Subgroups — C₂²×#336×C₆₇ #30699583

Order 17152.

3086 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	C₂⁴×C₈×C₆₇	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#336×C₆₇	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₆₇	C₄	1	2	yes	—
5	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₆₇	—	2	8	no	—
7	C₂³×C₈×C₆₇	—	2	8	no	—
8	C₂³×C₈×C₆₇	C₄	1	16	yes	complemented
9	C₂³×C₈×C₆₇	C₂²	1	12	yes	—
10	C₂²×C₄²×C₆₇	—	2	8	no	—
11	C₂²×C₄²×C₆₇	C₄	1	6	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	6	yes	—
13	#336×C₆₇	C₂²	1	16	yes	direct product factor
14	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₆₇	—	2	8	no	—
16	C₂³×C₄×C₆₇	—	2	8	no	—
17	C₂³×C₄×C₆₇	C₈	1	16	yes	complemented
18	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
19	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₆₇	D₄	1	4	yes	—
22	C₂³×C₄×C₆₇	D₄	1	8	yes	—
23	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
24	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
25	C₂²×C₈×C₆₇	—	2	64	no	—
26	C₂²×C₈×C₆₇	—	2	24	no	—
27	C₂²×C₈×C₆₇	—	2	24	no	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	48	yes	complemented
29	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
30	C₂×C₄²×C₆₇	—	2	24	no	—
31	C₂×C₄²×C₆₇	—	2	24	no	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
33	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
34	C₂⁴×C₆₇	C₄²	1	4	yes	—
35	C₂⁴×C₆₇	C₂²⋊C₄	1	4	yes	—
36	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
37	C₂⁴×C₆₇	C₂×C₈	1	8	yes	—
38	C₂⁴×C₆₇	M₁₆	1	8	yes	—
39	C₂⁴×C₆₇	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₆₇	—	2	64	no	—
41	C₂²×C₄×C₆₇	—	2	24	no	—
42	C₂²×C₄×C₆₇	—	2	24	no	—
43	C₂²×C₄×C₆₇	C₄²	1	12	yes	—
44	C₂²×C₄×C₆₇	C₂²⋊C₄	1	12	yes	—
45	C₂²×C₄×C₆₇	C₂²⋊C₄	1	32	yes	—
46	C₂²×C₄×C₆₇	C₄⋊C₄	1	12	yes	—
47	C₂²×C₄×C₆₇	C₄⋊C₄	1	32	yes	—
48	C₂²×C₄×C₆₇	C₂×C₈	1	48	yes	complemented
49	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
54	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
55	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₆₇	—	2	96	no	—
57	C₂×C₈×C₆₇	—	2	8	no	—
58	C₂×C₈×C₆₇	—	2	8	no	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	16	yes	complemented
60	C₄²×C₆₇	—	2	24	no	—
61	C₄²×C₆₇	—	2	8	no	—
62	C₂³×C₆₇	#95	1	16	yes	—
63	C₂³×C₆₇	C₄×C₈	1	16	yes	—
64	C₂³×C₆₇	#97	1	16	yes	—
65	C₂³×C₆₇	C₂²⋊C₈	1	32	yes	—
66	C₂³×C₆₇	C₄⋊C₈	1	32	yes	—
67	C₂³×C₆₇	C₂×C₄²	1	6	yes	—
68	C₂³×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
69	C₂³×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
70	C₂³×C₆₇	C₂²×C₈	1	12	yes	—
71	C₂³×C₆₇	C₂×M₁₆	1	12	yes	—
72	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	—	2	8	no	—
74	C₂×C₄×C₆₇	—	2	8	no	—
75	C₂×C₄×C₆₇	—	2	96	no	—
76	C₂×C₄×C₆₇	#95	1	48	yes	—
77	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
78	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	24	yes	—
79	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
80	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
81	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×C₄×C₆₇	C₂²×C₈	1	16	yes	complemented
