Subgroups — C₆₇×#21440 #30700691

Order 17152.

1708 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#435×C₆₇	C₂	1	3	yes	maximal, complemented
3	C₄×#120×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1869	C₂	1	3	yes	maximal, complemented
5	C₆₇×#1850	C₂	1	3	yes	maximal, complemented
6	C₆₇×#1779	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1752	C₂	1	3	yes	maximal, complemented
8	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₈×C₆₇	—	2	12	no	—
11	C₂³×C₈×C₆₇	C₂²	1	3	yes	—
12	C₂³×C₈×C₆₇	C₂²	1	3	yes	—
13	C₂×#120×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
15	#435×C₆₇	—	2	12	no	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	3	yes	complemented
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	3	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	3	yes	—
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	3	yes	complemented
20	C₂×#97×C₆₇	C₂²	1	3	yes	complemented
21	#386×C₆₇	C₄	1	3	yes	—
22	#385×C₆₇	C₄	1	3	yes	—
23	#379×C₆₇	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊C₄×C₆₇	C₂²	1	3	yes	—
25	#336×C₆₇	C₂²	1	6	yes	complemented
26	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₄×C₆₇	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₆₇	—	2	6	no	—
29	C₂³×C₄×C₆₇	—	2	6	no	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
33	C₂×M₁₆×C₆₇	—	2	6	no	—
34	C₂²×C₈×C₆₇	—	2	32	no	—
35	C₂²×C₈×C₆₇	—	2	6	no	—
36	C₂²×C₈×C₆₇	—	2	12	no	—
37	C₂²×C₈×C₆₇	—	2	12	no	—
38	C₂²×C₈×C₆₇	D₄	1	12	yes	complemented
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#120×C₆₇	—	2	4	no	—
44	C₄×D₄×C₆₇	—	2	6	no	—
45	C₄×D₄×C₆₇	—	2	6	no	—
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
47	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
48	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
49	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
50	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
52	C₄⋊C₈×C₆₇	—	2	6	no	—
53	C₂²⋊C₈×C₆₇	—	2	6	no	—
54	C₂²⋊C₈×C₆₇	—	2	6	no	—
55	C₂²⋊C₈×C₆₇	—	2	6	no	—
56	#97×C₆₇	—	2	6	no	—
57	#95×C₆₇	C₂×C₄	1	6	yes	—
58	C₂⁴×C₆₇	—	2	6	no	—
59	C₂⁴×C₆₇	—	2	6	no	—
60	C₂⁴×C₆₇	M₁₆	1	4	yes	complemented
61	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂⁴×C₆₇	C₂²×C₄	1	3	yes	—
63	C₂×D₄×C₆₇	—	2	6	no	—
64	C₂×D₄×C₆₇	—	2	6	no	—
65	C₂×D₄×C₆₇	—	2	6	no	—
66	C₂²×C₄×C₆₇	—	2	8	no	—
67	C₂²×C₄×C₆₇	—	2	24	no	—
68	C₂²×C₄×C₆₇	—	2	6	no	—
69	C₂²×C₄×C₆₇	—	2	12	no	—
70	C₂²×C₄×C₆₇	—	2	6	no	—
71	C₂²×C₄×C₆₇	—	2	6	no	—
72	C₂²×C₄×C₆₇	—	2	3	no	—
73	C₂²×C₄×C₆₇	—	2	3	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
79	C₂²×C₄×C₆₇	#41	1	6	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₆₇	—	4	4	no	—
82	C₂×C₈×C₆₇	—	2	24	no	—
83	C₂×C₈×C₆₇	—	2	6	no	—
84	C₂×C₈×C₆₇	—	2	24	no	—
85	C₂×C₈×C₆₇	—	2	3	no	—
