Subgroups — C₆₇×#2555 #22556986

Order 8576.

696 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#124×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#121×C₆₇	C₂	1	2	yes	maximal, complemented
3	#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#394×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#392×C₆₇	C₂	1	2	yes	maximal, complemented
6	#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#388×C₆₇	C₂	1	2	yes	maximal, complemented
8	#386×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₆₇	C₂²	1	2	yes	—
16	#124×C₆₇	—	2	8	no	—
17	#121×C₆₇	—	2	8	no	—
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊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	2	yes	—
25	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	#95×C₆₇	C₂²	1	2	yes	—
30	#95×C₆₇	C₂²	1	2	yes	—
31	#95×C₆₇	C₂²	1	1	yes	characteristic
32	#95×C₆₇	C₂²	1	1	yes	characteristic
33	C₂⁴×C₆₇	C₂³	1	2	yes	—
34	C₂⁴×C₆₇	C₂³	1	2	yes	—
35	C₂×Q₈×C₆₇	—	2	4	no	—
36	C₂×Q₈×C₆₇	—	4	1	no	—
37	C₂×D₄×C₆₇	—	2	4	no	—
38	C₂×D₄×C₆₇	—	2	4	no	—
39	C₂×D₄×C₆₇	—	2	4	no	—
40	C₂×D₄×C₆₇	—	2	4	no	—
41	C₂×D₄×C₆₇	—	2	2	no	—
42	C₂×D₄×C₆₇	—	2	4	no	—
43	C₂×D₄×C₆₇	—	4	1	no	—
44	C₂×D₄×C₆₇	—	4	1	no	—
45	C₂×D₄×C₆₇	—	4	2	no	—
46	C₂×D₄×C₆₇	—	4	2	no	—
47	C₂×D₄×C₆₇	—	4	1	no	—
48	C₂²×C₄×C₆₇	—	2	4	no	—
49	C₂²×C₄×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
58	C₄⋊C₄×C₆₇	—	2	4	no	—
59	C₄⋊C₄×C₆₇	—	4	1	no	—
60	C₂²⋊C₄×C₆₇	—	2	4	no	—
61	C₂²⋊C₄×C₆₇	—	2	4	no	—
62	C₂²⋊C₄×C₆₇	—	2	4	no	—
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	D₄	1	8	yes	complemented
67	C₄²×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	—	4	1	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	—	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	C₂×D₄	1	4	yes	—
83	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
84	Q₈×C₆₇	—	4	4	no	—
85	D₄×C₆₇	—	4	8	no	—
86	D₄×C₆₇	—	4	8	no	—
87	D₄×C₆₇	—	4	4	no	—
88	D₄×C₆₇	—	4	4	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	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	2	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	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	4	8	no	—
105	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
106	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
107	C₂×C₄×C₆₇	#41	1	2	yes	—
108	C₂×C₄×C₆₇	#41	1	4	yes	—
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	—	2	4	no	—
111	C₂²×C₆₇	—	2	4	no	—
112	C₂²×C₆₇	—	4	2	no	—
113	C₂²×C₆₇	—	4	2	no	—
114	C₂²×C₆₇	—	4	2	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	C₂²×D₄	1	2	yes	—
124	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#41	1	2	yes	—
126	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	—	2	4	no	—
130	C₄×C₆₇	—	2	4	no	—
131	C₄×C₆₇	—	2	4	no	—
132	C₄×C₆₇	—	4	2	no	—
133	C₄×C₆₇	—	4	1	no	—
134	C₄×C₆₇	—	4	1	no	—
135	C₂×C₆₇	—	4	4	no	—
136	C₂×C₆₇	—	8	2	no	—
137	C₂×C₆₇	#532	1	2	yes	—
138	C₂×C₆₇	#538	1	1	yes	characteristic
139	C₂×C₆₇	D₄²	1	1	yes	characteristic
140	C₂×C₆₇	#546	1	2	yes	—
141	C₂×C₆₇	#555	1	1	yes	characteristic
142	#2555	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₆₇	#2555	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#124	C₂×C₆₇	1	2	yes	complemented
145	C₂×#121	C₂×C₆₇	1	2	yes	complemented
