Subgroups — C₆₇×#3340 #22557771

Order 8576.

352 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#555×C₆₇	C₂	1	1	yes	maximal, characteristic
3	#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#485×C₆₇	C₂	1	2	yes	maximal, complemented
5	#482×C₆₇	C₂	1	2	yes	maximal, complemented
6	#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#474×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#465×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#427×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	#126×C₆₇	—	2	2	no	—
17	#124×C₆₇	—	2	1	no	—
18	#124×C₆₇	—	2	2	no	—
19	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	#121×C₆₇	—	2	1	no	—
21	#121×C₆₇	C₂²	1	2	yes	—
22	C₄×Q₈×C₆₇	—	2	1	no	—
23	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	#117×C₆₇	—	2	1	no	—
27	#117×C₆₇	C₂²	1	2	yes	—
28	#117×C₆₇	C₂²	1	2	yes	—
29	#107×C₆₇	C₂²	1	2	yes	—
30	#107×C₆₇	C₂²	1	2	yes	—
31	#106×C₆₇	C₂²	1	2	yes	—
32	#106×C₆₇	C₂²	1	2	yes	—
33	#106×C₆₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#103×C₆₇	C₂²	1	2	yes	—
36	#103×C₆₇	C₂²	1	1	yes	characteristic
37	#102×C₆₇	C₂²	1	2	yes	—
38	#102×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
40	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₆₇	—	2	1	no	—
42	C₂×Q₈×C₆₇	—	2	1	no	—
43	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₆₇	—	2	2	no	—
45	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
49	SD₁₆×C₆₇	—	2	2	no	—
50	C₂×C₈×C₆₇	—	2	4	no	—
51	C₂×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
62	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂²⋊C₄×C₆₇	—	2	2	no	—
65	C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂²⋊C₄×C₆₇	—	2	1	no	—
68	C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	1	no	—
72	C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₆₇	—	2	1	no	—
74	C₂³×C₆₇	C₂×D₄	1	2	yes	—
75	Q₈×C₆₇	—	2	1	no	—
76	Q₈×C₆₇	—	2	1	no	—
77	Q₈×C₆₇	—	4	2	no	—
78	Q₈×C₆₇	—	8	1	no	—
79	D₄×C₆₇	—	2	2	no	—
80	D₄×C₆₇	—	2	1	no	—
81	D₄×C₆₇	—	2	1	no	—
82	D₄×C₆₇	—	8	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	1	no	—
86	C₂×C₄×C₆₇	—	2	1	no	—
87	C₂×C₄×C₆₇	—	2	1	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₈×C₆₇	—	2	2	no	—
96	C₈×C₆₇	—	2	1	no	—
97	C₈×C₆₇	#41	1	2	yes	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	2	2	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	4	1	no	—
102	C₂²×C₆₇	—	4	1	no	—
103	C₂²×C₆₇	—	4	1	no	—
104	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₆₇	—	2	2	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	4	1	no	—
108	C₄×C₆₇	—	4	1	no	—
109	C₄×C₆₇	—	4	2	no	—
110	C₄×C₆₇	—	4	1	no	—
111	C₄×C₆₇	—	4	2	no	—
112	C₄×C₆₇	#135	1	2	yes	—
113	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂×C₆₇	—	4	2	no	—
116	C₂×C₆₇	—	8	1	no	—
117	C₂×C₆₇	#547	1	1	yes	characteristic
118	C₂×C₆₇	C₂×#135	1	1	yes	characteristic
119	C₂×C₆₇	#577	1	1	yes	characteristic
120	#3340	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₆₇	#3340	1	1	yes	characteristic, direct product factor, Sylow
122	#556	C₂×C₆₇	1	1	yes	characteristic, complemented
123	#555	C₂×C₆₇	1	1	yes	characteristic
124	#498	C₂×C₆₇	1	1	yes	characteristic, complemented
125	#485	C₂×C₆₇	1	2	yes	complemented
126	#482	C₂×C₆₇	1	2	yes	complemented
127	#477	C₂×C₆₇	1	1	yes	characteristic, complemented
128	#474	C₂×C₆₇	1	1	yes	characteristic, complemented
129	#465	C₂×C₆₇	1	2	yes	complemented
130	C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