86	C₂×C₈×C₆₇	—	2	3	no	—
87	C₂×C₈×C₆₇	—	2	6	no	—
88	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
89	C₂×C₈×C₆₇	C₂×D₄	1	6	yes	—
90	C₄⋊C₄×C₆₇	—	2	6	no	—
91	C₂²⋊C₄×C₆₇	—	2	6	no	—
92	C₂²⋊C₄×C₆₇	—	2	6	no	—
93	C₂²⋊C₄×C₆₇	—	2	6	no	—
94	C₂²⋊C₄×C₆₇	—	2	6	no	—
95	C₄²×C₆₇	—	2	3	no	—
96	C₄²×C₆₇	—	2	3	no	—
97	C₂³×C₆₇	—	2	8	no	—
98	C₂³×C₆₇	—	2	24	no	—
99	C₂³×C₆₇	—	2	6	no	—
100	C₂³×C₆₇	—	2	6	no	—
101	C₂³×C₆₇	—	2	24	no	—
102	C₂³×C₆₇	—	2	3	no	—
103	C₂³×C₆₇	—	2	1	no	—
104	C₂³×C₆₇	—	2	3	no	—
105	C₂³×C₆₇	C₄×D₄	1	6	yes	—
106	C₂³×C₆₇	C₂×M₁₆	1	6	yes	—
107	C₂³×C₆₇	#131	1	6	yes	—
108	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
109	D₄×C₆₇	—	4	12	no	—
110	C₂×C₄×C₆₇	—	2	24	no	—
111	C₂×C₄×C₆₇	—	2	6	no	—
112	C₂×C₄×C₆₇	—	2	6	no	—
113	C₂×C₄×C₆₇	—	2	6	no	—
114	C₂×C₄×C₆₇	—	2	12	no	—
115	C₂×C₄×C₆₇	—	2	3	no	—
116	C₂×C₄×C₆₇	—	2	3	no	—
117	C₂×C₄×C₆₇	—	2	3	no	—
118	C₂×C₄×C₆₇	—	2	3	no	—
119	C₂×C₄×C₆₇	—	2	6	no	—
120	C₂×C₄×C₆₇	—	2	3	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	—	4	3	no	—
123	C₂×C₄×C₆₇	C₄×D₄	1	6	yes	—
124	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
125	C₂×C₄×C₆₇	C₂×#41	1	3	yes	—
126	C₈×C₆₇	—	2	6	no	—
127	C₈×C₆₇	—	4	1	no	—
128	C₈×C₆₇	—	4	3	no	—
129	C₈×C₆₇	#120	1	4	yes	complemented
130	C₂²×C₆₇	—	2	32	no	—
131	C₂²×C₆₇	—	2	12	no	—
132	C₂²×C₆₇	—	2	6	no	—
133	C₂²×C₆₇	—	2	12	no	—
134	C₂²×C₆₇	—	2	6	no	—
135	C₂²×C₆₇	—	4	3	no	—
136	C₂²×C₆₇	—	4	3	no	—
137	C₂²×C₆₇	—	4	1	no	—
138	C₂²×C₆₇	#435	1	12	yes	complemented
139	C₂²×C₆₇	C₂×C₄×D₄	1	3	yes	—
140	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
141	C₂²×C₆₇	C₂×#131	1	3	yes	—
142	C₄×C₆₇	—	2	6	no	—
143	C₄×C₆₇	—	4	3	no	—
144	C₄×C₆₇	—	4	3	no	—
145	C₄×C₆₇	—	4	1	no	—
146	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
147	C₄×C₆₇	#525	1	3	yes	—
148	#21440	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₆₇	—	2	12	no	—
150	C₂×C₆₇	—	8	1	no	—
151	C₂×C₆₇	C₄×#120	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#435	1	3	yes	—
153	C₂×C₆₇	#2929	1	3	yes	—
154	C₂⁴×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
155	C₂×#435	C₂×C₆₇	1	3	yes	complemented
156	C₄×#120	C₂×C₆₇	1	1	yes	characteristic
157	#1869	C₂×C₆₇	1	3	yes	complemented
158	#1850	C₂×C₆₇	1	3	yes	complemented
159	#1779	C₂×C₆₇	1	1	yes	characteristic, complemented
160	#1752	C₂×C₆₇	1	3	yes	complemented
161	C₆₇	#21440	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
163	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
164	C₂³×C₈	—	2	12	no	—
165	C₂³×C₈	C₂²×C₆₇	1	3	yes	—
166	C₂³×C₈	C₂²×C₆₇	1	3	yes	—
167	C₂×#120	C₄×C₆₇	1	1	yes	characteristic
168	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
169	#435	—	2	12	no	—
170	C₂×C₄⋊C₈	C₂²×C₆₇	1	3	yes	complemented
171	C₂×C₂²⋊C₈	C₂²×C₆₇	1	3	yes	—
