Subgroups — C₆₇×#20776 #30700027

Order 17152.

684 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1841	C₂	1	8	yes	maximal, complemented
3	C₂×#344×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#1733	C₂	1	2	yes	maximal
5	C₂×#338×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×H(3,ℤ/4)×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
12	#405×C₆₇	C₄	1	4	yes	complemented
13	#405×C₆₇	C₄	1	4	yes	complemented
14	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
15	#344×C₆₇	C₂²	1	8	yes	—
16	#339×C₆₇	C₂²	1	8	yes	—
17	#338×C₆₇	C₂²	1	4	yes	—
18	H(3,ℤ/4)×C₆₇	C₂²	1	4	yes	—
19	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₆₇	—	2	2	no	—
21	C₂×M₁₆×C₆₇	—	2	2	no	—
22	C₂×M₁₆×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂×M₁₆×C₆₇	—	2	2	no	—
27	C₂×M₁₆×C₆₇	—	2	2	no	—
28	C₂×M₁₆×C₆₇	—	2	2	no	—
29	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
32	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
33	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	#117×C₆₇	—	2	1	no	—
38	#117×C₆₇	—	2	1	no	—
39	#117×C₆₇	—	2	2	no	—
40	#117×C₆₇	—	2	2	no	—
41	#117×C₆₇	—	2	2	no	—
42	#117×C₆₇	C₂×C₄	1	2	yes	—
43	#117×C₆₇	C₂×C₄	1	2	yes	—
44	#117×C₆₇	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
51	#97×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
53	C₂⁴×C₆₇	#41	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	#41	1	2	yes	—
60	C₂²×C₄×C₆₇	#41	1	2	yes	—
61	C₂²×C₄×C₆₇	#41	1	2	yes	—
62	C₂²×C₄×C₆₇	#41	1	2	yes	—
63	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₆₇	—	2	4	no	—
67	M₁₆×C₆₇	—	2	2	no	—
68	M₁₆×C₆₇	—	2	2	no	—
69	M₁₆×C₆₇	—	2	4	no	—
70	M₁₆×C₆₇	—	2	4	no	—
71	M₁₆×C₆₇	—	2	4	no	—
72	M₁₆×C₆₇	—	2	4	no	—
73	M₁₆×C₆₇	—	4	2	no	—
74	M₁₆×C₆₇	—	4	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	—	4	2	no	—
80	C₂×C₈×C₆₇	—	4	2	no	—
81	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	—	4	2	no	—
87	C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂²⋊C₄×C₆₇	—	2	4	no	—
90	C₂²⋊C₄×C₆₇	—	4	2	no	—
91	C₄²×C₆₇	—	2	2	no	—
92	C₄²×C₆₇	—	2	4	no	—
93	C₄²×C₆₇	—	4	2	no	—
94	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂³×C₆₇	—	8	1	no	—
96	C₂³×C₆₇	#117	1	1	yes	characteristic
97	C₂³×C₆₇	#117	1	1	yes	characteristic
98	C₂³×C₆₇	#126	1	2	yes	—
99	C₂³×C₆₇	#126	1	2	yes	—
100	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	4	4	no	—
103	C₂×C₄×C₆₇	—	8	2	no	—
104	C₂×C₄×C₆₇	—	8	4	no	—
105	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#117	1	2	yes	—
108	C₂×C₄×C₆₇	#117	1	2	yes	—
109	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#126	1	2	yes	—
114	C₂×C₄×C₆₇	#126	1	2	yes	—
115	C₂×C₄×C₆₇	#126	1	2	yes	—
116	C₂×C₄×C₆₇	#126	1	2	yes	—
117	C₂×C₄×C₆₇	#126	1	2	yes	—
118	C₂×C₄×C₆₇	#126	1	2	yes	—
119	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₈×C₆₇	—	2	4	no	—
124	C₈×C₆₇	—	2	4	no	—
125	C₈×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	4	2	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	—	4	1	no	—
129	C₂²×C₆₇	—	4	1	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	#383	1	1	yes	characteristic
132	C₂²×C₆₇	#383	1	1	yes	characteristic
133	C₂²×C₆₇	#383	1	2	yes	—
134	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
135	C₂²×C₆₇	C₂×#126	1	2	yes	—
136	C₄×C₆₇	—	4	4	no	—
137	C₄×C₆₇	—	8	4	no	—
138	C₄×C₆₇	#383	1	1	yes	characteristic
139	C₄×C₆₇	#383	1	1	yes	characteristic
140	C₄×C₆₇	#383	1	2	yes	—
141	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
142	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
143	C₄×C₆₇	C₂×#126	1	2	yes	—
144	#20776	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₆₇	—	2	1	no	—
146	C₂×C₆₇	—	2	2	no	—
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	C₂×#383	1	1	yes	characteristic
151	#2873	C₂×C₆₇	1	1	yes	characteristic
152	#1841	C₂×C₆₇	1	8	yes	complemented
153	C₂×#344	C₂×C₆₇	1	2	yes	—
154	#1733	C₂×C₆₇	1	2	yes	—
