Subgroups — C₆₇×#2828 #22557259

Order 8576.

508 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#400×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#397×C₆₇	C₂	1	2	yes	maximal, complemented
3	#394×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#394×C₆₇	C₂	1	2	yes	maximal, complemented
5	#394×C₆₇	C₂	1	2	yes	maximal, complemented
6	#392×C₆₇	C₂	1	2	yes	maximal, complemented
7	#392×C₆₇	C₂	1	2	yes	maximal, complemented
8	#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#388×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
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	2	yes	—
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	2	yes	—
25	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	#95×C₆₇	C₂²	1	2	yes	—
27	#95×C₆₇	C₂²	1	2	yes	—
28	#95×C₆₇	C₂²	1	2	yes	—
29	#95×C₆₇	C₂²	1	1	yes	characteristic
30	#95×C₆₇	C₂²	1	1	yes	characteristic
31	C₂⁴×C₆₇	C₂³	1	2	yes	—
32	C₂⁴×C₆₇	C₂³	1	2	yes	—
33	C₂×D₄×C₆₇	—	2	2	no	—
34	C₂×D₄×C₆₇	—	4	1	no	—
35	C₂×D₄×C₆₇	—	4	1	no	—
36	C₂×D₄×C₆₇	—	4	2	no	—
37	C₂×D₄×C₆₇	—	4	2	no	—
38	C₂×D₄×C₆₇	—	4	2	no	—
39	C₂×D₄×C₆₇	—	4	2	no	—
40	C₂×D₄×C₆₇	—	4	2	no	—
41	C₂×D₄×C₆₇	—	4	1	no	—
42	C₂×D₄×C₆₇	—	4	2	no	—
43	C₂×D₄×C₆₇	—	4	2	no	—
44	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₆₇	—	4	1	no	—
52	C₄⋊C₄×C₆₇	—	4	2	no	—
53	C₄⋊C₄×C₆₇	—	4	1	no	—
54	C₂²⋊C₄×C₆₇	—	2	4	no	—
55	C₂²⋊C₄×C₆₇	—	4	2	no	—
56	C₂²⋊C₄×C₆₇	—	4	2	no	—
57	C₂²⋊C₄×C₆₇	—	4	2	no	—
58	C₂²⋊C₄×C₆₇	—	4	2	no	—
59	C₂²⋊C₄×C₆₇	—	4	2	no	—
60	C₂²⋊C₄×C₆₇	—	4	2	no	—
61	C₂²⋊C₄×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	—	4	1	no	—
63	C₂³×C₆₇	—	2	2	no	—
64	C₂³×C₆₇	—	2	2	no	—
65	C₂³×C₆₇	—	2	2	no	—
66	C₂³×C₆₇	—	2	2	no	—
67	C₂³×C₆₇	—	2	2	no	—
68	C₂³×C₆₇	—	2	2	no	—
69	C₂³×C₆₇	—	2	2	no	—
70	C₂³×C₆₇	—	2	2	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	C₂⁴	1	1	yes	characteristic
78	D₄×C₆₇	—	4	4	no	—
79	D₄×C₆₇	—	4	8	no	—
80	D₄×C₆₇	—	4	4	no	—
81	D₄×C₆₇	—	4	8	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	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	2	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	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	#41	1	2	yes	—
101	C₂²×C₆₇	—	4	2	no	—
102	C₂²×C₆₇	—	4	2	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	2	no	—
108	C₂²×C₆₇	—	4	2	no	—
109	C₂²×C₆₇	—	4	2	no	—
110	C₂²×C₆₇	—	4	2	no	—
111	C₂²×C₆₇	—	4	2	no	—
112	C₂²×C₆₇	—	4	2	no	—
113	C₂²×C₆₇	—	4	2	no	—
114	C₂²×C₆₇	—	4	2	no	—
115	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
116	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
119	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
120	C₄×C₆₇	—	2	2	no	—
121	C₄×C₆₇	—	4	2	no	—
122	C₄×C₆₇	—	4	2	no	—
123	C₄×C₆₇	—	4	1	no	—
124	C₄×C₆₇	—	4	2	no	—
125	C₄×C₆₇	—	4	1	no	—
126	C₄×C₆₇	—	4	2	no	—
127	C₂×C₆₇	—	8	2	no	—
128	C₂×C₆₇	—	8	2	no	—
129	C₂×C₆₇	#538	1	1	yes	characteristic
130	C₂×C₆₇	#540	1	2	yes	—
131	C₂×C₆₇	#560	1	2	yes	—
132	C₂×C₆₇	#560	1	1	yes	characteristic
133	C₂×C₆₇	H(3,𝔽₄)	1	1	yes	characteristic
134	#2828	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₆₇	#2828	1	1	yes	characteristic, direct product factor, Sylow
136	#400	C₂×C₆₇	1	1	yes	characteristic, complemented
