1	1	yes	characteristic
56	C₂×C₉×C₃₂	C₂×C₃×C₄	1	2	yes	—
57	C₂×C₉×C₃₂	C₂×C₃×C₄	1	4	yes	—
58	C₂×C₉×C₃₂	C₂³×C₃	1	1	yes	characteristic
59	C₄×C₉×C₁₆	C₃×C₈	1	2	yes	—
60	C₄×C₉×C₁₆	C₂×C₃×C₄	1	2	yes	—
61	C₉×C₆₄	C₂×C₃×C₄	1	8	yes	—
62	C₂×C₄×C₆₄	C₂₇	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²×C₄×C₂₇	C₃₂	1	2	yes	—
64	C₂²×C₄×C₂₇	C₂×C₁₆	1	1	yes	characteristic
65	C₂×C₈×C₂₇	C₃₂	1	4	yes	—
66	C₂×C₈×C₂₇	C₄×C₈	1	2	yes	—
67	C₂×C₈×C₂₇	C₂×C₁₆	1	1	yes	characteristic
68	C₂×C₈×C₂₇	C₂×C₁₆	1	2	yes	—
69	C₂×C₈×C₂₇	C₂×C₁₆	1	2	yes	—
70	C₂×C₈×C₂₇	C₂²×C₈	1	1	yes	characteristic
71	C₄²×C₂₇	C₃₂	1	2	yes	—
72	C₄²×C₂₇	C₂×C₁₆	1	2	yes	—
73	C₁₆×C₂₇	C₄×C₈	1	2	yes	—
74	C₁₆×C₂₇	C₂×C₁₆	1	4	yes	—
75	C₁₆×C₂₇	C₂×C₄²	1	1	yes	characteristic
76	C₁₆×C₂₇	C₂²×C₈	1	1	yes	characteristic
77	C₂²×C₃×C₃₂	C₄×C₉	1	1	yes	characteristic
78	C₂²×C₃×C₃₂	C₂²×C₉	1	1	yes	characteristic
79	C₂×C₃×C₄×C₁₆	C₄×C₉	1	1	yes	characteristic
80	C₂×C₃×C₆₄	C₄×C₉	1	8	yes	—
81	C₂×C₃×C₆₄	C₂²×C₉	1	4	yes	—
82	C₃×C₄×C₃₂	C₄×C₉	1	2	yes	—
83	C₃×C₄×C₃₂	C₂²×C₉	1	2	yes	—
84	C₂²×C₈×C₉	C₃×C₁₆	1	1	yes	characteristic
85	C₂²×C₈×C₉	C₂×C₃×C₈	1	1	yes	characteristic
86	C₂×C₄²×C₉	C₃×C₁₆	1	1	yes	characteristic
87	C₂×C₉×C₁₆	C₃×C₁₆	1	4	yes	—
88	C₂×C₉×C₁₆	C₃×C₄²	1	2	yes	—
89	C₂×C₉×C₁₆	C₂×C₃×C₈	1	1	yes	characteristic
90	C₂×C₉×C₁₆	C₂×C₃×C₈	1	2	yes	—
91	C₂×C₉×C₁₆	C₂×C₃×C₈	1	2	yes	—
92	C₂×C₉×C₁₆	C₂²×C₃×C₄	1	1	yes	characteristic
93	C₄×C₈×C₉	C₃×C₁₆	1	2	yes	—
94	C₄×C₈×C₉	C₂×C₃×C₈	1	2	yes	—
95	C₉×C₃₂	C₃×C₄²	1	2	yes	—
96	C₉×C₃₂	C₂×C₃×C₈	1	4	yes	—
97	C₉×C₃₂	C₂²×C₃×C₄	1	2	yes	—
98	C₂²×C₆₄	C₂×C₂₇	1	2	yes	—
99	C₂×C₄×C₃₂	C₂×C₂₇	1	1	yes	characteristic
100	C₄×C₆₄	C₂×C₂₇	1	4	yes	direct product factor
101	C₂³×C₂₇	C₂×C₃₂	1	1	yes	characteristic
102	C₂×C₄×C₂₇	C₆₄	1	8	yes	direct product factor
103	C₂×C₄×C₂₇	C₄×C₁₆	1	2	yes	—
104	C₂×C₄×C₂₇	C₂×C₃₂	1	1	yes	characteristic
105	C₂×C₄×C₂₇	C₂×C₃₂	1	2	yes	—
106	C₂×C₄×C₂₇	C₂×C₃₂	1	4	yes	—
107	C₂×C₄×C₂₇	C₂²×C₁₆	1	1	yes	characteristic
