Subgroups — C₂×C₆₇×#2644 #30713868

Order 17152.

2222 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	C₆₇×#2644	C₂	1	16	yes	maximal, direct product factor
2	C₂×#397×C₆₇	C₂	1	6	yes	maximal, complemented
3	C₂×#396×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#390×C₆₇	C₂	1	3	yes	maximal
5	C₂×#389×C₆₇	C₂	1	3	yes	maximal, complemented
6	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
10	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	6	yes	—
11	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	6	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
13	#397×C₆₇	C₂²	1	48	yes	complemented
14	#396×C₆₇	C₂²	1	16	yes	complemented
15	#390×C₆₇	C₂²	1	24	yes	—
16	#389×C₆₇	C₂²	1	24	yes	complemented
17	#376×C₆₇	C₂²	1	8	yes	complemented
18	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#95×C₆₇	C₂²	1	3	yes	—
20	C₂×#95×C₆₇	C₂²	1	6	yes	—
21	C₂⁵×C₆₇	C₂³	1	2	yes	—
22	C₂²×D₄×C₆₇	—	2	12	no	—
23	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
24	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
29	C₂×C₄⋊C₄×C₆₇	—	4	12	no	—
30	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
31	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
32	C₂×C₂²⋊C₄×C₆₇	—	4	12	no	—
33	C₂×C₂²⋊C₄×C₆₇	—	4	12	no	—
34	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
35	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
36	C₂×C₄²×C₆₇	—	2	12	no	—
37	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
38	#95×C₆₇	C₂³	1	4	yes	—
39	#95×C₆₇	C₂³	1	12	yes	—
40	#95×C₆₇	C₂³	1	24	yes	—
41	C₂⁴×C₆₇	—	2	12	no	—
42	C₂⁴×C₆₇	—	2	12	no	—
43	C₂⁴×C₆₇	—	2	4	no	—
44	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
45	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
46	C₂×D₄×C₆₇	—	2	24	no	—
47	C₂×D₄×C₆₇	—	4	16	no	—
48	C₂²×C₄×C₆₇	—	2	6	no	—
49	C₂²×C₄×C₆₇	—	2	6	no	—
50	C₂²×C₄×C₆₇	—	2	12	no	—
51	C₂²×C₄×C₆₇	—	2	6	no	—
52	C₂²×C₄×C₆₇	—	2	6	no	—
53	C₂²×C₄×C₆₇	—	2	12	no	—
54	C₂²×C₄×C₆₇	—	2	12	no	—
55	C₂²×C₄×C₆₇	—	2	6	no	—
56	C₂²×C₄×C₆₇	—	2	6	no	—
57	C₂²×C₄×C₆₇	#41	1	12	yes	—
58	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
60	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
62	C₄⋊C₄×C₆₇	—	2	24	no	—
63	C₄⋊C₄×C₆₇	—	4	24	no	—
64	C₂²⋊C₄×C₆₇	—	4	24	no	—
65	C₂²⋊C₄×C₆₇	—	4	24	no	—
66	C₄²×C₆₇	—	2	24	no	—
67	C₂³×C₆₇	—	2	12	no	—
68	C₂³×C₆₇	—	2	12	no	—
69	C₂³×C₆₇	—	2	4	no	—
70	C₂³×C₆₇	—	4	8	no	—
71	C₂³×C₆₇	—	4	24	no	—
72	C₂³×C₆₇	—	4	24	no	—
73	C₂³×C₆₇	C₂×#41	1	6	yes	—
74	C₂³×C₆₇	H(5,𝔽₂)	1	6	yes	—
75	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
76	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
77	D₄×C₆₇	—	4	16	no	—
78	C₂×C₄×C₆₇	—	2	6	no	—
79	C₂×C₄×C₆₇	—	2	6	no	—
80	C₂×C₄×C₆₇	—	2	12	no	—
81	C₂×C₄×C₆₇	—	2	6	no	—
82	C₂×C₄×C₆₇	—	2	6	no	—
83	C₂×C₄×C₆₇	—	2	12	no	—
84	C₂×C₄×C₆₇	—	2	12	no	—
85	C₂×C₄×C₆₇	—	2	6	no	—
86	C₂×C₄×C₆₇	—	2	6	no	—
87	C₂×C₄×C₆₇	—	4	12	no	—
88	C₂×C₄×C₆₇	—	4	24	no	—
89	C₂×C₄×C₆₇	—	4	12	no	—
90	C₂×C₄×C₆₇	#126	1	16	yes	—
91	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
92	C₂²×C₆₇	—	4	4	no	—
93	C₂²×C₆₇	—	4	12	no	—
94	C₂²×C₆₇	—	4	12	no	—
95	C₂²×C₆₇	—	8	16	no	—
96	C₂²×C₆₇	C₂×#126	1	4	yes	—
97	C₂²×C₆₇	#540	1	12	yes	—
98	C₂²×C₆₇	#542	1	12	yes	—
99	C₂²×C₆₇	C₂²×#41	1	3	yes	—
100	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	3	yes	—
101	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
102	C₄×C₆₇	—	4	6	no	—
103	C₄×C₆₇	—	4	12	no	—
104	C₄×C₆₇	—	4	6	no	—
105	C₄×C₆₇	C₂×#126	1	8	yes	—
106	C₂×#2644	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₆₇	—	8	4	no	—
108	C₂×C₆₇	#2644	1	8	yes	direct product factor
109	C₂×C₆₇	C₂²×#126	1	1	yes	characteristic
110	C₂×C₆₇	C₂×#540	1	3	yes	—
111	C₂×C₆₇	C₂×#542	1	3	yes	—
112	#2644	C₂×C₆₇	1	16	yes	direct product factor
113	C₂×#397	C₂×C₆₇	1	6	yes	complemented
114	C₂×#396	C₂×C₆₇	1	2	yes	complemented
115	C₂×#390	C₂×C₆₇	1	3	yes	—
116	C₂×#389	C₂×C₆₇	1	3	yes	complemented
117	C₂×#376	C₂×C₆₇	1	1	yes	characteristic, complemented
118	C₆₇	C₂×#2644	1	1	yes	characteristic, direct product factor, Sylow
119	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
120	C₂²×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
121	C₂²×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
122	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	6	yes	—
