Subgroups — C₆₇×#2735 #22557166

Order 8576.

552 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#399×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#397×C₆₇	C₂	1	2	yes	maximal, complemented
3	#397×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#396×C₆₇	C₂	1	2	yes	maximal, complemented
5	#388×C₆₇	C₂	1	2	yes	maximal, complemented
6	#388×C₆₇	C₂	1	2	yes	maximal, complemented
7	#388×C₆₇	C₂	1	2	yes	maximal, complemented
8	#379×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#379×C₆₇	C₂	1	2	yes	maximal
10	C₂⁵×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂²	1	2	yes	complemented
26	#95×C₆₇	C₂²	1	2	yes	complemented
27	#95×C₆₇	C₂²	1	2	yes	complemented
28	#95×C₆₇	C₂²	1	1	yes	characteristic, complemented
29	#95×C₆₇	C₂²	1	2	yes	complemented
30	#95×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	C₂⁴×C₆₇	—	2	2	no	—
32	C₂⁴×C₆₇	—	2	4	no	—
33	C₂⁴×C₆₇	—	4	2	no	—
34	C₂⁴×C₆₇	—	4	1	no	—
35	C₂⁴×C₆₇	—	4	1	no	—
36	C₂⁴×C₆₇	C₂³	1	2	yes	—
37	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₆₇	—	4	1	no	—
46	C₄⋊C₄×C₆₇	—	4	2	no	—
47	C₄⋊C₄×C₆₇	—	4	2	no	—
48	C₄⋊C₄×C₆₇	—	4	1	no	—
49	C₂²⋊C₄×C₆₇	—	2	4	no	—
50	C₂²⋊C₄×C₆₇	—	2	4	no	—
51	C₂²⋊C₄×C₆₇	—	2	4	no	—
52	C₂²⋊C₄×C₆₇	—	4	2	no	—
53	C₂²⋊C₄×C₆₇	—	4	2	no	—
54	C₂²⋊C₄×C₆₇	—	4	2	no	—
55	C₂²⋊C₄×C₆₇	—	4	1	no	—
56	C₂²⋊C₄×C₆₇	—	4	2	no	—
57	C₂²⋊C₄×C₆₇	—	4	2	no	—
58	C₂²⋊C₄×C₆₇	—	4	1	no	—
59	C₄²×C₆₇	—	4	2	no	—
60	C₂³×C₆₇	—	2	2	no	—
61	C₂³×C₆₇	—	2	2	no	—
62	C₂³×C₆₇	—	2	2	no	—
63	C₂³×C₆₇	—	2	1	no	—
64	C₂³×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	—	2	2	no	—
66	C₂³×C₆₇	—	2	1	no	—
67	C₂³×C₆₇	—	2	2	no	—
68	C₂³×C₆₇	—	2	2	no	—
69	C₂³×C₆₇	—	2	2	no	—
70	C₂³×C₆₇	—	4	8	no	—
71	C₂³×C₆₇	—	4	4	no	—
72	C₂³×C₆₇	—	4	8	no	—
73	C₂³×C₆₇	—	4	4	no	—
74	C₂³×C₆₇	—	4	4	no	—
75	C₂³×C₆₇	#41	1	4	yes	—
76	C₂³×C₆₇	#41	1	2	yes	—
77	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₆₇	—	2	4	no	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	2	4	no	—
100	C₂²×C₆₇	—	2	4	no	—
101	C₂²×C₆₇	—	2	4	no	—
102	C₂²×C₆₇	—	4	16	no	—
103	C₂²×C₆₇	—	4	2	no	—
104	C₂²×C₆₇	—	4	2	no	—
105	C₂²×C₆₇	—	4	2	no	—
106	C₂²×C₆₇	—	4	2	no	—
107	C₂²×C₆₇	—	4	1	no	—
108	C₂²×C₆₇	—	4	2	no	—
109	C₂²×C₆₇	—	4	1	no	—
110	C₂²×C₆₇	C₂×#41	1	2	yes	—
111	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
112	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
114	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₆₇	—	4	2	no	—
116	C₄×C₆₇	—	4	2	no	—
117	C₄×C₆₇	—	4	2	no	—
118	C₄×C₆₇	—	4	2	no	—
119	C₄×C₆₇	—	4	1	no	—
120	C₄×C₆₇	—	4	2	no	—
121	C₄×C₆₇	—	4	1	no	—
122	C₂×C₆₇	—	4	4	no	—
123	C₂×C₆₇	—	4	2	no	—
124	C₂×C₆₇	#538	1	2	yes	—
125	C₂×C₆₇	#538	1	1	yes	characteristic
126	C₂×C₆₇	#551	1	2	yes	—
127	C₂×C₆₇	#551	1	1	yes	characteristic
128	C₂×C₆₇	#560	1	1	yes	characteristic
129	#2735	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₆₇	#2735	1	1	yes	characteristic, direct product factor, Sylow
131	#399	C₂×C₆₇	1	1	yes	characteristic, complemented
132	#397	C₂×C₆₇	1	1	yes	characteristic, complemented
133	#397	C₂×C₆₇	1	2	yes	complemented
134	#396	C₂×C₆₇	1	2	yes	complemented
135	#388	C₂×C₆₇	1	2	yes	complemented
136	#388	C₂×C₆₇	1	2	yes	complemented
137	#388	C₂×C₆₇	1	2	yes	complemented
138	#379	C₂×C₆₇	1	1	yes	characteristic
139	#379	C₂×C₆₇	1	2	yes	—
140	C₂⁵	C₂²×C₆₇	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
142	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
143	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