83	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
87	C₈×C₆₇	—	2	16	no	—
88	C₂²×C₆₇	#336	1	64	yes	direct product factor
89	C₂²×C₆₇	C₂×#95	1	12	yes	—
90	C₂²×C₆₇	C₂×C₄×C₈	1	12	yes	—
91	C₂²×C₆₇	C₂×#97	1	12	yes	—
92	C₂²×C₆₇	C₂×C₂²⋊C₈	1	24	yes	—
93	C₂²×C₆₇	C₂×C₄⋊C₈	1	24	yes	—
94	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
95	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₆₇	C₂³×C₈	1	2	yes	—
98	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
99	C₄×C₆₇	—	2	16	no	—
100	C₄×C₆₇	C₂×#95	1	12	yes	—
101	C₄×C₆₇	C₂²×C₂²⋊C₄	1	2	yes	—
102	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
103	C₂²×#336	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₆₇	C₂×#336	1	24	yes	direct product factor
105	C₂×C₆₇	C₂²×#95	1	1	yes	characteristic
106	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
107	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
108	C₂×C₆₇	C₂²×C₂²⋊C₈	1	2	yes	—
109	C₂×C₆₇	C₂²×C₄⋊C₈	1	2	yes	—
110	C₂⁴×C₈	C₂×C₆₇	1	2	yes	—
111	C₂³×C₄²	C₂×C₆₇	1	1	yes	characteristic
112	C₂×#336	C₂×C₆₇	1	12	yes	direct product factor
113	C₆₇	C₂²×#336	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴×C₄	C₄×C₆₇	1	2	yes	—
115	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
116	C₂³×C₈	—	2	8	no	—
117	C₂³×C₈	—	2	8	no	—
118	C₂³×C₈	C₄×C₆₇	1	16	yes	complemented
119	C₂³×C₈	C₂²×C₆₇	1	12	yes	—
120	C₂²×C₄²	—	2	8	no	—
121	C₂²×C₄²	C₄×C₆₇	1	6	yes	—
122	C₂²×C₄²	C₂²×C₆₇	1	6	yes	—
123	#336	C₂²×C₆₇	1	16	yes	direct product factor
124	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
125	C₂³×C₄	—	2	8	no	—
126	C₂³×C₄	—	2	8	no	—
127	C₂³×C₄	C₈×C₆₇	1	16	yes	complemented
128	C₂³×C₄	C₂×C₄×C₆₇	1	12	yes	—
129	C₂³×C₄	C₂×C₄×C₆₇	1	3	yes	—
130	C₂³×C₄	C₂×C₄×C₆₇	1	8	yes	—
131	C₂³×C₄	D₄×C₆₇	1	4	yes	—
132	C₂³×C₄	D₄×C₆₇	1	8	yes	—
133	C₂³×C₄	Q₈×C₆₇	1	4	yes	—
134	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
135	C₂²×C₈	—	2	64	no	—
136	C₂²×C₈	—	2	24	no	—
137	C₂²×C₈	—	2	24	no	—
138	C₂²×C₈	C₂×C₄×C₆₇	1	48	yes	complemented
139	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
140	C₂×C₄²	—	2	24	no	—
141	C₂×C₄²	—	2	24	no	—
142	C₂×C₄²	C₂×C₄×C₆₇	1	12	yes	—
143	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
144	C₂⁴	C₄²×C₆₇	1	4	yes	—
145	C₂⁴	C₂²⋊C₄×C₆₇	1	4	yes	—
146	C₂⁴	C₄⋊C₄×C₆₇	1	4	yes	—
147	C₂⁴	C₂×C₈×C₆₇	1	8	yes	—
148	C₂⁴	M₁₆×C₆₇	1	8	yes	—
149	C₂⁴	C₂²×C₄×C₆₇	1	3	yes	—
150	C₂²×C₄	—	2	24	no	—
151	C₂²×C₄	—	2	24	no	—
152	C₂²×C₄	—	2	64	no	—
153	C₂²×C₄	C₄²×C₆₇	1	12	yes	—
154	C₂²×C₄	C₂²⋊C₄×C₆₇	1	32	yes	—
155	C₂²×C₄	C₂²⋊C₄×C₆₇	1	12	yes	—
156	C₂²×C₄	C₄⋊C₄×C₆₇	1	32	yes	—
157	C₂²×C₄	C₄⋊C₄×C₆₇	1	12	yes	—
158	C₂²×C₄	C₂×C₈×C₆₇	1	48	yes	complemented
159	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
160	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
161	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
162	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
163	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
164	C₂²×C₄	C₂×Q₈×C₆₇	1	6	yes	—
165	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
166	C₂×C₈	—	2	96	no	—
