Subgroups — C₂×C₆₇×#2113 #30712583

Order 17152.

1190 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	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2113	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic
7	C₂×#131×C₆₇	—	2	4	no	—
8	C₂×#131×C₆₇	—	2	4	no	—
9	C₂×#131×C₆₇	C₄	1	4	yes	—
10	C₂×#131×C₆₇	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₆₇	—	2	4	no	—
16	C₂×M₃₂×C₆₇	C₂²	1	4	yes	complemented
17	C₂²×C₁₆×C₆₇	—	2	4	no	—
18	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₁₆×C₆₇	C₂²	1	4	yes	complemented
21	C₂²⋊C₁₆×C₆₇	C₂²	1	16	yes	complemented
22	C₂×#41×C₆₇	—	2	4	no	—
23	C₂×#41×C₆₇	—	2	4	no	—
24	C₂×#41×C₆₇	C₈	1	4	yes	—
25	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
26	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
27	C₂²×D₄×C₆₇	C₈	1	2	yes	—
28	C₂²×D₄×C₆₇	C₈	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	#131×C₆₇	—	2	16	no	—
32	#131×C₆₇	—	2	8	no	—
33	#131×C₆₇	—	2	8	no	—
34	C₂×M₁₆×C₆₇	—	2	4	no	—
35	C₂×M₁₆×C₆₇	—	2	8	no	—
36	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	—	2	4	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	4	yes	—
45	C₂²×C₈×C₆₇	D₄	1	4	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	M₃₂×C₆₇	—	2	8	no	—
50	C₂×C₁₆×C₆₇	—	2	4	no	—
51	C₂×C₁₆×C₆₇	—	2	8	no	—
52	C₂×C₁₆×C₆₇	—	2	4	no	—
53	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
54	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
55	C₂⁴×C₆₇	C₂×C₈	1	2	yes	—
56	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
57	#41×C₆₇	—	2	16	no	—
58	#41×C₆₇	—	2	8	no	—
59	#41×C₆₇	—	2	8	no	—
60	C₂×Q₈×C₆₇	—	2	4	no	—
61	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
62	C₂×D₄×C₆₇	—	2	8	no	—
63	C₂×D₄×C₆₇	—	2	4	no	—
64	C₂×D₄×C₆₇	C₂×C₈	1	8	yes	—
65	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
71	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	M₁₆×C₆₇	—	2	8	no	—
78	M₁₆×C₆₇	—	2	16	no	—
79	C₂×C₈×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	8	no	—
81	C₂×C₈×C₆₇	—	2	4	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
84	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₁₆×C₆₇	—	2	4	no	—
94	C₁₆×C₆₇	—	2	4	no	—
95	C₂³×C₆₇	—	2	4	no	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	—	2	8	no	—
98	C₂³×C₆₇	C₂²⋊C₈	1	4	yes	—
99	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
100	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
101	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
102	Q₈×C₆₇	—	2	8	no	—
103	D₄×C₆₇	—	2	16	no	—
104	D₄×C₆₇	—	2	8	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	8	no	—
109	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
110	C₂×C₄×C₆₇	C₂²⋊C₈	1	8	yes	—
111	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
114	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₆₇	—	2	4	no	—
120	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
121	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
122	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂²×C₆₇	—	2	8	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	2	8	no	—
128	C₂²×C₆₇	—	4	8	no	—
129	C₂²×C₆₇	C₂×C₂²⋊C₈	1	4	yes	—
130	C₂²×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
131	C₂²×C₆₇	#504	1	4	yes	—
132	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₆₇	—	2	4	no	—
134	C₄×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
135	C₄×C₆₇	C₂×C₂²⋊C₈	1	4	yes	—
136	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
138	C₂×#2113	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₆₇	—	2	4	no	—
140	C₂×C₆₇	—	4	4	no	—
141	C₂×C₆₇	#2113	1	4	yes	direct product factor
142	C₂×C₆₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
143	C₂×C₆₇	C₂×#504	1	2	yes	—
144	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic
145	C₂²×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
146	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
147	#2113	C₂×C₆₇	1	8	yes	direct product factor
148	C₂×C₂²⋊C₁₆	C₂×C₆₇	1	4	yes	complemented
149	C₆₇	C₂×#2113	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²×#41	C₄×C₆₇	1	1	yes	characteristic
151	C₂×#131	—	2	4	no	—
152	C₂×#131	—	2	4	no	—
153	C₂×#131	C₄×C₆₇	1	4	yes	—
154	C₂×#131	C₂²×C₆₇	1	4	yes	—
155	C₂²×M₁₆	C₄×C₆₇	1	1	yes	characteristic
156	C₂²×M₁₆	C₄×C₆₇	1	2	yes	—
157	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
158	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
159	C₂×M₃₂	—	2	4	no	—
160	C₂×M₃₂	C₂²×C₆₇	1	4	yes	complemented
161	C₂²×C₁₆	—	2	4	no	—
162	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
163	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
164	C₂²×C₁₆	C₂²×C₆₇	1	4	yes	complemented
165	C₂²⋊C₁₆	C₂²×C₆₇	1	16	yes	complemented
166	C₂×#41	—	2	4	no	—
167	C₂×#41	—	2	4	no	—
168	C₂×#41	C₈×C₆₇	1	4	yes	—
169	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
