Subgroups — C₂²×#432×C₆₇ #30739366

Order 17152.

2830 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₆₇	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal
5	C₂²×#97×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×C₆₇	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₆₇	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
10	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₄	1	12	yes	—
12	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₆₇	C₂²	1	12	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#432×C₆₇	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	24	yes	complemented
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	24	yes	—
19	C₂×#97×C₆₇	C₂²	1	24	yes	complemented
20	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₆₇	—	2	8	no	—
22	C₂³×C₄×C₆₇	—	2	8	no	—
23	C₂³×C₄×C₆₇	—	2	8	no	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂×C₄	1	12	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	—	2	16	no	—
35	C₂²×C₈×C₆₇	C₂³	1	24	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	48	yes	—
38	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	16	yes	—
44	C₄⋊C₈×C₆₇	C₂³	1	32	yes	complemented
45	C₂²⋊C₈×C₆₇	C₂³	1	32	yes	—
46	#97×C₆₇	C₂³	1	32	yes	complemented
47	C₂⁴×C₆₇	—	2	4	no	—
48	C₂⁴×C₆₇	—	2	4	no	—
49	C₂⁴×C₆₇	M₁₆	1	8	yes	—
50	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
51	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	—	2	48	no	—
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	2	4	no	—
57	C₂²×C₄×C₆₇	—	2	24	no	—
58	C₂²×C₄×C₆₇	—	2	48	no	—
59	C₂²×C₄×C₆₇	—	2	24	no	—
60	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
70	C₂²×C₄×C₆₇	#41	1	8	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
73	C₂×C₈×C₆₇	—	2	48	no	—
74	C₂×C₈×C₆₇	C₂⁴	1	16	yes	—
75	C₄²×C₆₇	C₂²×C₄	1	16	yes	—
76	C₄²×C₆₇	C₂²×C₄	1	16	yes	—
77	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
78	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
79	C₄²×C₆₇	C₂⁴	1	4	yes	—
80	C₄²×C₆₇	C₂⁴	1	4	yes	—
81	C₂³×C₆₇	—	2	12	no	—
82	C₂³×C₆₇	—	2	12	no	—
83	C₂³×C₆₇	—	2	32	no	—
84	C₂³×C₆₇	#117	1	4	yes	—
85	C₂³×C₆₇	C₂×M₁₆	1	24	yes	—
86	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
87	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
88	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
89	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₆₇	—	2	12	no	—
91	C₂×C₄×C₆₇	—	2	32	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	24	no	—
96	C₂×C₄×C₆₇	—	2	32	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	#117	1	12	yes	—
99	C₂×C₄×C₆₇	C₂×M₁₆	1	24	yes	—
100	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
101	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
102	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
107	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
108	C₈×C₆₇	—	2	16	no	—
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	—	2	48	no	—
111	C₂²×C₆₇	—	2	4	no	—
112	C₂²×C₆₇	#432	1	16	yes	direct product factor
113	C₂²×C₆₇	C₂×#117	1	6	yes	—
114	C₂²×C₆₇	C₂²×M₁₆	1	8	yes	—
115	C₂²×C₆₇	C₂²×M₁₆	1	6	yes	—
116	C₂²×C₆₇	C₂²×M₁₆	1	6	yes	—
117	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
118	C₄×C₆₇	—	2	4	no	—
119	C₄×C₆₇	—	2	4	no	—
120	C₄×C₆₇	—	2	4	no	—
121	C₄×C₆₇	—	2	4	no	—
122	C₄×C₆₇	C₂×#117	1	6	yes	—
123	C₄×C₆₇	C₂²×M₁₆	1	8	yes	—
124	C₄×C₆₇	C₂²×#41	1	2	yes	—
125	C₂²×#432	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₆₇	—	2	8	no	—
127	C₂×C₆₇	C₂×#432	1	12	yes	direct product factor
128	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
129	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
130	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
131	C₂³×C₄²	C₂×C₆₇	1	1	yes	characteristic
132	C₂×#432	C₂×C₆₇	1	24	yes	direct product factor
133	C₂²×C₄⋊C₈	C₂×C₆₇	1	2	yes	complemented
134	C₂²×C₂²⋊C₈	C₂×C₆₇	1	2	yes	—
135	C₂²×#97	C₂×C₆₇	1	2	yes	complemented
136	C₆₇	C₂²×#432	1	1	yes	characteristic, direct product factor, Sylow
137	C₂⁴×C₄	C₄×C₆₇	1	1	yes	characteristic
138	C₂⁴×C₄	C₄×C₆₇	1	1	yes	characteristic
139	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
140	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
141	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
142	C₂²×C₄²	C₄×C₆₇	1	12	yes	—
143	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
144	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
145	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
146	C₂²×C₄²	C₂²×C₆₇	1	12	yes	—
147	#432	C₂²×C₆₇	1	64	yes	direct product factor
148	C₂×C₄⋊C₈	C₂²×C₆₇	1	24	yes	complemented
149	C₂×C₂²⋊C₈	C₂²×C₆₇	1	24	yes	—
150	C₂×#97	C₂²×C₆₇	1	24	yes	complemented
151	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
152	C₂³×C₄	—	2	8	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	C₂×C₄×C₆₇	1	12	yes	—
156	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
157	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