137	#397	C₂×C₆₇	1	2	yes	complemented
138	#394	C₂×C₆₇	1	2	yes	complemented
139	#394	C₂×C₆₇	1	2	yes	complemented
140	#394	C₂×C₆₇	1	1	yes	characteristic, complemented
141	#392	C₂×C₆₇	1	2	yes	complemented
142	#392	C₂×C₆₇	1	2	yes	complemented
143	#390	C₂×C₆₇	1	1	yes	characteristic, complemented
144	#388	C₂×C₆₇	1	2	yes	complemented
145	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
146	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
147	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
148	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
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	2	yes	—
155	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
160	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
161	#95	C₂²×C₆₇	1	2	yes	—
162	#95	C₂²×C₆₇	1	2	yes	—
163	#95	C₂²×C₆₇	1	2	yes	—
164	#95	C₂²×C₆₇	1	1	yes	characteristic
165	#95	C₂²×C₆₇	1	1	yes	characteristic
166	C₂⁴	C₂³×C₆₇	1	2	yes	—
167	C₂⁴	C₂³×C₆₇	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	4	1	no	—
170	C₂×D₄	—	4	1	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	2	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	1	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
180	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
181	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
183	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
184	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
185	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	2	4	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	4	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₄²	—	4	1	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
213	D₄	—	4	4	no	—
214	D₄	—	4	8	no	—
215	D₄	—	4	4	no	—
216	D₄	—	4	8	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	#41×C₆₇	1	2	yes	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
251	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
252	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
253	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
254	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	2	no	—
262	C₂	—	8	2	no	—
263	C₂	—	8	2	no	—
264	C₂	#538×C₆₇	1	1	yes	characteristic
265	C₂	#540×C₆₇	1	2	yes	—
266	C₂	#560×C₆₇	1	2	yes	—
267	C₂	#560×C₆₇	1	1	yes	characteristic
268	C₂	H(3,𝔽₄)×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. #400×C₆₇	C₂	semidirect
2. #397×C₆₇	C₂	semidirect
3. #394×C₆₇	C₂	semidirect
4. #394×C₆₇	C₂	semidirect
5. #394×C₆₇	C₂	semidirect
6. #392×C₆₇	C₂	semidirect
7. #392×C₆₇	C₂	semidirect
8. #390×C₆₇	C₂	semidirect
9. #388×C₆₇	C₂	semidirect
134. #2828	C₆₇	direct (N × H)
135. C₆₇	#2828	direct (N × H)
136. #400	C₂×C₆₇	semidirect
137. #397	C₂×C₆₇	semidirect
138. #394	C₂×C₆₇	semidirect
139. #394	C₂×C₆₇	semidirect
140. #394	C₂×C₆₇	semidirect
141. #392	C₂×C₆₇	semidirect
142. #392	C₂×C₆₇	semidirect
143. #390	C₂×C₆₇	semidirect
144. #388	C₂×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
#400×C₆₇	2
#397×C₆₇	2
#394×C₆₇	2
#392×C₆₇	2
#390×C₆₇	2
#388×C₆₇	2
#2828	67

Subgroups — C67×#2828 #22557259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2828 #22557259