Subgroups — C₆₇×#2682 #22557113

Order 8576.

668 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#121×C₆₇	C₂	1	2	yes	maximal, complemented
3	#396×C₆₇	C₂	1	2	yes	maximal, complemented
4	#394×C₆₇	C₂	1	2	yes	maximal, complemented
5	#392×C₆₇	C₂	1	2	yes	maximal, complemented
6	#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#389×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#385×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
11	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₆₇	C₂²	1	2	yes	—
15	#123×C₆₇	—	2	8	no	—
16	#121×C₆₇	—	2	8	no	—
17	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
31	#95×C₆₇	C₂²	1	2	yes	—
32	#95×C₆₇	C₂²	1	2	yes	—
33	C₂⁴×C₆₇	C₂³	1	2	yes	—
34	C₂⁴×C₆₇	C₂³	1	2	yes	—
35	C₂×D₄×C₆₇	—	2	4	no	—
36	C₂×D₄×C₆₇	—	2	4	no	—
37	C₂×D₄×C₆₇	—	2	4	no	—
38	C₂×D₄×C₆₇	—	2	2	no	—
39	C₂×D₄×C₆₇	—	4	1	no	—
40	C₂×D₄×C₆₇	—	4	1	no	—
41	C₂×D₄×C₆₇	—	4	1	no	—
42	C₂×D₄×C₆₇	—	4	1	no	—
43	C₂×D₄×C₆₇	—	4	2	no	—
44	C₂×D₄×C₆₇	—	4	2	no	—
45	C₂×D₄×C₆₇	—	4	2	no	—
46	C₂×D₄×C₆₇	D₄	1	4	yes	complemented
47	C₂²×C₄×C₆₇	—	2	4	no	—
48	C₂²×C₄×C₆₇	—	2	4	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	2	4	no	—
59	C₄⋊C₄×C₆₇	—	4	1	no	—
60	C₄⋊C₄×C₆₇	—	4	1	no	—
61	C₄⋊C₄×C₆₇	—	4	1	no	—
62	C₄⋊C₄×C₆₇	D₄	1	4	yes	complemented
63	C₂²⋊C₄×C₆₇	—	2	4	no	—
64	C₂²⋊C₄×C₆₇	—	2	4	no	—
65	C₂²⋊C₄×C₆₇	—	2	4	no	—
66	C₂²⋊C₄×C₆₇	—	2	4	no	—
67	C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₂²⋊C₄×C₆₇	—	2	4	no	—
69	C₂²⋊C₄×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	—	4	1	no	—
71	C₂³×C₆₇	—	2	2	no	—
72	C₂³×C₆₇	—	2	2	no	—
73	C₂³×C₆₇	—	2	2	no	—
74	C₂³×C₆₇	—	2	2	no	—
75	C₂³×C₆₇	—	2	2	no	—
76	C₂³×C₆₇	—	2	2	no	—
77	C₂³×C₆₇	—	2	2	no	—
78	C₂³×C₆₇	—	2	2	no	—
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	2	2	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	—	2	2	no	—
83	C₂³×C₆₇	—	2	2	no	—
84	C₂³×C₆₇	C₂×D₄	1	4	yes	—
85	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
86	D₄×C₆₇	—	4	4	no	—
87	D₄×C₆₇	—	4	8	no	—
88	D₄×C₆₇	—	4	4	no	—
89	D₄×C₆₇	—	4	8	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	4	8	no	—
108	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
111	C₂×C₄×C₆₇	#41	1	2	yes	—
112	C₂²×C₆₇	—	2	4	no	—
113	C₂²×C₆₇	—	2	4	no	—
114	C₂²×C₆₇	—	2	4	no	—
115	C₂²×C₆₇	—	4	2	no	—
116	C₂²×C₆₇	—	4	2	no	—
117	C₂²×C₆₇	—	4	2	no	—
118	C₂²×C₆₇	—	4	2	no	—
119	C₂²×C₆₇	—	4	2	no	—
120	C₂²×C₆₇	—	4	2	no	—
121	C₂²×C₆₇	—	4	2	no	—
122	C₂²×C₆₇	—	4	2	no	—
123	C₂²×C₆₇	—	4	2	no	—
124	C₂²×C₆₇	—	4	2	no	—
125	C₂²×C₆₇	—	4	2	no	—
126	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂²×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	2	no	—
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	2	4	no	—
137	C₄×C₆₇	—	4	2	no	—
138	C₄×C₆₇	—	4	2	no	—
139	C₄×C₆₇	—	4	2	no	—
140	C₂×C₆₇	—	4	4	no	—
141	C₂×C₆₇	—	8	2	no	—
142	C₂×C₆₇	#534	1	1	yes	characteristic
143	C₂×C₆₇	#537	1	1	yes	characteristic
144	C₂×C₆₇	#538	1	1	yes	characteristic
145	C₂×C₆₇	D₄²	1	1	yes	characteristic
146	C₂×C₆₇	#547	1	1	yes	characteristic
147	C₂×C₆₇	#550	1	1	yes	characteristic
148	C₂×C₆₇	#562	1	1	yes	characteristic
149	#2682	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₆₇	#2682	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#123	C₂×C₆₇	1	2	yes	complemented
152	C₂×#121	C₂×C₆₇	1	2	yes	complemented
153	#396	C₂×C₆₇	1	2	yes	complemented