167	C₂×C₈	—	2	8	no	—
168	C₂×C₈	—	2	8	no	—
169	C₂×C₈	C₂²×C₄×C₆₇	1	16	yes	complemented
170	C₄²	—	2	24	no	—
171	C₄²	—	2	8	no	—
172	C₂³	#95×C₆₇	1	16	yes	—
173	C₂³	C₄×C₈×C₆₇	1	16	yes	—
174	C₂³	#97×C₆₇	1	16	yes	—
175	C₂³	C₂²⋊C₈×C₆₇	1	32	yes	—
176	C₂³	C₄⋊C₈×C₆₇	1	32	yes	—
177	C₂³	C₂×C₄²×C₆₇	1	6	yes	—
178	C₂³	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
179	C₂³	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
180	C₂³	C₂²×C₈×C₆₇	1	12	yes	—
181	C₂³	C₂×M₁₆×C₆₇	1	12	yes	—
182	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	96	no	—
186	C₂×C₄	#95×C₆₇	1	48	yes	—
187	C₂×C₄	C₂×C₄²×C₆₇	1	6	yes	—
188	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	24	yes	—
189	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	6	yes	—
190	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	24	yes	—
191	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	6	yes	—
192	C₂×C₄	C₂²×C₈×C₆₇	1	16	yes	complemented
193	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
194	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
196	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
197	C₈	—	2	16	no	—
198	C₂²	#336×C₆₇	1	64	yes	direct product factor
199	C₂²	C₂×#95×C₆₇	1	12	yes	—
200	C₂²	C₂×C₄×C₈×C₆₇	1	12	yes	—
201	C₂²	C₂×#97×C₆₇	1	12	yes	—
202	C₂²	C₂×C₂²⋊C₈×C₆₇	1	24	yes	—
203	C₂²	C₂×C₄⋊C₈×C₆₇	1	24	yes	—
204	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
205	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
206	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
207	C₂²	C₂³×C₈×C₆₇	1	2	yes	—
208	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
209	C₄	—	2	16	no	—
210	C₄	C₂×#95×C₆₇	1	12	yes	—
211	C₄	C₂²×C₂²⋊C₄×C₆₇	1	2	yes	—
212	C₄	C₂²×C₄⋊C₄×C₆₇	1	2	yes	—
213	C₂	C₂×#336×C₆₇	1	24	yes	direct product factor
214	C₂	C₂²×#95×C₆₇	1	1	yes	characteristic
215	C₂	C₂²×C₄×C₈×C₆₇	1	1	yes	characteristic
216	C₂	C₂²×#97×C₆₇	1	1	yes	characteristic
217	C₂	C₂²×C₂²⋊C₈×C₆₇	1	2	yes	—
218	C₂	C₂²×C₄⋊C₈×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄×C₆₇	C₂²
commutator	C₂	C₂²×C₄×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂⁵×C₆₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#336×C₆₇	C₂	direct (N × H)
8. C₂³×C₈×C₆₇	C₄	semidirect
13. #336×C₆₇	C₂²	direct (N × H)
17. C₂³×C₄×C₆₇	C₈	semidirect
28. C₂²×C₈×C₆₇	C₂×C₄	semidirect
48. C₂²×C₄×C₆₇	C₂×C₈	semidirect
59. C₂×C₈×C₆₇	C₂²×C₄	semidirect
82. C₂×C₄×C₆₇	C₂²×C₈	semidirect
88. C₂²×C₆₇	#336	direct (N × H)
103. C₂²×#336	C₆₇	direct (N × H)
104. C₂×C₆₇	C₂×#336	direct (N × H)
112. C₂×#336	C₂×C₆₇	direct (N × H)
113. C₆₇	C₂²×#336	direct (N × H)
118. C₂³×C₈	C₄×C₆₇	semidirect
123. #336	C₂²×C₆₇	direct (N × H)
127. C₂³×C₄	C₈×C₆₇	semidirect
138. C₂²×C₈	C₂×C₄×C₆₇	semidirect
158. C₂²×C₄	C₂×C₈×C₆₇	semidirect
169. C₂×C₈	C₂²×C₄×C₆₇	semidirect
192. C₂×C₄	C₂²×C₈×C₆₇	semidirect
198. C₂²	#336×C₆₇	direct (N × H)
213. C₂	C₂×#336×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₆₇	2
C₂³×C₄²×C₆₇	2
C₂×#336×C₆₇	2
C₂²×#336	67

Subgroups — C22×#336×C67 #30699583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#336×C₆₇ #30699583