108	C₈×C₂₇	C₄×C₁₆	1	2	yes	—
109	C₈×C₂₇	C₂×C₃₂	1	4	yes	—
110	C₈×C₂₇	C₂×C₄×C₈	1	1	yes	characteristic
111	C₈×C₂₇	C₂²×C₁₆	1	1	yes	characteristic
112	C₂²×C₃×C₁₆	C₈×C₉	1	1	yes	characteristic
113	C₂²×C₃×C₁₆	C₂×C₄×C₉	1	1	yes	characteristic
114	C₂×C₃×C₄×C₈	C₈×C₉	1	1	yes	characteristic
115	C₂×C₃×C₃₂	C₈×C₉	1	4	yes	—
116	C₂×C₃×C₃₂	C₂×C₄×C₉	1	1	yes	characteristic
117	C₂×C₃×C₃₂	C₂×C₄×C₉	1	2	yes	—
118	C₂×C₃×C₃₂	C₂×C₄×C₉	1	4	yes	—
119	C₂×C₃×C₃₂	C₂³×C₉	1	1	yes	characteristic
120	C₃×C₄×C₁₆	C₈×C₉	1	2	yes	—
121	C₃×C₄×C₁₆	C₂×C₄×C₉	1	2	yes	—
122	C₃×C₆₄	C₂×C₄×C₉	1	8	yes	—
123	C₂²×C₄×C₉	C₃×C₃₂	1	2	yes	—
124	C₂²×C₄×C₉	C₂×C₃×C₁₆	1	1	yes	characteristic
125	C₂×C₈×C₉	C₃×C₃₂	1	4	yes	—
126	C₂×C₈×C₉	C₃×C₄×C₈	1	2	yes	—
127	C₂×C₈×C₉	C₂×C₃×C₁₆	1	1	yes	characteristic
128	C₂×C₈×C₉	C₂×C₃×C₁₆	1	2	yes	—
129	C₂×C₈×C₉	C₂×C₃×C₁₆	1	2	yes	—
130	C₂×C₈×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
131	C₄²×C₉	C₃×C₃₂	1	2	yes	—
132	C₄²×C₉	C₂×C₃×C₁₆	1	2	yes	—
133	C₉×C₁₆	C₃×C₄×C₈	1	2	yes	—
134	C₉×C₁₆	C₂×C₃×C₁₆	1	4	yes	—
135	C₉×C₁₆	C₂×C₃×C₄²	1	1	yes	characteristic
136	C₉×C₁₆	C₂²×C₃×C₈	1	1	yes	characteristic
137	C₂²×C₃₂	C₄×C₂₇	1	1	yes	characteristic
138	C₂²×C₃₂	C₂²×C₂₇	1	1	yes	characteristic
139	C₂×C₄×C₁₆	C₄×C₂₇	1	1	yes	characteristic
140	C₂×C₆₄	C₄×C₂₇	1	8	yes	direct product factor
141	C₂×C₆₄	C₂²×C₂₇	1	4	yes	—
142	C₄×C₃₂	C₄×C₂₇	1	2	yes	—
143	C₄×C₃₂	C₂²×C₂₇	1	2	yes	—
144	C₂²×C₂₇	C₄×C₃₂	1	2	yes	—
145	C₂²×C₂₇	C₂×C₆₄	1	4	yes	—
146	C₂²×C₂₇	C₂²×C₃₂	1	1	yes	characteristic
147	C₄×C₂₇	C₄×C₃₂	1	2	yes	—
148	C₄×C₂₇	C₂×C₆₄	1	8	yes	direct product factor
149	C₄×C₂₇	C₂×C₄×C₁₆	1	1	yes	characteristic
150	C₄×C₂₇	C₂²×C₃₂	1	1	yes	characteristic
151	C₂²×C₃×C₈	C₉×C₁₆	1	1	yes	characteristic
152	C₂²×C₃×C₈	C₂×C₈×C₉	1	1	yes	characteristic
153	C₂×C₃×C₄²	C₉×C₁₆	1	1	yes	characteristic
154	C₂×C₃×C₁₆	C₉×C₁₆	1	4	yes	—
155	C₂×C₃×C₁₆	C₄²×C₉	1	2	yes	—
156	C₂×C₃×C₁₆	C₂×C₈×C₉	1	1	yes	characteristic
