Subgroups — C₂×C₄²×D₄×C₆₇ #30713242

Order 17152.

5694 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₆₇	C₂	1	3	yes	maximal
2	C₂³×C₄²×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₄²×D₄×C₆₇	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	6	yes	maximal, complemented
6	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	C₂²	1	6	yes	—
9	C₂×C₄×D₄×C₆₇	C₄	1	48	yes	direct product factor
10	C₂×C₄×D₄×C₆₇	C₂²	1	24	yes	—
11	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	6	yes	—
13	C₂²×C₄²×C₆₇	—	2	16	no	—
14	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	6	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	C₂²	1	16	yes	complemented
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	24	yes	complemented
20	C₄×C₂²⋊C₄×C₆₇	C₂²	1	48	yes	complemented
21	C₂×#95×C₆₇	C₂²	1	6	yes	—
22	C₄³×C₆₇	C₂²	1	8	yes	complemented
23	C₂⁵×C₆₇	C₂³	1	2	yes	—
24	C₂²×D₄×C₆₇	C₂×C₄	1	24	yes	—
25	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₆₇	—	2	24	no	—
27	C₂³×C₄×C₆₇	—	2	24	no	—
28	C₂³×C₄×C₆₇	C₂×C₄	1	48	yes	complemented
29	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	24	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
34	C₄×D₄×C₆₇	C₂×C₄	1	192	yes	direct product factor
35	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	24	yes	complemented
36	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
37	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	48	yes	complemented
38	C₂×C₂²⋊C₄×C₆₇	C₂³	1	24	yes	—
39	C₂×C₄²×C₆₇	—	2	24	no	—
40	C₂×C₄²×C₆₇	—	2	12	no	—
41	C₂×C₄²×C₆₇	—	2	64	no	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	24	yes	complemented
43	C₂×C₄²×C₆₇	D₄	1	8	yes	direct product factor
44	C₂×C₄²×C₆₇	C₂³	1	24	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
46	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
48	#95×C₆₇	C₂³	1	24	yes	—
49	C₂⁴×C₆₇	—	2	4	no	—
50	C₂⁴×C₆₇	—	2	12	no	—
51	C₂⁴×C₆₇	C₂²×C₄	1	24	yes	—
52	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
53	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
54	C₂×D₄×C₆₇	C₄²	1	64	yes	direct product factor
55	C₂×D₄×C₆₇	C₂²×C₄	1	48	yes	—
56	C₂²×C₄×C₆₇	—	2	12	no	—
57	C₂²×C₄×C₆₇	—	2	12	no	—
58	C₂²×C₄×C₆₇	—	2	12	no	—
59	C₂²×C₄×C₆₇	—	2	192	no	—
60	C₂²×C₄×C₆₇	—	2	48	no	—
61	C₂²×C₄×C₆₇	—	2	48	no	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	96	yes	complemented
67	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
68	C₂²×C₄×C₆₇	#41	1	12	yes	—
69	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
73	C₄⋊C₄×C₆₇	C₂²×C₄	1	48	yes	complemented
74	C₂²⋊C₄×C₆₇	C₂²×C₄	1	96	yes	complemented
75	C₄²×C₆₇	—	2	48	no	—
76	C₄²×C₆₇	—	2	24	no	—
77	C₄²×C₆₇	C₂²×C₄	1	48	yes	complemented
78	C₄²×C₆₇	C₂×D₄	1	16	yes	direct product factor
79	C₂³×C₆₇	—	2	48	no	—
80	C₂³×C₆₇	—	2	48	no	—
81	C₂³×C₆₇	—	2	4	no	—
82	C₂³×C₆₇	—	2	12	no	—
83	C₂³×C₆₇	C₂×C₄²	1	32	yes	complemented
84	C₂³×C₆₇	C₂³×C₄	1	24	yes	—
85	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
86	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
87	C₂³×C₆₇	C₂×#41	1	6	yes	—
88	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