154	#394	C₂×C₆₇	1	2	yes	complemented
155	#392	C₂×C₆₇	1	2	yes	complemented
156	#390	C₂×C₆₇	1	1	yes	characteristic, complemented
157	#390	C₂×C₆₇	1	1	yes	characteristic, complemented
158	#389	C₂×C₆₇	1	1	yes	characteristic, complemented
159	#385	C₂×C₆₇	1	2	yes	complemented
160	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
161	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
162	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
163	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
164	C₂³×C₄	C₂²×C₆₇	1	2	yes	—
165	#123	—	2	8	no	—
166	#121	—	2	8	no	—
167	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
179	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
180	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
181	#95	C₂²×C₆₇	1	2	yes	—
182	#95	C₂²×C₆₇	1	2	yes	—
183	C₂⁴	C₂³×C₆₇	1	2	yes	—
184	C₂⁴	C₂³×C₆₇	1	2	yes	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	D₄×C₆₇	1	4	yes	complemented
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
201	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
202	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
203	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
204	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	D₄×C₆₇	1	4	yes	complemented
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	—	4	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₂×D₄×C₆₇	1	4	yes	—
235	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
236	D₄	—	4	4	no	—
237	D₄	—	4	8	no	—
238	D₄	—	4	4	no	—
239	D₄	—	4	8	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
259	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₆₇	1	4	yes	—
261	C₂×C₄	#41×C₆₇	1	2	yes	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
277	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
278	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
279	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
280	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
281	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
282	C₂²	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₂	—	4	4	no	—
291	C₂	—	8	2	no	—
292	C₂	#534×C₆₇	1	1	yes	characteristic
293	C₂	#537×C₆₇	1	1	yes	characteristic
294	C₂	#538×C₆₇	1	1	yes	characteristic
295	C₂	D₄²×C₆₇	1	1	yes	characteristic
296	C₂	#547×C₆₇	1	1	yes	characteristic
297	C₂	#550×C₆₇	1	1	yes	characteristic
298	C₂	#562×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×#123×C₆₇	C₂	semidirect
2. C₂×#121×C₆₇	C₂	semidirect
3. #396×C₆₇	C₂	semidirect
4. #394×C₆₇	C₂	semidirect
5. #392×C₆₇	C₂	semidirect
6. #390×C₆₇	C₂	semidirect
7. #390×C₆₇	C₂	semidirect
8. #389×C₆₇	C₂	semidirect
9. #385×C₆₇	C₂	semidirect
46. C₂×D₄×C₆₇	D₄	semidirect
62. C₄⋊C₄×C₆₇	D₄	semidirect
149. #2682	C₆₇	direct (N × H)
150. C₆₇	#2682	direct (N × H)
151. C₂×#123	C₂×C₆₇	semidirect
152. C₂×#121	C₂×C₆₇	semidirect
153. #396	C₂×C₆₇	semidirect
154. #394	C₂×C₆₇	semidirect
155. #392	C₂×C₆₇	semidirect
156. #390	C₂×C₆₇	semidirect
157. #390	C₂×C₆₇	semidirect
158. #389	C₂×C₆₇	semidirect
159. #385	C₂×C₆₇	semidirect
196. C₂×D₄	D₄×C₆₇	semidirect
212. C₄⋊C₄	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₆₇	2
C₂×#121×C₆₇	2
#396×C₆₇	2
#394×C₆₇	2
#392×C₆₇	2
#390×C₆₇	2
#389×C₆₇	2
#385×C₆₇	2
#2682	67

Subgroups — C67×#2682 #22557113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2682 #22557113