Subgroups — C₆₇×#2326 #22556757

Order 8576.

612 conjugacy classes of subgroups.

The subgroup lattice has 224 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	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#390×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#388×C₆₇	C₂	1	4	yes	maximal, complemented
7	#385×C₆₇	C₂	1	4	yes	maximal, complemented
8	#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₆₇	C₂²	1	2	yes	—
13	#124×C₆₇	C₄	1	8	yes	complemented
14	C₄×Q₈×C₆₇	—	2	4	no	—
15	C₄×D₄×C₆₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#95×C₆₇	C₂²	1	2	yes	—
28	#95×C₆₇	C₂²	1	2	yes	—
29	#95×C₆₇	C₂²	1	4	yes	—
30	C₂⁴×C₆₇	C₂³	1	2	yes	—
31	C₂×Q₈×C₆₇	—	2	2	no	—
32	C₂×Q₈×C₆₇	—	2	2	no	—
33	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
34	C₂×D₄×C₆₇	—	2	2	no	—
35	C₂×D₄×C₆₇	—	2	2	no	—
36	C₂×D₄×C₆₇	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₆₇	—	2	4	no	—
38	C₂²×C₄×C₆₇	—	4	4	no	—
39	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
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	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₄⋊C₄×C₆₇	—	2	4	no	—
49	C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₄⋊C₄×C₆₇	—	4	4	no	—
51	C₂²⋊C₄×C₆₇	—	2	4	no	—
52	C₂²⋊C₄×C₆₇	—	4	4	no	—
53	C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	complemented
54	C₄²×C₆₇	—	2	4	no	—
55	C₄²×C₆₇	—	2	2	no	—
56	C₄²×C₆₇	C₂×C₄	1	4	yes	complemented
57	C₄²×C₆₇	D₄	1	4	yes	—
58	C₂³×C₆₇	—	2	2	no	—
59	C₂³×C₆₇	—	2	2	no	—
60	C₂³×C₆₇	—	2	4	no	—
61	C₂³×C₆₇	—	2	4	no	—
62	C₂³×C₆₇	C₂²×C₄	1	4	yes	—
63	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
64	Q₈×C₆₇	—	2	8	no	—
65	D₄×C₆₇	—	2	8	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	4	8	no	—
80	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₆₇	#41	1	2	yes	—
87	C₂²×C₆₇	—	2	4	no	—
88	C₂²×C₆₇	—	2	8	no	—
89	C₂²×C₆₇	—	4	2	no	—
90	C₂²×C₆₇	—	4	2	no	—
91	C₂²×C₆₇	—	4	4	no	—
92	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
95	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
96	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
97	C₄×C₆₇	—	2	4	no	—
98	C₄×C₆₇	—	2	4	no	—
99	C₄×C₆₇	—	2	2	no	—
100	C₄×C₆₇	—	2	2	no	—
101	C₄×C₆₇	—	4	2	no	—
102	C₄×C₆₇	—	4	2	no	—
103	C₄×C₆₇	—	4	2	no	—
104	C₄×C₆₇	C₄×D₄	1	4	yes	—
105	C₂×C₆₇	—	4	4	no	—
106	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂×C₆₇	#520	1	2	yes	—
108	C₂×C₆₇	#535	1	1	yes	characteristic
109	C₂×C₆₇	#536	1	1	yes	characteristic
110	C₂×C₆₇	#542	1	2	yes	—
111	#2326	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₆₇	#2326	1	1	yes	characteristic, direct product factor, Sylow
113	C₂×#124	C₂×C₆₇	1	1	yes	characteristic
114	C₂×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
115	C₂×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
116	#391	C₂×C₆₇	1	1	yes	characteristic, complemented
117	#390	C₂×C₆₇	1	1	yes	characteristic
118	#388	C₂×C₆₇	1	4	yes	complemented
119	#385	C₂×C₆₇	1	4	yes	complemented
120	#384	C₂×C₆₇	1	1	yes	characteristic, complemented
121	#376	C₂×C₆₇	1	1	yes	characteristic, complemented
122	C₂²×Q₈	C₂²×C₆₇	1	1	yes	characteristic
123	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
124	C₂³×C₄	C₂²×C₆₇	1	2	yes	—
125	#124	C₄×C₆₇	1	8	yes	complemented
126	C₄×Q₈	—	2	4	no	—
127	C₄×D₄	—	2	4	no	—
128	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
130	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
131	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
135	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
136	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
138	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
139	#95	C₂²×C₆₇	1	2	yes	—
140	#95	C₂²×C₆₇	1	2	yes	—
141	#95	C₂²×C₆₇	1	4	yes	—
142	C₂⁴	C₂³×C₆₇	1	2	yes	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	C₂×C₄×C₆₇	1	4	yes	—
149	C₂²×C₄	—	2	4	no	—
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
152	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
154	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
155	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
156	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	4	4	no	—
165	C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	complemented
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	C₂×C₄×C₆₇	1	4	yes	complemented
169	C₄²	D₄×C₆₇	1	4	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	4	no	—
173	C₂³	—	2	4	no	—
174	C₂³	C₂²×C₄×C₆₇	1	4	yes	—
175	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
176	Q₈	—	2	8	no	—
177	D₄	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	C₂²×C₄×C₆₇	1	4	yes	—
193	C₂×C₄	C₂²×C₄×C₆₇	1	4	yes	—
194	C₂×C₄	C₂²×C₄×C₆₇	1	2	yes	—
195	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
196	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
197	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
198	C₂×C₄	#41×C₆₇	1	2	yes	—
199	C₂²	—	2	4	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	4	no	—
204	C₂²	C₂³×C₄×C₆₇	1	1	yes	characteristic
205	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
206	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
207	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
208	C₂²	2₋¹⁺⁴×C₆₇	1	2	yes	—
209	C₄	—	2	4	no	—
210	C₄	—	2	4	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	C₄×D₄×C₆₇	1	4	yes	—
217	C₂	—	4	4	no	—
218	C₂	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
219	C₂	#520×C₆₇	1	2	yes	—
220	C₂	#535×C₆₇	1	1	yes	characteristic
221	C₂	#536×C₆₇	1	1	yes	characteristic
222	C₂	#542×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₆₇	C₂	semidirect
4. #391×C₆₇	C₂	semidirect
6. #388×C₆₇	C₂	semidirect
7. #385×C₆₇	C₂	semidirect
8. #384×C₆₇	C₂	semidirect
9. #376×C₆₇	C₂	semidirect
13. #124×C₆₇	C₄	semidirect
33. C₂×Q₈×C₆₇	C₂×C₄	semidirect
53. C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
56. C₄²×C₆₇	C₂×C₄	semidirect
111. #2326	C₆₇	direct (N × H)
112. C₆₇	#2326	direct (N × H)
114. C₂×C₄×Q₈	C₂×C₆₇	semidirect
116. #391	C₂×C₆₇	semidirect
118. #388	C₂×C₆₇	semidirect
119. #385	C₂×C₆₇	semidirect
120. #384	C₂×C₆₇	semidirect
121. #376	C₂×C₆₇	semidirect
125. #124	C₄×C₆₇	semidirect
145. C₂×Q₈	C₂×C₄×C₆₇	semidirect
165. C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
168. C₄²	C₂×C₄×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×C₄×Q₈×C₆₇	2
C₂×C₄×D₄×C₆₇	2
#391×C₆₇	2
#390×C₆₇	2
#388×C₆₇	2
#385×C₆₇	2
#384×C₆₇	2
#376×C₆₇	2
#2326	67

Subgroups — C67×#2326 #22556757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2326 #22556757