131	C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
132	#427	C₂×C₆₇	1	2	yes	complemented
133	C₂×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₆₇	1	2	yes	—
135	C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
136	C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
137	#126	—	2	2	no	—
138	#124	—	2	1	no	—
139	#124	—	2	2	no	—
140	C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
141	#121	—	2	1	no	—
142	#121	C₂²×C₆₇	1	2	yes	—
143	C₄×Q₈	—	2	1	no	—
144	C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
146	C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
147	#117	—	2	1	no	—
148	#117	C₂²×C₆₇	1	2	yes	—
149	#117	C₂²×C₆₇	1	2	yes	—
150	#107	C₂²×C₆₇	1	2	yes	—
151	#107	C₂²×C₆₇	1	2	yes	—
152	#106	C₂²×C₆₇	1	1	yes	characteristic
153	#106	C₂²×C₆₇	1	2	yes	—
154	#106	C₂²×C₆₇	1	2	yes	—
155	C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
156	#103	C₂²×C₆₇	1	1	yes	characteristic
157	#103	C₂²×C₆₇	1	2	yes	—
158	#102	C₂²×C₆₇	1	1	yes	characteristic
159	#102	C₂²×C₆₇	1	2	yes	—
160	C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
161	C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	C₂³×C₆₇	1	1	yes	characteristic
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	D₄×C₆₇	1	1	yes	characteristic
167	C₂×D₄	C₂³×C₆₇	1	1	yes	characteristic
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
170	SD₁₆	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₆₇	1	2	yes	—
174	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
180	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
181	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
183	C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	C₂³×C₆₇	1	1	yes	characteristic
194	C₂³	—	2	1	no	—
195	C₂³	C₂×D₄×C₆₇	1	2	yes	—
196	Q₈	—	2	1	no	—
197	Q₈	—	2	1	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	1	no	—
200	D₄	—	2	2	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	1	no	—
203	D₄	—	8	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
215	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	#41×C₆₇	1	2	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	#135×C₆₇	1	2	yes	—
234	C₄	C₂×#41×C₆₇	1	1	yes	characteristic
235	C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
236	C₂	—	4	2	no	—
237	C₂	—	8	1	no	—
238	C₂	#547×C₆₇	1	1	yes	characteristic
239	C₂	C₂×#135×C₆₇	1	1	yes	characteristic
240	C₂	#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₇
prime radical	#555×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #556×C₆₇	C₂	semidirect
3. #498×C₆₇	C₂	semidirect
4. #485×C₆₇	C₂	semidirect
5. #482×C₆₇	C₂	semidirect
6. #477×C₆₇	C₂	semidirect
7. #474×C₆₇	C₂	semidirect
8. #465×C₆₇	C₂	semidirect
9. C₄×SD₁₆×C₆₇	C₂	semidirect
10. C₈×D₄×C₆₇	C₂	semidirect
11. #427×C₆₇	C₂	semidirect
120. #3340	C₆₇	direct (N × H)
121. C₆₇	#3340	direct (N × H)
122. #556	C₂×C₆₇	semidirect
124. #498	C₂×C₆₇	semidirect
125. #485	C₂×C₆₇	semidirect
126. #482	C₂×C₆₇	semidirect
127. #477	C₂×C₆₇	semidirect
128. #474	C₂×C₆₇	semidirect
129. #465	C₂×C₆₇	semidirect
130. C₄×SD₁₆	C₂×C₆₇	semidirect
131. C₈×D₄	C₂×C₆₇	semidirect
132. #427	C₂×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
#556×C₆₇	2
#555×C₆₇	2
#498×C₆₇	2
#485×C₆₇	2
#482×C₆₇	2
#477×C₆₇	2
#474×C₆₇	2
#465×C₆₇	2
C₄×SD₁₆×C₆₇	2
C₈×D₄×C₆₇	2
#427×C₆₇	2
#3340	67

Subgroups — C67×#3340 #22557771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#3340 #22557771