Subgroups — C₂×C₆₇×#2834 #30715907

Order 17152.

1958 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2834	C₂	1	16	yes	maximal, direct product factor
2	C₂×#401×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#399×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#398×C₆₇	C₂	1	6	yes	maximal, complemented
5	C₂×#397×C₆₇	C₂	1	3	yes	maximal
6	C₂×#385×C₆₇	C₂	1	3	yes	maximal
7	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
9	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
11	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	3	yes	—
13	#401×C₆₇	C₂²	1	16	yes	complemented
14	#399×C₆₇	C₂²	1	8	yes	—
15	#398×C₆₇	C₂²	1	48	yes	complemented
16	#397×C₆₇	C₂²	1	24	yes	—
17	#385×C₆₇	C₂²	1	24	yes	—
18	C₂×#95×C₆₇	C₂²	1	6	yes	—
19	C₂×#95×C₆₇	C₂²	1	3	yes	—
20	C₂×#95×C₆₇	C₂²	1	6	yes	—
21	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
22	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	—	4	6	no	—
24	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
25	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
27	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
28	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
29	C₂×C₄⋊C₄×C₆₇	—	4	6	no	—
30	C₂×C₄⋊C₄×C₆₇	—	4	6	no	—
31	C₂×C₄⋊C₄×C₆₇	—	4	12	no	—
32	C₂×C₄⋊C₄×C₆₇	C₂³	1	24	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
34	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
35	C₂×C₂²⋊C₄×C₆₇	—	2	12	no	—
36	C₂×C₂²⋊C₄×C₆₇	—	4	6	no	—
37	C₂×C₂²⋊C₄×C₆₇	C₂³	1	12	yes	—
38	C₂×C₂²⋊C₄×C₆₇	C₂³	1	12	yes	—
39	#95×C₆₇	C₂³	1	24	yes	—
40	#95×C₆₇	C₂³	1	12	yes	—
41	#95×C₆₇	C₂³	1	24	yes	—
42	#95×C₆₇	C₂³	1	4	yes	—
43	C₂⁴×C₆₇	—	2	6	no	—
44	C₂⁴×C₆₇	—	2	6	no	—
45	C₂⁴×C₆₇	#41	1	4	yes	—
46	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
47	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	—	2	12	no	—
49	C₂²×C₄×C₆₇	—	2	6	no	—
50	C₂²×C₄×C₆₇	—	2	6	no	—
51	C₂²×C₄×C₆₇	—	2	12	no	—
52	C₂²×C₄×C₆₇	—	2	12	no	—
53	C₂²×C₄×C₆₇	—	2	12	no	—
54	C₂²×C₄×C₆₇	—	2	12	no	—
55	C₂²×C₄×C₆₇	—	2	12	no	—
56	C₂²×C₄×C₆₇	—	4	12	no	—
57	C₂²×C₄×C₆₇	—	4	16	no	—
58	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
59	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
60	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
62	C₄⋊C₄×C₆₇	—	4	12	no	—
63	C₄⋊C₄×C₆₇	—	4	12	no	—
64	C₄⋊C₄×C₆₇	—	4	24	no	—
65	C₂²⋊C₄×C₆₇	—	2	24	no	—
66	C₂²⋊C₄×C₆₇	—	4	12	no	—
67	C₂³×C₆₇	—	2	24	no	—
68	C₂³×C₆₇	—	2	6	no	—
69	C₂³×C₆₇	—	2	6	no	—
70	C₂³×C₆₇	—	4	4	no	—
71	C₂³×C₆₇	—	4	12	no	—
72	C₂³×C₆₇	C₂×#41	1	4	yes	—
73	C₂³×C₆₇	C₂×#41	1	2	yes	—
74	C₂³×C₆₇	H(5,𝔽₂)	1	6	yes	—
75	C₂³×C₆₇	2₋¹⁺⁴	1	6	yes	—
76	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	12	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	12	no	—
82	C₂×C₄×C₆₇	—	2	12	no	—
83	C₂×C₄×C₆₇	—	2	12	no	—
84	C₂×C₄×C₆₇	—	2	12	no	—
85	C₂×C₄×C₆₇	—	4	16	no	—
86	C₂×C₄×C₆₇	—	4	24	no	—
87	C₂×C₄×C₆₇	—	4	12	no	—
88	C₂×C₄×C₆₇	—	4	8	no	—
89	C₂×C₄×C₆₇	—	4	12	no	—
90	C₂²×C₆₇	—	2	12	no	—
91	C₂²×C₆₇	—	4	16	no	—
92	C₂²×C₆₇	—	4	2	no	—
93	C₂²×C₆₇	—	4	6	no	—
94	C₂²×C₆₇	#539	1	12	yes	—
95	C₂²×C₆₇	#560	1	4	yes	—
96	C₂²×C₆₇	#563	1	12	yes	—
97	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
98	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	3	yes	—
99	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	3	yes	—
100	C₄×C₆₇	—	4	12	no	—
101	C₄×C₆₇	—	4	6	no	—
102	C₄×C₆₇	—	4	4	no	—
103	C₄×C₆₇	—	4	6	no	—
104	C₂×#2834	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂×C₆₇	—	4	4	no	—
106	C₂×C₆₇	#2834	1	8	yes	direct product factor
107	C₂×C₆₇	C₂×#539	1	3	yes	—
108	C₂×C₆₇	C₂×#560	1	1	yes	characteristic
109	C₂×C₆₇	C₂×#563	1	3	yes	—
110	#2834	C₂×C₆₇	1	16	yes	direct product factor
111	C₂×#401	C₂×C₆₇	1	2	yes	complemented
112	C₂×#399	C₂×C₆₇	1	1	yes	characteristic
113	C₂×#398	C₂×C₆₇	1	6	yes	complemented
114	C₂×#397	C₂×C₆₇	1	3	yes	—
115	C₂×#385	C₂×C₆₇	1	3	yes	—
116	C₆₇	C₂×#2834	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
118	C₂²×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
119	C₂²×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