172	C₂×C₂²⋊C₈	C₂²×C₆₇	1	3	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₆₇	1	3	yes	complemented
174	C₂×#97	C₂²×C₆₇	1	3	yes	complemented
175	#386	C₄×C₆₇	1	3	yes	—
176	#385	C₄×C₆₇	1	3	yes	—
177	#379	C₄×C₆₇	1	1	yes	characteristic
178	C₄×C₂²⋊C₄	C₂²×C₆₇	1	3	yes	—
179	#336	C₂²×C₆₇	1	6	yes	complemented
180	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
181	C₂²×D₄	C₂×C₄×C₆₇	1	3	yes	—
182	C₂³×C₄	—	2	6	no	—
183	C₂³×C₄	—	2	6	no	—
184	C₂³×C₄	C₂×C₄×C₆₇	1	3	yes	—
185	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
186	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
187	C₂×M₁₆	—	2	6	no	—
188	C₂²×C₈	—	2	32	no	—
189	C₂²×C₈	—	2	6	no	—
190	C₂²×C₈	—	2	12	no	—
191	C₂²×C₈	—	2	12	no	—
192	C₂²×C₈	D₄×C₆₇	1	12	yes	complemented
193	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
195	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
196	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
197	#120	—	2	4	no	—
198	C₄×D₄	—	2	6	no	—
199	C₄×D₄	—	2	6	no	—
200	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
201	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
202	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
203	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
204	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	3	yes	—
205	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
206	C₄⋊C₈	—	2	6	no	—
207	C₂²⋊C₈	—	2	6	no	—
208	C₂²⋊C₈	—	2	6	no	—
209	C₂²⋊C₈	—	2	6	no	—
210	#97	—	2	6	no	—
211	#95	C₂×C₄×C₆₇	1	6	yes	—
212	C₂⁴	—	2	6	no	—
213	C₂⁴	—	2	6	no	—
214	C₂⁴	M₁₆×C₆₇	1	4	yes	complemented
215	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
216	C₂⁴	C₂²×C₄×C₆₇	1	3	yes	—
217	C₂×D₄	—	2	6	no	—
218	C₂×D₄	—	2	6	no	—
219	C₂×D₄	—	2	6	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	6	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	6	no	—
225	C₂²×C₄	—	2	6	no	—
226	C₂²×C₄	—	2	3	no	—
227	C₂²×C₄	—	2	3	no	—
228	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
229	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
230	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
232	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
233	C₂²×C₄	#41×C₆₇	1	6	yes	—
234	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
235	M₁₆	—	4	4	no	—
236	C₂×C₈	—	2	24	no	—
237	C₂×C₈	—	2	6	no	—
238	C₂×C₈	—	2	24	no	—
239	C₂×C₈	—	2	3	no	—
240	C₂×C₈	—	2	3	no	—
241	C₂×C₈	—	2	6	no	—
242	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
243	C₂×C₈	C₂×D₄×C₆₇	1	6	yes	—
244	C₄⋊C₄	—	2	6	no	—
245	C₂²⋊C₄	—	2	6	no	—
246	C₂²⋊C₄	—	2	6	no	—
247	C₂²⋊C₄	—	2	6	no	—
248	C₂²⋊C₄	—	2	6	no	—
249	C₄²	—	2	3	no	—
250	C₄²	—	2	3	no	—
251	C₂³	—	2	8	no	—
252	C₂³	—	2	24	no	—
253	C₂³	—	2	6	no	—
254	C₂³	—	2	6	no	—
255	C₂³	—	2	24	no	—
256	C₂³	—	2	3	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	3	no	—