89	D₄×C₆₇	C₂×C₄²	1	64	yes	direct product factor
90	C₂×C₄×C₆₇	—	2	48	no	—
91	C₂×C₄×C₆₇	—	2	24	no	—
92	C₂×C₄×C₆₇	—	2	12	no	—
93	C₂×C₄×C₆₇	—	2	12	no	—
94	C₂×C₄×C₆₇	—	2	12	no	—
95	C₂×C₄×C₆₇	—	2	192	no	—
96	C₂×C₄×C₆₇	C₂×C₄²	1	16	yes	complemented
97	C₂×C₄×C₆₇	C₄×D₄	1	48	yes	direct product factor
98	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
99	C₂×C₄×C₆₇	C₂³×C₄	1	24	yes	—
100	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
101	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
102	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
103	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
104	C₂²×C₆₇	—	2	24	no	—
105	C₂²×C₆₇	—	2	24	no	—
106	C₂²×C₆₇	—	2	64	no	—
107	C₂²×C₆₇	C₂²×C₄²	1	4	yes	—
108	C₂²×C₆₇	C₂²×C₄²	1	16	yes	complemented
109	C₂²×C₆₇	C₂×C₄×D₄	1	12	yes	—
110	C₂²×C₆₇	C₄×#41	1	12	yes	—
111	C₂²×C₆₇	C₂⁴×C₄	1	3	yes	—
112	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
113	C₂²×C₆₇	C₂²×#41	1	3	yes	—
114	C₄×C₆₇	—	2	24	no	—
115	C₄×C₆₇	—	2	12	no	—
116	C₄×C₆₇	C₂²×C₄²	1	8	yes	complemented
117	C₄×C₆₇	C₂×C₄×D₄	1	24	yes	direct product factor
118	C₂×C₄²×D₄	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₆₇	—	2	16	no	—
120	C₂×C₆₇	C₄²×D₄	1	8	yes	direct product factor
121	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
122	C₂×C₆₇	C₂²×C₄×D₄	1	3	yes	—
123	C₂×C₆₇	C₂×C₄×#41	1	3	yes	—
124	C₂²×C₄×D₄	C₂×C₆₇	1	3	yes	—
125	C₂³×C₄²	C₂×C₆₇	1	2	yes	complemented
126	C₄²×D₄	C₂×C₆₇	1	16	yes	direct product factor
127	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	3	yes	complemented
128	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	6	yes	complemented
129	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic, complemented
130	C₆₇	C₂×C₄²×D₄	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
132	C₂⁴×C₄	C₂²×C₆₇	1	6	yes	—
133	C₂×C₄×D₄	C₄×C₆₇	1	48	yes	direct product factor
134	C₂×C₄×D₄	C₂²×C₆₇	1	24	yes	—
135	C₂²×C₄⋊C₄	C₂²×C₆₇	1	3	yes	—
136	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	6	yes	—
137	C₂²×C₄²	—	2	16	no	—
138	C₂²×C₄²	C₂²×C₆₇	1	6	yes	—
139	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
140	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
141	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
142	C₂²×C₄²	C₂²×C₆₇	1	16	yes	complemented
143	C₄×C₄⋊C₄	C₂²×C₆₇	1	24	yes	complemented
144	C₄×C₂²⋊C₄	C₂²×C₆₇	1	48	yes	complemented
145	C₂×#95	C₂²×C₆₇	1	6	yes	—
146	C₄³	C₂²×C₆₇	1	8	yes	complemented
147	C₂⁵	C₂³×C₆₇	1	2	yes	—
148	C₂²×D₄	C₂×C₄×C₆₇	1	24	yes	—
149	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
150	C₂³×C₄	—	2	24	no	—
151	C₂³×C₄	—	2	24	no	—
152	C₂³×C₄	C₂×C₄×C₆₇	1	48	yes	complemented
153	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
154	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
155	C₂³×C₄	C₂³×C₆₇	1	24	yes	—
156	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
157	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
158	C₄×D₄	C₂×C₄×C₆₇	1	192	yes	direct product factor
159	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	24	yes	complemented