157	C₂×C₃×C₁₆	C₂×C₈×C₉	1	2	yes	—
158	C₂×C₃×C₁₆	C₂×C₈×C₉	1	2	yes	—
159	C₂×C₃×C₁₆	C₂²×C₄×C₉	1	1	yes	characteristic
160	C₃×C₄×C₈	C₉×C₁₆	1	2	yes	—
161	C₃×C₄×C₈	C₂×C₈×C₉	1	2	yes	—
162	C₃×C₃₂	C₄²×C₉	1	2	yes	—
163	C₃×C₃₂	C₂×C₈×C₉	1	4	yes	—
164	C₃×C₃₂	C₂²×C₄×C₉	1	2	yes	—
165	C₂³×C₉	C₂×C₃×C₃₂	1	1	yes	characteristic
166	C₂×C₄×C₉	C₃×C₆₄	1	8	yes	—
167	C₂×C₄×C₉	C₃×C₄×C₁₆	1	2	yes	—
168	C₂×C₄×C₉	C₂×C₃×C₃₂	1	1	yes	characteristic
169	C₂×C₄×C₉	C₂×C₃×C₃₂	1	2	yes	—
170	C₂×C₄×C₉	C₂×C₃×C₃₂	1	4	yes	—
171	C₂×C₄×C₉	C₂²×C₃×C₁₆	1	1	yes	characteristic
172	C₈×C₉	C₃×C₄×C₁₆	1	2	yes	—
173	C₈×C₉	C₂×C₃×C₃₂	1	4	yes	—
174	C₈×C₉	C₂×C₃×C₄×C₈	1	1	yes	characteristic
175	C₈×C₉	C₂²×C₃×C₁₆	1	1	yes	characteristic
176	C₂²×C₁₆	C₈×C₂₇	1	1	yes	characteristic
177	C₂²×C₁₆	C₂×C₄×C₂₇	1	1	yes	characteristic
178	C₂×C₄×C₈	C₈×C₂₇	1	1	yes	characteristic
179	C₂×C₃₂	C₈×C₂₇	1	4	yes	—
180	C₂×C₃₂	C₂×C₄×C₂₇	1	1	yes	characteristic
181	C₂×C₃₂	C₂×C₄×C₂₇	1	2	yes	—
182	C₂×C₃₂	C₂×C₄×C₂₇	1	4	yes	—
183	C₂×C₃₂	C₂³×C₂₇	1	1	yes	characteristic
184	C₄×C₁₆	C₈×C₂₇	1	2	yes	—
185	C₄×C₁₆	C₂×C₄×C₂₇	1	2	yes	—
186	C₆₄	C₂×C₄×C₂₇	1	8	yes	direct product factor
187	C₂×C₂₇	C₄×C₆₄	1	4	yes	direct product factor
188	C₂×C₂₇	C₂×C₄×C₃₂	1	1	yes	characteristic
189	C₂×C₂₇	C₂²×C₆₄	1	2	yes	—
190	C₂²×C₃×C₄	C₉×C₃₂	1	2	yes	—
191	C₂²×C₃×C₄	C₂×C₉×C₁₆	1	1	yes	characteristic
192	C₂×C₃×C₈	C₉×C₃₂	1	4	yes	—
193	C₂×C₃×C₈	C₄×C₈×C₉	1	2	yes	—
194	C₂×C₃×C₈	C₂×C₉×C₁₆	1	1	yes	characteristic
195	C₂×C₃×C₈	C₂×C₉×C₁₆	1	2	yes	—
196	C₂×C₃×C₈	C₂×C₉×C₁₆	1	2	yes	—
197	C₂×C₃×C₈	C₂²×C₈×C₉	1	1	yes	characteristic
198	C₃×C₄²	C₉×C₃₂	1	2	yes	—
199	C₃×C₄²	C₂×C₉×C₁₆	1	2	yes	—
200	C₃×C₁₆	C₄×C₈×C₉	1	2	yes	—
201	C₃×C₁₆	C₂×C₉×C₁₆	1	4	yes	—
202	C₃×C₁₆	C₂×C₄²×C₉	1	1	yes	characteristic
203	C₃×C₁₆	C₂²×C₈×C₉	1	1	yes	characteristic
204	C₂²×C₉	C₃×C₄×C₃₂	1	2	yes	—
205	C₂²×C₉	C₂×C₃×C₆₄	1	4	yes	—
206	C₂²×C₉	C₂²×C₃×C₃₂	1	1	yes	characteristic
207	C₄×C₉	C₃×C₄×C₃₂	1	2	yes	—