146	#395	C₂×C₆₇	1	1	yes	characteristic, complemented
147	#394	C₂×C₆₇	1	1	yes	characteristic, complemented
148	#392	C₂×C₆₇	1	2	yes	complemented
149	#390	C₂×C₆₇	1	1	yes	characteristic, complemented
150	#388	C₂×C₆₇	1	2	yes	complemented
151	#386	C₂×C₆₇	1	2	yes	complemented
152	C₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
153	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
154	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
155	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
156	C₂²×D₄	C₂²×C₆₇	1	2	yes	—
157	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₆₇	1	2	yes	—
159	#124	—	2	8	no	—
160	#121	—	2	8	no	—
161	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
170	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
171	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
172	#95	C₂²×C₆₇	1	2	yes	—
173	#95	C₂²×C₆₇	1	2	yes	—
174	#95	C₂²×C₆₇	1	1	yes	characteristic
175	#95	C₂²×C₆₇	1	1	yes	characteristic
176	C₂⁴	C₂³×C₆₇	1	2	yes	—
177	C₂⁴	C₂³×C₆₇	1	2	yes	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	4	1	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
195	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
198	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
199	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
200	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	D₄×C₆₇	1	8	yes	complemented
210	C₄²	—	2	4	no	—
211	C₄²	—	4	1	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂×D₄×C₆₇	1	4	yes	—
226	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
227	Q₈	—	4	4	no	—
228	D₄	—	4	8	no	—
229	D₄	—	4	8	no	—
230	D₄	—	4	4	no	—
231	D₄	—	4	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	C₂×D₄×C₆₇	1	4	yes	—
249	C₂×C₄	C₂×D₄×C₆₇	1	4	yes	—
250	C₂×C₄	#41×C₆₇	1	2	yes	—
251	C₂×C₄	#41×C₆₇	1	4	yes	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×D₄×C₆₇	1	2	yes	—
267	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
268	C₂²	C₂×#41×C₆₇	1	2	yes	—
269	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
270	C₂²	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₂	—	4	4	no	—
279	C₂	—	8	2	no	—
280	C₂	#532×C₆₇	1	2	yes	—
281	C₂	#538×C₆₇	1	1	yes	characteristic
282	C₂	D₄²×C₆₇	1	1	yes	characteristic
283	C₂	#546×C₆₇	1	2	yes	—
284	C₂	#555×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×#124×C₆₇	C₂	semidirect
2. C₂×#121×C₆₇	C₂	semidirect
3. #395×C₆₇	C₂	semidirect
4. #394×C₆₇	C₂	semidirect
5. #392×C₆₇	C₂	semidirect
6. #390×C₆₇	C₂	semidirect
7. #388×C₆₇	C₂	semidirect
8. #386×C₆₇	C₂	semidirect
9. C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
66. C₂²⋊C₄×C₆₇	D₄	semidirect
142. #2555	C₆₇	direct (N × H)
143. C₆₇	#2555	direct (N × H)
144. C₂×#124	C₂×C₆₇	semidirect
145. C₂×#121	C₂×C₆₇	semidirect
146. #395	C₂×C₆₇	semidirect
147. #394	C₂×C₆₇	semidirect
148. #392	C₂×C₆₇	semidirect
149. #390	C₂×C₆₇	semidirect
150. #388	C₂×C₆₇	semidirect
151. #386	C₂×C₆₇	semidirect
152. C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
209. C₂²⋊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×#124×C₆₇	2
C₂×#121×C₆₇	2
#395×C₆₇	2
#394×C₆₇	2
#392×C₆₇	2
#390×C₆₇	2
#388×C₆₇	2
#386×C₆₇	2
C₄×C₂²⋊C₄×C₆₇	2
#2555	67

Subgroups — C67×#2555 #22556986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2555 #22556986