160	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
161	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	48	yes	complemented
162	C₂×C₂²⋊C₄	C₂³×C₆₇	1	24	yes	—
163	C₂×C₄²	—	2	24	no	—
164	C₂×C₄²	—	2	12	no	—
165	C₂×C₄²	—	2	64	no	—
166	C₂×C₄²	C₂×C₄×C₆₇	1	24	yes	complemented
167	C₂×C₄²	D₄×C₆₇	1	8	yes	direct product factor
168	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
169	C₂×C₄²	C₂³×C₆₇	1	24	yes	—
170	C₂×C₄²	C₂³×C₆₇	1	12	yes	—
171	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
172	#95	C₂³×C₆₇	1	24	yes	—
173	C₂⁴	—	2	4	no	—
174	C₂⁴	—	2	12	no	—
175	C₂⁴	C₂²×C₄×C₆₇	1	24	yes	—
176	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
177	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
178	C₂×D₄	C₄²×C₆₇	1	64	yes	direct product factor
179	C₂×D₄	C₂²×C₄×C₆₇	1	48	yes	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	2	12	no	—
183	C₂²×C₄	—	2	192	no	—
184	C₂²×C₄	—	2	48	no	—
185	C₂²×C₄	—	2	48	no	—
186	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
187	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
188	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
189	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
190	C₂²×C₄	C₂²×C₄×C₆₇	1	96	yes	complemented
191	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
192	C₂²×C₄	#41×C₆₇	1	12	yes	—
193	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
194	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
195	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
196	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
197	C₄⋊C₄	C₂²×C₄×C₆₇	1	48	yes	complemented
198	C₂²⋊C₄	C₂²×C₄×C₆₇	1	96	yes	complemented
199	C₄²	—	2	48	no	—
200	C₄²	—	2	24	no	—
201	C₄²	C₂²×C₄×C₆₇	1	48	yes	complemented
202	C₄²	C₂×D₄×C₆₇	1	16	yes	direct product factor
203	C₂³	—	2	48	no	—
204	C₂³	—	2	48	no	—
205	C₂³	—	2	4	no	—
206	C₂³	—	2	12	no	—
207	C₂³	C₂×C₄²×C₆₇	1	32	yes	complemented
208	C₂³	C₂³×C₄×C₆₇	1	24	yes	—
209	C₂³	C₂³×C₄×C₆₇	1	6	yes	—
210	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
211	C₂³	C₂×#41×C₆₇	1	6	yes	—
212	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
213	D₄	C₂×C₄²×C₆₇	1	64	yes	direct product factor
214	C₂×C₄	—	2	48	no	—
215	C₂×C₄	—	2	24	no	—
216	C₂×C₄	—	2	12	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	192	no	—
220	C₂×C₄	C₂×C₄²×C₆₇	1	16	yes	complemented
221	C₂×C₄	C₄×D₄×C₆₇	1	48	yes	direct product factor
222	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
223	C₂×C₄	C₂³×C₄×C₆₇	1	24	yes	—
224	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
225	C₂×C₄	C₂³×C₄×C₆₇	1	12	yes	—
226	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
227	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
228	C₂²	—	2	64	no	—
229	C₂²	—	2	24	no	—
230	C₂²	—	2	24	no	—
231	C₂²	C₂²×C₄²×C₆₇	1	4	yes	—
232	C₂²	C₂²×C₄²×C₆₇	1	16	yes	complemented
233	C₂²	C₂×C₄×D₄×C₆₇	1	12	yes	—
234	C₂²	C₄×#41×C₆₇	1	12	yes	—
235	C₂²	C₂⁴×C₄×C₆₇	1	3	yes	—
236	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
237	C₂²	C₂²×#41×C₆₇	1	3	yes	—
238	C₄	—	2	24	no	—
239	C₄	—	2	12	no	—
240	C₄	C₂²×C₄²×C₆₇	1	8	yes	complemented