208	C₄×C₉	C₂×C₃×C₆₄	1	8	yes	—
209	C₄×C₉	C₂×C₃×C₄×C₁₆	1	1	yes	characteristic
210	C₄×C₉	C₂²×C₃×C₃₂	1	1	yes	characteristic
211	C₂²×C₈	C₁₆×C₂₇	1	1	yes	characteristic
212	C₂²×C₈	C₂×C₈×C₂₇	1	1	yes	characteristic
213	C₂×C₄²	C₁₆×C₂₇	1	1	yes	characteristic
214	C₂×C₁₆	C₁₆×C₂₇	1	4	yes	—
215	C₂×C₁₆	C₄²×C₂₇	1	2	yes	—
216	C₂×C₁₆	C₂×C₈×C₂₇	1	1	yes	characteristic
217	C₂×C₁₆	C₂×C₈×C₂₇	1	2	yes	—
218	C₂×C₁₆	C₂×C₈×C₂₇	1	2	yes	—
219	C₂×C₁₆	C₂²×C₄×C₂₇	1	1	yes	characteristic
220	C₄×C₈	C₁₆×C₂₇	1	2	yes	—
221	C₄×C₈	C₂×C₈×C₂₇	1	2	yes	—
222	C₃₂	C₄²×C₂₇	1	2	yes	—
223	C₃₂	C₂×C₈×C₂₇	1	4	yes	—
224	C₃₂	C₂²×C₄×C₂₇	1	2	yes	—
225	C₂₇	C₂×C₄×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³×C₃	C₂×C₉×C₃₂	1	1	yes	characteristic
227	C₂×C₃×C₄	C₉×C₆₄	1	8	yes	—
228	C₂×C₃×C₄	C₄×C₉×C₁₆	1	2	yes	—
229	C₂×C₃×C₄	C₂×C₉×C₃₂	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₉×C₃₂	1	2	yes	—
231	C₂×C₃×C₄	C₂×C₉×C₃₂	1	4	yes	—
232	C₂×C₃×C₄	C₂²×C₉×C₁₆	1	1	yes	characteristic
233	C₃×C₈	C₄×C₉×C₁₆	1	2	yes	—
234	C₃×C₈	C₂×C₉×C₃₂	1	4	yes	—
235	C₃×C₈	C₂×C₄×C₈×C₉	1	1	yes	characteristic
236	C₃×C₈	C₂²×C₉×C₁₆	1	1	yes	characteristic
237	C₂×C₉	C₃×C₄×C₆₄	1	4	yes	—
238	C₂×C₉	C₂×C₃×C₄×C₃₂	1	1	yes	characteristic
239	C₂×C₉	C₂²×C₃×C₆₄	1	2	yes	—
240	C₂²×C₄	C₂₇×C₃₂	1	2	yes	—
241	C₂²×C₄	C₂×C₁₆×C₂₇	1	1	yes	characteristic
242	C₂×C₈	C₂₇×C₃₂	1	4	yes	—
243	C₂×C₈	C₄×C₈×C₂₇	1	2	yes	—
244	C₂×C₈	C₂×C₁₆×C₂₇	1	1	yes	characteristic
245	C₂×C₈	C₂×C₁₆×C₂₇	1	2	yes	—
246	C₂×C₈	C₂×C₁₆×C₂₇	1	2	yes	—
247	C₂×C₈	C₂²×C₈×C₂₇	1	1	yes	characteristic
248	C₄²	C₂₇×C₃₂	1	2	yes	—
249	C₄²	C₂×C₁₆×C₂₇	1	2	yes	—
250	C₁₆	C₄×C₈×C₂₇	1	2	yes	—
251	C₁₆	C₂×C₁₆×C₂₇	1	4	yes	—
252	C₁₆	C₂×C₄²×C₂₇	1	1	yes	characteristic
253	C₁₆	C₂²×C₈×C₂₇	1	1	yes	characteristic
254	C₂²×C₃	C₄×C₉×C₃₂	1	2	yes	—
255	C₂²×C₃	C₂×C₉×C₆₄	1	4	yes	—
256	C₂²×C₃	C₂²×C₉×C₃₂	1	1	yes	characteristic
257	C₃×C₄	C₄×C₉×C₃₂	1	2	yes	—
258	C₃×C₄	C₂×C₉×C₆₄	1	8	yes	—
259	C₃×C₄	C₂×C₄×C₉×C₁₆	1	1	yes	characteristic