120	C₂²×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
121	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	3	yes	—
122	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	3	yes	—
123	#401	C₂²×C₆₇	1	16	yes	complemented
124	#399	C₂²×C₆₇	1	8	yes	—
125	#398	C₂²×C₆₇	1	48	yes	complemented
126	#397	C₂²×C₆₇	1	24	yes	—
127	#385	C₂²×C₆₇	1	24	yes	—
128	C₂×#95	C₂²×C₆₇	1	3	yes	—
129	C₂×#95	C₂²×C₆₇	1	6	yes	—
130	C₂×#95	C₂²×C₆₇	1	6	yes	—
131	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
132	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
133	C₂³×C₄	—	4	6	no	—
134	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
135	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
136	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
137	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
138	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
139	C₂×C₄⋊C₄	—	4	6	no	—
140	C₂×C₄⋊C₄	—	4	6	no	—
141	C₂×C₄⋊C₄	—	4	12	no	—
142	C₂×C₄⋊C₄	C₂³×C₆₇	1	24	yes	—
143	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
144	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
145	C₂×C₂²⋊C₄	—	2	12	no	—
146	C₂×C₂²⋊C₄	—	4	6	no	—
147	C₂×C₂²⋊C₄	C₂³×C₆₇	1	12	yes	—
148	C₂×C₂²⋊C₄	C₂³×C₆₇	1	12	yes	—
149	#95	C₂³×C₆₇	1	24	yes	—
150	#95	C₂³×C₆₇	1	12	yes	—
151	#95	C₂³×C₆₇	1	24	yes	—
152	#95	C₂³×C₆₇	1	4	yes	—
153	C₂⁴	—	2	6	no	—
154	C₂⁴	—	2	6	no	—
155	C₂⁴	#41×C₆₇	1	4	yes	—
156	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
157	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
158	C₂²×C₄	—	2	12	no	—
159	C₂²×C₄	—	2	6	no	—
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	2	12	no	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	2	12	no	—
164	C₂²×C₄	—	2	12	no	—
165	C₂²×C₄	—	2	12	no	—
166	C₂²×C₄	—	4	12	no	—
167	C₂²×C₄	—	4	16	no	—
168	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
169	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
170	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
171	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
172	C₄⋊C₄	—	4	12	no	—
173	C₄⋊C₄	—	4	12	no	—
174	C₄⋊C₄	—	4	24	no	—
175	C₂²⋊C₄	—	2	24	no	—
176	C₂²⋊C₄	—	4	12	no	—
177	C₂³	—	2	24	no	—
178	C₂³	—	2	6	no	—
179	C₂³	—	2	6	no	—
180	C₂³	—	4	4	no	—
181	C₂³	—	4	12	no	—
182	C₂³	C₂×#41×C₆₇	1	4	yes	—
183	C₂³	C₂×#41×C₆₇	1	2	yes	—
184	C₂³	H(5,𝔽₂)×C₆₇	1	6	yes	—
185	C₂³	2₋¹⁺⁴×C₆₇	1	6	yes	—
186	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
187	C₂×C₄	—	2	12	no	—
188	C₂×C₄	—	2	6	no	—
189	C₂×C₄	—	2	6	no	—
190	C₂×C₄	—	2	12	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	4	16	no	—
196	C₂×C₄	—	4	24	no	—
197	C₂×C₄	—	4	8	no	—
198	C₂×C₄	—	4	12	no	—
199	C₂×C₄	—	4	12	no	—
200	C₂²	—	2	12	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	6	no	—
203	C₂²	—	4	16	no	—
204	C₂²	#539×C₆₇	1	12	yes	—
205	C₂²	#560×C₆₇	1	4	yes	—
206	C₂²	#563×C₆₇	1	12	yes	—
207	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
208	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	3	yes	—
209	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	3	yes	—
210	C₄	—	4	4	no	—
211	C₄	—	4	6	no	—
212	C₄	—	4	6	no	—
213	C₄	—	4	12	no	—
214	C₂	—	4	4	no	—
215	C₂	C₆₇×#2834	1	8	yes	direct product factor
216	C₂	C₂×#539×C₆₇	1	3	yes	—
217	C₂	C₂×#560×C₆₇	1	1	yes	characteristic
218	C₂	C₂×#563×C₆₇	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
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×#2834	C₂	direct (N × H)
2. C₂×#401×C₆₇	C₂	semidirect
4. C₂×#398×C₆₇	C₂	semidirect
13. #401×C₆₇	C₂²	semidirect
15. #398×C₆₇	C₂²	semidirect
104. C₂×#2834	C₆₇	direct (N × H)
106. C₂×C₆₇	#2834	direct (N × H)
110. #2834	C₂×C₆₇	direct (N × H)
111. C₂×#401	C₂×C₆₇	semidirect
113. C₂×#398	C₂×C₆₇	semidirect
116. C₆₇	C₂×#2834	direct (N × H)
123. #401	C₂²×C₆₇	semidirect
125. #398	C₂²×C₆₇	semidirect
215. C₂	C₆₇×#2834	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×#2834	2
C₂×#401×C₆₇	2
C₂×#399×C₆₇	2
C₂×#398×C₆₇	2
C₂×#397×C₆₇	2
C₂×#385×C₆₇	2
C₂×#2834	67

Subgroups — C2×C67×#2834 #30715907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2834 #30715907