170	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
171	C₂²×D₄	C₈×C₆₇	1	2	yes	—
172	C₂²×D₄	C₈×C₆₇	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
174	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
175	#131	—	2	16	no	—
176	#131	—	2	8	no	—
177	#131	—	2	8	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	2	8	no	—
180	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
181	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
185	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
186	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
188	C₂²×C₈	D₄×C₆₇	1	4	yes	—
189	C₂²×C₈	D₄×C₆₇	1	4	yes	—
190	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
191	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
192	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
193	M₃₂	—	2	8	no	—
194	C₂×C₁₆	—	2	4	no	—
195	C₂×C₁₆	—	2	8	no	—
196	C₂×C₁₆	—	2	4	no	—
197	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
198	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
199	C₂⁴	C₂×C₈×C₆₇	1	2	yes	—
200	C₂⁴	C₂×C₈×C₆₇	1	1	yes	characteristic
201	#41	—	2	8	no	—
202	#41	—	2	8	no	—
203	#41	—	2	16	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	—
206	C₂×D₄	—	2	8	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	C₂×C₈×C₆₇	1	8	yes	—
209	C₂×D₄	C₂×C₈×C₆₇	1	4	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
213	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
214	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
215	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₈×C₆₇	1	2	yes	—
217	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
218	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
219	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
221	M₁₆	—	2	8	no	—
222	M₁₆	—	2	16	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
228	C₂×C₈	C₂²⋊C₄×C₆₇	1	8	yes	—
229	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
230	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
231	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
235	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
236	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
237	C₁₆	—	2	4	no	—
238	C₁₆	—	2	4	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	4	no	—
242	C₂³	C₂²⋊C₈×C₆₇	1	4	yes	—
243	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
244	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
245	C₂³	C₂²×C₈×C₆₇	1	1	yes	characteristic
246	Q₈	—	2	8	no	—
247	D₄	—	2	16	no	—
248	D₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	C₂²⋊C₈×C₆₇	1	4	yes	—
254	C₂×C₄	C₂²⋊C₈×C₆₇	1	8	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
257	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
258	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
260	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
261	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
262	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
263	C₈	—	2	4	no	—
264	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
265	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
266	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
267	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
268	C₂²	—	2	8	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	4	8	no	—
273	C₂²	C₂×C₂²⋊C₈×C₆₇	1	4	yes	—
274	C₂²	C₂×C₂²⋊C₈×C₆₇	1	2	yes	—
275	C₂²	#504×C₆₇	1	4	yes	—
276	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	C₂×C₂²⋊C₈×C₆₇	1	2	yes	—
279	C₄	C₂×C₂²⋊C₈×C₆₇	1	4	yes	—
280	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
281	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	—	4	4	no	—
284	C₂	C₆₇×#2113	1	4	yes	direct product factor
285	C₂	C₂²×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
286	C₂	C₂×#504×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₆₇	C₂³
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂²×#41×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₆₇	C₂	semidirect
3. C₂³×C₁₆×C₆₇	C₂	semidirect
4. C₆₇×#2113	C₂	direct (N × H)
5. C₂×C₂²⋊C₁₆×C₆₇	C₂	semidirect
16. C₂×M₃₂×C₆₇	C₂²	semidirect
20. C₂²×C₁₆×C₆₇	C₂²	semidirect
21. C₂²⋊C₁₆×C₆₇	C₂²	semidirect
138. C₂×#2113	C₆₇	direct (N × H)
141. C₂×C₆₇	#2113	direct (N × H)
145. C₂²×M₃₂	C₂×C₆₇	semidirect
146. C₂³×C₁₆	C₂×C₆₇	semidirect
147. #2113	C₂×C₆₇	direct (N × H)
148. C₂×C₂²⋊C₁₆	C₂×C₆₇	semidirect
149. C₆₇	C₂×#2113	direct (N × H)
160. C₂×M₃₂	C₂²×C₆₇	semidirect
164. C₂²×C₁₆	C₂²×C₆₇	semidirect
165. C₂²⋊C₁₆	C₂²×C₆₇	semidirect
284. C₂	C₆₇×#2113	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×#131×C₆₇	2
C₂²×M₃₂×C₆₇	2
C₂³×C₁₆×C₆₇	2
C₆₇×#2113	2
C₂×C₂²⋊C₁₆×C₆₇	2
C₂×#2113	67

Subgroups — C2×C67×#2113 #30712583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2113 #30712583