241	C₄	C₂×C₄×D₄×C₆₇	1	24	yes	direct product factor
242	C₂	—	2	16	no	—
243	C₂	C₄²×D₄×C₆₇	1	8	yes	direct product factor
244	C₂	C₂³×C₄²×C₆₇	1	1	yes	characteristic
245	C₂	C₂²×C₄×D₄×C₆₇	1	3	yes	—
246	C₂	C₂×C₄×#41×C₆₇	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄²×C₆₇	C₂²
commutator	C₂	C₂³×C₄²×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂⁴×C₆₇	C₂⁴
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄²×C₆₇	C₂	semidirect
3. C₄²×D₄×C₆₇	C₂	direct (N × H)
4. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
5. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
6. C₂×C₄³×C₆₇	C₂	semidirect
9. C₂×C₄×D₄×C₆₇	C₄	direct (N × H)
18. C₂²×C₄²×C₆₇	C₂²	semidirect
19. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
20. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
22. C₄³×C₆₇	C₂²	semidirect
28. C₂³×C₄×C₆₇	C₂×C₄	semidirect
34. C₄×D₄×C₆₇	C₂×C₄	direct (N × H)
35. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
37. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
42. C₂×C₄²×C₆₇	C₂×C₄	semidirect
43. C₂×C₄²×C₆₇	D₄	direct (N × H)
54. C₂×D₄×C₆₇	C₄²	direct (N × H)
66. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
73. C₄⋊C₄×C₆₇	C₂²×C₄	semidirect
74. C₂²⋊C₄×C₆₇	C₂²×C₄	semidirect
77. C₄²×C₆₇	C₂²×C₄	semidirect
78. C₄²×C₆₇	C₂×D₄	direct (N × H)
83. C₂³×C₆₇	C₂×C₄²	semidirect
89. D₄×C₆₇	C₂×C₄²	direct (N × H)
96. C₂×C₄×C₆₇	C₂×C₄²	semidirect
97. C₂×C₄×C₆₇	C₄×D₄	direct (N × H)
108. C₂²×C₆₇	C₂²×C₄²	semidirect
116. C₄×C₆₇	C₂²×C₄²	semidirect
117. C₄×C₆₇	C₂×C₄×D₄	direct (N × H)
118. C₂×C₄²×D₄	C₆₇	direct (N × H)
120. C₂×C₆₇	C₄²×D₄	direct (N × H)
125. C₂³×C₄²	C₂×C₆₇	semidirect
126. C₄²×D₄	C₂×C₆₇	direct (N × H)
127. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
128. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
129. C₂×C₄³	C₂×C₆₇	semidirect
130. C₆₇	C₂×C₄²×D₄	direct (N × H)
133. C₂×C₄×D₄	C₄×C₆₇	direct (N × H)
142. C₂²×C₄²	C₂²×C₆₇	semidirect
143. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
144. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
146. C₄³	C₂²×C₆₇	semidirect
152. C₂³×C₄	C₂×C₄×C₆₇	semidirect
158. C₄×D₄	C₂×C₄×C₆₇	direct (N × H)
159. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
161. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
166. C₂×C₄²	C₂×C₄×C₆₇	semidirect
167. C₂×C₄²	D₄×C₆₇	direct (N × H)
178. C₂×D₄	C₄²×C₆₇	direct (N × H)
190. C₂²×C₄	C₂²×C₄×C₆₇	semidirect
197. C₄⋊C₄	C₂²×C₄×C₆₇	semidirect
198. C₂²⋊C₄	C₂²×C₄×C₆₇	semidirect
201. C₄²	C₂²×C₄×C₆₇	semidirect
202. C₄²	C₂×D₄×C₆₇	direct (N × H)
207. C₂³	C₂×C₄²×C₆₇	semidirect
213. D₄	C₂×C₄²×C₆₇	direct (N × H)
220. C₂×C₄	C₂×C₄²×C₆₇	semidirect
221. C₂×C₄	C₄×D₄×C₆₇	direct (N × H)
232. C₂²	C₂²×C₄²×C₆₇	semidirect
240. C₄	C₂²×C₄²×C₆₇	semidirect
241. C₄	C₂×C₄×D₄×C₆₇	direct (N × H)
243. C₂	C₄²×D₄×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
C₂²×C₄×D₄×C₆₇	2
C₂³×C₄²×C₆₇	2
C₄²×D₄×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×C₄³×C₆₇	2
C₂×C₄²×D₄	67

Subgroups — C2×C42×D4×C67 #30713242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×D₄×C₆₇ #30713242