260	C₃×C₄	C₂²×C₉×C₃₂	1	1	yes	characteristic
261	C₉	C₂×C₃×C₄×C₆₄	1	1	yes	characteristic
262	C₂³	C₂×C₂₇×C₃₂	1	1	yes	characteristic
263	C₂×C₄	C₂₇×C₆₄	1	8	yes	direct product factor
264	C₂×C₄	C₄×C₁₆×C₂₇	1	2	yes	—
265	C₂×C₄	C₂×C₂₇×C₃₂	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂₇×C₃₂	1	2	yes	—
267	C₂×C₄	C₂×C₂₇×C₃₂	1	4	yes	—
268	C₂×C₄	C₂²×C₁₆×C₂₇	1	1	yes	characteristic
269	C₈	C₄×C₁₆×C₂₇	1	2	yes	—
270	C₈	C₂×C₂₇×C₃₂	1	4	yes	—
271	C₈	C₂×C₄×C₈×C₂₇	1	1	yes	characteristic
272	C₈	C₂²×C₁₆×C₂₇	1	1	yes	characteristic
273	C₂×C₃	C₄×C₉×C₆₄	1	4	yes	—
274	C₂×C₃	C₂²×C₉×C₆₄	1	2	yes	—
275	C₂×C₃	C₂×C₄×C₉×C₃₂	1	1	yes	characteristic
276	C₂²	C₂×C₂₇×C₆₄	1	4	yes	—
277	C₂²	C₄×C₂₇×C₃₂	1	2	yes	—
278	C₂²	C₂²×C₂₇×C₃₂	1	1	yes	characteristic
279	C₄	C₂×C₂₇×C₆₄	1	8	yes	direct product factor
280	C₄	C₄×C₂₇×C₃₂	1	2	yes	—
281	C₄	C₂²×C₂₇×C₃₂	1	1	yes	characteristic
282	C₄	C₂×C₄×C₁₆×C₂₇	1	1	yes	characteristic
283	C₃	C₂×C₄×C₉×C₆₄	1	1	yes	characteristic
284	C₂	C₄×C₂₇×C₆₄	1	4	yes	direct product factor
285	C₂	C₂²×C₂₇×C₆₄	1	2	yes	—
286	C₂	C₂×C₄×C₂₇×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₉×C₃₂	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×C₉×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₂₇×C₆₄	C₂	direct (N × H)
10. C₂×C₂₇×C₆₄	C₄	direct (N × H)
26. C₂₇×C₆₄	C₂×C₄	direct (N × H)
62. C₂×C₄×C₆₄	C₂₇	direct (N × H)
100. C₄×C₆₄	C₂×C₂₇	direct (N × H)
102. C₂×C₄×C₂₇	C₆₄	direct (N × H)
140. C₂×C₆₄	C₄×C₂₇	direct (N × H)
148. C₄×C₂₇	C₂×C₆₄	direct (N × H)
186. C₆₄	C₂×C₄×C₂₇	direct (N × H)
187. C₂×C₂₇	C₄×C₆₄	direct (N × H)
225. C₂₇	C₂×C₄×C₆₄	direct (N × H)
263. C₂×C₄	C₂₇×C₆₄	direct (N × H)
279. C₄	C₂×C₂₇×C₆₄	direct (N × H)
284. C₂	C₄×C₂₇×C₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂₇×C₃₂	2
C₂²×C₂₇×C₆₄	2
C₄×C₂₇×C₆₄	2
C₂×C₄×C₉×C₆₄	3

Subgroups — C2×C4×C27×C64 #72191997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₂₇×C₆₄ #72191997