155	C₂×#338	C₂×C₆₇	1	1	yes	characteristic
156	C₂×H(3,ℤ/4)	C₂×C₆₇	1	1	yes	characteristic
157	C₆₇	#20776	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
159	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
160	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
161	C₂×#117	C₂²×C₆₇	1	2	yes	—
162	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
163	#405	C₄×C₆₇	1	4	yes	complemented
164	#405	C₄×C₆₇	1	4	yes	complemented
165	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
166	#344	C₂²×C₆₇	1	8	yes	—
167	#339	C₂²×C₆₇	1	8	yes	—
168	#338	C₂²×C₆₇	1	4	yes	—
169	H(3,ℤ/4)	C₂²×C₆₇	1	4	yes	—
170	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
181	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
182	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
183	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
184	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
187	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	C₂×C₄×C₆₇	1	2	yes	—
194	#117	C₂×C₄×C₆₇	1	2	yes	—
195	#117	C₂³×C₆₇	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
202	#97	C₂×C₄×C₆₇	1	8	yes	complemented
203	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
204	C₂⁴	#41×C₆₇	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
209	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
210	C₂²×C₄	#41×C₆₇	1	2	yes	—
211	C₂²×C₄	#41×C₆₇	1	2	yes	—
212	C₂²×C₄	#41×C₆₇	1	2	yes	—
213	C₂²×C₄	#41×C₆₇	1	2	yes	—
214	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
233	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	2	no	—
245	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
246	C₂³	—	8	1	no	—
247	C₂³	#117×C₆₇	1	1	yes	characteristic
248	C₂³	#117×C₆₇	1	1	yes	characteristic
249	C₂³	#126×C₆₇	1	2	yes	—
250	C₂³	#126×C₆₇	1	2	yes	—
251	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
257	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
258	C₂×C₄	#117×C₆₇	1	2	yes	—
259	C₂×C₄	#117×C₆₇	1	2	yes	—
260	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
261	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
262	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
263	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
264	C₂×C₄	#126×C₆₇	1	2	yes	—
265	C₂×C₄	#126×C₆₇	1	2	yes	—
266	C₂×C₄	#126×C₆₇	1	2	yes	—
267	C₂×C₄	#126×C₆₇	1	2	yes	—
268	C₂×C₄	#126×C₆₇	1	2	yes	—
269	C₂×C₄	#126×C₆₇	1	2	yes	—
270	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
272	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
273	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	—	4	4	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#383×C₆₇	1	1	yes	characteristic
283	C₂²	#383×C₆₇	1	2	yes	—
284	C₂²	#383×C₆₇	1	1	yes	characteristic
285	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
286	C₂²	C₂×#126×C₆₇	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	#383×C₆₇	1	1	yes	characteristic
290	C₄	#383×C₆₇	1	2	yes	—
291	C₄	#383×C₆₇	1	1	yes	characteristic
292	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
293	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
294	C₄	C₂×#126×C₆₇	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	1	no	—
300	C₂	C₂×#383×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂×#383
prime radical	C₂⁴×C₆₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1841	C₂	semidirect
12. #405×C₆₇	C₄	semidirect
13. #405×C₆₇	C₄	semidirect
51. #97×C₆₇	C₂×C₄	semidirect
52. C₄×C₈×C₆₇	C₂×C₄	semidirect
144. #20776	C₆₇	direct (N × H)
152. #1841	C₂×C₆₇	semidirect
157. C₆₇	#20776	direct (N × H)
163. #405	C₄×C₆₇	semidirect
164. #405	C₄×C₆₇	semidirect
202. #97	C₂×C₄×C₆₇	semidirect
203. C₄×C₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2873	2
C₆₇×#1841	2
C₂×#344×C₆₇	2
C₆₇×#1733	2
C₂×#338×C₆₇	2
C₂×H(3,ℤ/4)×C₆₇	2
#20776	67

Subgroups — C67×#20776 #30700027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20776 #30700027