123	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	6	yes	—
124	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
125	#397	C₂²×C₆₇	1	48	yes	complemented
126	#396	C₂²×C₆₇	1	16	yes	complemented
127	#390	C₂²×C₆₇	1	24	yes	—
128	#389	C₂²×C₆₇	1	24	yes	complemented
129	#376	C₂²×C₆₇	1	8	yes	complemented
130	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
131	C₂×#95	C₂²×C₆₇	1	3	yes	—
132	C₂×#95	C₂²×C₆₇	1	6	yes	—
133	C₂⁵	C₂³×C₆₇	1	2	yes	—
134	C₂²×D₄	—	2	12	no	—
135	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
136	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
137	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
138	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
139	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	2	12	no	—
141	C₂×C₄⋊C₄	—	4	12	no	—
142	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
143	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
144	C₂×C₂²⋊C₄	—	4	12	no	—
145	C₂×C₂²⋊C₄	—	4	12	no	—
146	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
147	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
148	C₂×C₄²	—	2	12	no	—
149	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
150	#95	C₂³×C₆₇	1	4	yes	—
151	#95	C₂³×C₆₇	1	12	yes	—
152	#95	C₂³×C₆₇	1	24	yes	—
153	C₂⁴	—	2	12	no	—
154	C₂⁴	—	2	12	no	—
155	C₂⁴	—	2	4	no	—
156	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
157	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
158	C₂×D₄	—	2	24	no	—
159	C₂×D₄	—	4	16	no	—
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	2	6	no	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	12	no	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	2	6	no	—
169	C₂²×C₄	#41×C₆₇	1	12	yes	—
170	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
171	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
172	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
173	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
174	C₄⋊C₄	—	2	24	no	—
175	C₄⋊C₄	—	4	24	no	—
176	C₂²⋊C₄	—	4	24	no	—
177	C₂²⋊C₄	—	4	24	no	—
178	C₄²	—	2	24	no	—
179	C₂³	—	2	12	no	—
180	C₂³	—	2	12	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	4	8	no	—
183	C₂³	—	4	24	no	—
184	C₂³	—	4	24	no	—
185	C₂³	C₂×#41×C₆₇	1	6	yes	—
186	C₂³	H(5,𝔽₂)×C₆₇	1	6	yes	—
187	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
188	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
189	D₄	—	4	16	no	—
190	C₂×C₄	—	2	6	no	—
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	12	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	4	12	no	—
200	C₂×C₄	—	4	24	no	—
201	C₂×C₄	—	4	12	no	—
202	C₂×C₄	#126×C₆₇	1	16	yes	—
203	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
204	C₂²	—	4	4	no	—
205	C₂²	—	4	12	no	—
206	C₂²	—	4	12	no	—
207	C₂²	—	8	16	no	—
208	C₂²	C₂×#126×C₆₇	1	4	yes	—
209	C₂²	#540×C₆₇	1	12	yes	—
210	C₂²	#542×C₆₇	1	12	yes	—
211	C₂²	C₂²×#41×C₆₇	1	3	yes	—
212	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	3	yes	—
213	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
214	C₄	—	4	6	no	—
215	C₄	—	4	12	no	—
216	C₄	—	4	6	no	—
217	C₄	C₂×#126×C₆₇	1	8	yes	—
218	C₂	—	8	4	no	—
219	C₂	C₆₇×#2644	1	8	yes	direct product factor
220	C₂	C₂²×#126×C₆₇	1	1	yes	characteristic
221	C₂	C₂×#540×C₆₇	1	3	yes	—
222	C₂	C₂×#542×C₆₇	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2644	C₂	direct (N × H)
2. C₂×#397×C₆₇	C₂	semidirect
3. C₂×#396×C₆₇	C₂	semidirect
5. C₂×#389×C₆₇	C₂	semidirect
6. C₂×#376×C₆₇	C₂	semidirect
13. #397×C₆₇	C₂²	semidirect
14. #396×C₆₇	C₂²	semidirect
16. #389×C₆₇	C₂²	semidirect
17. #376×C₆₇	C₂²	semidirect
106. C₂×#2644	C₆₇	direct (N × H)
108. C₂×C₆₇	#2644	direct (N × H)
112. #2644	C₂×C₆₇	direct (N × H)
113. C₂×#397	C₂×C₆₇	semidirect
114. C₂×#396	C₂×C₆₇	semidirect
116. C₂×#389	C₂×C₆₇	semidirect
117. C₂×#376	C₂×C₆₇	semidirect
118. C₆₇	C₂×#2644	direct (N × H)
125. #397	C₂²×C₆₇	semidirect
126. #396	C₂²×C₆₇	semidirect
128. #389	C₂²×C₆₇	semidirect
129. #376	C₂²×C₆₇	semidirect
219. C₂	C₆₇×#2644	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2644	2
C₂×#397×C₆₇	2
C₂×#396×C₆₇	2
C₂×#390×C₆₇	2
C₂×#389×C₆₇	2
C₂×#376×C₆₇	2
C₂×#2644	67

Subgroups — C2×C67×#2644 #30713868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2644 #30713868