158	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
159	C₂³×C₄	C₂×C₄×C₆₇	1	12	yes	—
160	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₆₇	1	6	yes	—
162	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
163	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
164	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
165	C₂²×C₈	—	2	16	no	—
166	C₂²×C₈	C₂³×C₆₇	1	24	yes	—
167	C₂×C₄²	C₂×C₄×C₆₇	1	12	yes	—
168	C₂×C₄²	C₂×C₄×C₆₇	1	48	yes	—
169	C₂×C₄²	C₂×C₄×C₆₇	1	12	yes	—
170	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
171	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
172	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
173	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
174	C₂×C₄²	C₂³×C₆₇	1	16	yes	—
175	C₄⋊C₈	C₂³×C₆₇	1	32	yes	complemented
176	C₂²⋊C₈	C₂³×C₆₇	1	32	yes	—
177	#97	C₂³×C₆₇	1	32	yes	complemented
178	C₂⁴	—	2	4	no	—
179	C₂⁴	—	2	4	no	—
180	C₂⁴	M₁₆×C₆₇	1	8	yes	—
181	C₂⁴	C₂²×C₄×C₆₇	1	6	yes	—
182	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	48	no	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	24	no	—
189	C₂²×C₄	—	2	48	no	—
190	C₂²×C₄	—	2	24	no	—
191	C₂²×C₄	M₁₆×C₆₇	1	8	yes	—
192	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
193	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
194	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
195	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
196	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
197	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
198	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
199	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
200	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
201	C₂²×C₄	#41×C₆₇	1	8	yes	—
202	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
203	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
204	C₂×C₈	—	2	48	no	—
205	C₂×C₈	C₂⁴×C₆₇	1	16	yes	—
206	C₄²	C₂²×C₄×C₆₇	1	16	yes	—
207	C₄²	C₂²×C₄×C₆₇	1	16	yes	—
208	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
209	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
210	C₄²	C₂⁴×C₆₇	1	4	yes	—
211	C₄²	C₂⁴×C₆₇	1	4	yes	—
212	C₂³	—	2	12	no	—
213	C₂³	—	2	12	no	—
214	C₂³	—	2	32	no	—
215	C₂³	#117×C₆₇	1	4	yes	—
216	C₂³	C₂×M₁₆×C₆₇	1	24	yes	—
217	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
218	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
219	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
220	C₂³	C₂³×C₄×C₆₇	1	3	yes	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	2	32	no	—
223	C₂×C₄	—	2	12	no	—
224	C₂×C₄	—	2	12	no	—
225	C₂×C₄	—	2	12	no	—
226	C₂×C₄	—	2	24	no	—
227	C₂×C₄	—	2	32	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	#117×C₆₇	1	12	yes	—
230	C₂×C₄	C₂×M₁₆×C₆₇	1	24	yes	—
231	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
232	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
233	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
234	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
235	C₂×C₄	C₂³×C₄×C₆₇	1	3	yes	—
236	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
237	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
238	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
239	C₈	—	2	16	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	48	no	—
243	C₂²	#432×C₆₇	1	16	yes	direct product factor
244	C₂²	C₂×#117×C₆₇	1	6	yes	—
245	C₂²	C₂²×M₁₆×C₆₇	1	8	yes	—
246	C₂²	C₂²×M₁₆×C₆₇	1	6	yes	—
247	C₂²	C₂²×M₁₆×C₆₇	1	6	yes	—
248	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	2	4	no	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	C₂×#117×C₆₇	1	6	yes	—
254	C₄	C₂²×M₁₆×C₆₇	1	8	yes	—
255	C₄	C₂²×#41×C₆₇	1	2	yes	—
256	C₂	—	2	8	no	—
257	C₂	C₂×#432×C₆₇	1	12	yes	direct product factor
258	C₂	C₂²×#117×C₆₇	1	1	yes	characteristic
259	C₂	C₂³×M₁₆×C₆₇	1	1	yes	characteristic
260	C₂	C₂³×M₁₆×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#432×C₆₇	C₂	direct (N × H)
3. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
5. C₂²×#97×C₆₇	C₂	semidirect
16. #432×C₆₇	C₂²	direct (N × H)
17. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
19. C₂×#97×C₆₇	C₂²	semidirect
44. C₄⋊C₈×C₆₇	C₂³	semidirect
46. #97×C₆₇	C₂³	semidirect
112. C₂²×C₆₇	#432	direct (N × H)
125. C₂²×#432	C₆₇	direct (N × H)
127. C₂×C₆₇	C₂×#432	direct (N × H)
132. C₂×#432	C₂×C₆₇	direct (N × H)
133. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
135. C₂²×#97	C₂×C₆₇	semidirect
136. C₆₇	C₂²×#432	direct (N × H)
147. #432	C₂²×C₆₇	direct (N × H)
148. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
150. C₂×#97	C₂²×C₆₇	semidirect
175. C₄⋊C₈	C₂³×C₆₇	semidirect
177. #97	C₂³×C₆₇	semidirect
243. C₂²	#432×C₆₇	direct (N × H)
257. C₂	C₂×#432×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
C₂³×C₄²×C₆₇	2
C₂×#432×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂²×#97×C₆₇	2
C₂²×#432	67

Subgroups — C22×#432×C67 #30739366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#432×C₆₇ #30739366