259	C₂³	C₄×D₄×C₆₇	1	6	yes	—
260	C₂³	C₂×M₁₆×C₆₇	1	6	yes	—
261	C₂³	#131×C₆₇	1	6	yes	—
262	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
263	D₄	—	4	12	no	—
264	C₂×C₄	—	2	24	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	2	6	no	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	2	3	no	—
270	C₂×C₄	—	2	3	no	—
271	C₂×C₄	—	2	3	no	—
272	C₂×C₄	—	2	3	no	—
273	C₂×C₄	—	2	6	no	—
274	C₂×C₄	—	2	3	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	3	no	—
277	C₂×C₄	C₄×D₄×C₆₇	1	6	yes	—
278	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
279	C₂×C₄	C₂×#41×C₆₇	1	3	yes	—
280	C₈	—	2	6	no	—
281	C₈	—	4	1	no	—
282	C₈	—	4	3	no	—
283	C₈	#120×C₆₇	1	4	yes	complemented
284	C₂²	—	2	32	no	—
285	C₂²	—	2	6	no	—
286	C₂²	—	2	12	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	12	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	3	no	—
291	C₂²	—	4	3	no	—
292	C₂²	#435×C₆₇	1	12	yes	complemented
293	C₂²	C₂×C₄×D₄×C₆₇	1	3	yes	—
294	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
295	C₂²	C₂×#131×C₆₇	1	3	yes	—
296	C₄	—	2	6	no	—
297	C₄	—	4	1	no	—
298	C₄	—	4	3	no	—
299	C₄	—	4	3	no	—
300	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
301	C₄	#525×C₆₇	1	3	yes	—
302	C₂	—	2	12	no	—
303	C₂	—	8	1	no	—
304	C₂	C₄×#120×C₆₇	1	1	yes	characteristic
305	C₂	C₂×#435×C₆₇	1	3	yes	—
306	C₂	C₆₇×#2929	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#120×C₆₇	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₈×C₆₇	C₂	semidirect
2. C₂×#435×C₆₇	C₂	semidirect
4. C₆₇×#1869	C₂	semidirect
5. C₆₇×#1850	C₂	semidirect
6. C₆₇×#1779	C₂	semidirect
7. C₆₇×#1752	C₂	semidirect
9. C₂²×M₁₆×C₆₇	C₂²	semidirect
16. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
19. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
20. C₂×#97×C₆₇	C₂²	semidirect
25. #336×C₆₇	C₂²	semidirect
38. C₂²×C₈×C₆₇	D₄	semidirect
60. C₂⁴×C₆₇	M₁₆	semidirect
129. C₈×C₆₇	#120	semidirect
138. C₂²×C₆₇	#435	semidirect
148. #21440	C₆₇	direct (N × H)
154. C₂⁴×C₈	C₂×C₆₇	semidirect
155. C₂×#435	C₂×C₆₇	semidirect
157. #1869	C₂×C₆₇	semidirect
158. #1850	C₂×C₆₇	semidirect
159. #1779	C₂×C₆₇	semidirect
160. #1752	C₂×C₆₇	semidirect
161. C₆₇	#21440	direct (N × H)
163. C₂²×M₁₆	C₂²×C₆₇	semidirect
170. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
173. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
174. C₂×#97	C₂²×C₆₇	semidirect
179. #336	C₂²×C₆₇	semidirect
192. C₂²×C₈	D₄×C₆₇	semidirect
214. C₂⁴	M₁₆×C₆₇	semidirect
283. C₈	#120×C₆₇	semidirect
292. C₂²	#435×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₆₇	2
C₂×#435×C₆₇	2
C₄×#120×C₆₇	2
C₆₇×#1869	2
C₆₇×#1850	2
C₆₇×#1779	2
C₆₇×#1752	2
#21440	67

Subgroups — C67×#21440 #30700691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21440 #30700691