144	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
145	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
155	C₂×C₄²	C₂²×C₆₇	1	2	yes	complemented
156	#95	C₂²×C₆₇	1	2	yes	complemented
157	#95	C₂²×C₆₇	1	1	yes	characteristic, complemented
158	#95	C₂²×C₆₇	1	2	yes	complemented
159	#95	C₂²×C₆₇	1	2	yes	complemented
160	#95	C₂²×C₆₇	1	1	yes	characteristic, complemented
161	C₂⁴	—	2	2	no	—
162	C₂⁴	—	2	4	no	—
163	C₂⁴	—	4	2	no	—
164	C₂⁴	—	4	1	no	—
165	C₂⁴	—	4	1	no	—
166	C₂⁴	C₂³×C₆₇	1	2	yes	—
167	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
169	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
171	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
172	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
173	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
174	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₂²⋊C₄	—	4	2	no	—
185	C₂²⋊C₄	—	4	1	no	—
186	C₂²⋊C₄	—	4	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₄²	—	4	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	4	8	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	4	8	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	4	4	no	—
205	C₂³	#41×C₆₇	1	4	yes	—
206	C₂³	#41×C₆₇	1	2	yes	—
207	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	16	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	1	no	—
240	C₂²	C₂×#41×C₆₇	1	2	yes	—
241	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
242	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
243	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
244	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
245	C₄	—	4	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	1	no	—
252	C₂	—	4	4	no	—
253	C₂	—	4	2	no	—
254	C₂	#538×C₆₇	1	2	yes	—
255	C₂	#538×C₆₇	1	1	yes	characteristic
256	C₂	#551×C₆₇	1	2	yes	—
257	C₂	#551×C₆₇	1	1	yes	characteristic
258	C₂	#560×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₆₇
prime radical	C₂⁵×C₆₇	C₂²
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. #399×C₆₇	C₂	semidirect
2. #397×C₆₇	C₂	semidirect
3. #397×C₆₇	C₂	semidirect
4. #396×C₆₇	C₂	semidirect
5. #388×C₆₇	C₂	semidirect
6. #388×C₆₇	C₂	semidirect
7. #388×C₆₇	C₂	semidirect
11. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
12. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
13. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
14. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
25. C₂×C₄²×C₆₇	C₂²	semidirect
26. #95×C₆₇	C₂²	semidirect
27. #95×C₆₇	C₂²	semidirect
28. #95×C₆₇	C₂²	semidirect
29. #95×C₆₇	C₂²	semidirect
30. #95×C₆₇	C₂²	semidirect
129. #2735	C₆₇	direct (N × H)
130. C₆₇	#2735	direct (N × H)
131. #399	C₂×C₆₇	semidirect
132. #397	C₂×C₆₇	semidirect
133. #397	C₂×C₆₇	semidirect
134. #396	C₂×C₆₇	semidirect
135. #388	C₂×C₆₇	semidirect
136. #388	C₂×C₆₇	semidirect
137. #388	C₂×C₆₇	semidirect
141. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
142. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
143. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
144. C₂×C₄⋊C₄	C₂²×C₆₇	semidirect
155. C₂×C₄²	C₂²×C₆₇	semidirect
156. #95	C₂²×C₆₇	semidirect
157. #95	C₂²×C₆₇	semidirect
158. #95	C₂²×C₆₇	semidirect
159. #95	C₂²×C₆₇	semidirect
160. #95	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
#399×C₆₇	2
#397×C₆₇	2
#396×C₆₇	2
#388×C₆₇	2
#379×C₆₇	2
#2735	67

Subgroups — C67×#2735 #22557166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2735 #22557166