Subgroups — C₂²×#436×C₆₇ #30739383

Order 17152.

3262 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#436×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂²⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₆₇	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₆₇	—	2	16	no	—
10	C₂²×M₁₆×C₆₇	C₂²	1	24	yes	complemented
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₄	1	12	yes	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	12	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
17	C₂²×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#436×C₆₇	C₂²	1	64	yes	direct product factor
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	12	yes	complemented
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	24	yes	complemented
22	C₂×C₄×C₈×C₆₇	C₂²	1	12	yes	complemented
23	C₂⁵×C₆₇	C₂×C₄	1	2	yes	—
24	C₂²×D₄×C₆₇	—	2	8	no	—
25	C₂²×D₄×C₆₇	C₂×C₄	1	12	yes	—
26	C₂³×C₄×C₆₇	—	2	8	no	—
27	C₂³×C₄×C₆₇	—	2	8	no	—
28	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
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	12	yes	—
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₆₇	—	2	96	no	—
35	C₂×M₁₆×C₆₇	C₂³	1	32	yes	complemented
36	C₂²×C₈×C₆₇	—	2	8	no	—
37	C₂²×C₈×C₆₇	—	2	4	no	—
38	C₂²×C₈×C₆₇	D₄	1	8	yes	complemented
39	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
42	C₄×D₄×C₆₇	C₂×C₄	1	48	yes	—
43	C₄×D₄×C₆₇	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	12	yes	—
45	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
48	C₄⋊C₈×C₆₇	C₂³	1	16	yes	complemented
49	C₂²⋊C₈×C₆₇	C₂³	1	32	yes	complemented
50	C₄×C₈×C₆₇	C₂³	1	16	yes	complemented
51	C₂⁴×C₆₇	—	2	8	no	—
52	C₂⁴×C₆₇	—	2	8	no	—
53	C₂⁴×C₆₇	—	2	8	no	—
54	C₂⁴×C₆₇	C₂²×C₄	1	12	yes	—
55	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×D₄×C₆₇	—	2	48	no	—
57	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
58	C₂²×C₄×C₆₇	—	2	8	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	—	2	48	no	—
61	C₂²×C₄×C₆₇	—	2	24	no	—
62	C₂²×C₄×C₆₇	—	2	24	no	—
63	C₂²×C₄×C₆₇	M₁₆	1	8	yes	complemented
64	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₆₇	#41	1	4	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
73	M₁₆×C₆₇	—	2	64	no	—
74	C₂×C₈×C₆₇	—	2	24	no	—
75	C₂×C₈×C₆₇	—	2	12	no	—
76	C₂×C₈×C₆₇	C₂×D₄	1	24	yes	complemented
77	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
78	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
79	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
80	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
81	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
82	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
83	C₄²×C₆₇	C₂⁴	1	4	yes	—
84	C₂³×C₆₇	—	2	24	no	—
85	C₂³×C₆₇	—	2	24	no	—
86	C₂³×C₆₇	—	2	24	no	—
87	C₂³×C₆₇	—	4	32	no	—
88	C₂³×C₆₇	C₄×D₄	1	4	yes	—
89	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
90	C₂³×C₆₇	#131	1	4	yes	—
91	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
92	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
93	D₄×C₆₇	—	2	32	no	—
94	C₂×C₄×C₆₇	—	2	24	no	—
95	C₂×C₄×C₆₇	—	2	12	no	—
96	C₂×C₄×C₆₇	—	2	48	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	2	8	no	—
99	C₂×C₄×C₆₇	C₄×D₄	1	12	yes	—
100	C₂×C₄×C₆₇	C₂×M₁₆	1	24	yes	complemented
101	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
102	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
104	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
106	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
107	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
108	C₈×C₆₇	—	2	8	no	—
109	C₈×C₆₇	—	2	4	no	—
110	C₈×C₆₇	C₂²×D₄	1	8	yes	complemented
111	C₂²×C₆₇	—	2	8	no	—
112	C₂²×C₆₇	—	2	8	no	—
113	C₂²×C₆₇	—	2	8	no	—
114	C₂²×C₆₇	—	4	48	no	—
115	C₂²×C₆₇	#436	1	16	yes	direct product factor
116	C₂²×C₆₇	C₂×C₄×D₄	1	6	yes	—
117	C₂²×C₆₇	C₂²×M₁₆	1	6	yes	—
118	C₂²×C₆₇	C₂×#131	1	6	yes	—
119	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
120	C₄×C₆₇	—	2	8	no	—
121	C₄×C₆₇	—	2	4	no	—
122	C₄×C₆₇	C₂×C₄×D₄	1	6	yes	—
123	C₄×C₆₇	C₂²×M₁₆	1	8	yes	complemented
124	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
125	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
126	C₂²×#436	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₆₇	—	4	8	no	—
128	C₂×C₆₇	C₂×#436	1	12	yes	direct product factor
129	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
130	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
131	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
132	C₂³×M₁₆	C₂×C₆₇	1	2	yes	complemented
133	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
134	C₂×#436	C₂×C₆₇	1	24	yes	direct product factor
135	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
136	C₂²×C₂²⋊C₈	C₂×C₆₇	1	2	yes	complemented
137	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
138	C₆₇	C₂²×#436	1	1	yes	characteristic, direct product factor, Sylow
139	C₂³×D₄	C₄×C₆₇	1	1	yes	characteristic
140	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
141	C₂²×M₁₆	—	2	16	no	—
142	C₂²×M₁₆	C₂²×C₆₇	1	24	yes	complemented
143	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
144	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
145	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
146	C₂×C₄×D₄	C₄×C₆₇	1	12	yes	—
147	C₂×C₄×D₄	C₂²×C₆₇	1	12	yes	—
148	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
149	C₂²×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
150	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
151	#436	C₂²×C₆₇	1	64	yes	direct product factor
152	C₂×C₄⋊C₈	C₂²×C₆₇	1	12	yes	complemented
153	C₂×C₂²⋊C₈	C₂²×C₆₇	1	24	yes	complemented
154	C₂×C₄×C₈	C₂²×C₆₇	1	12	yes	complemented
155	C₂⁵	C₂×C₄×C₆₇	1	2	yes	—
156	C₂²×D₄	—	2	8	no	—
157	C₂²×D₄	C₂×C₄×C₆₇	1	12	yes	—
158	C₂³×C₄	—	2	8	no	—
159	C₂³×C₄	—	2	8	no	—
160	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
162	C₂³×C₄	C₂×C₄×C₆₇	1	12	yes	—
163	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
164	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
165	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
166	C₂×M₁₆	—	2	96	no	—
167	C₂×M₁₆	C₂³×C₆₇	1	32	yes	complemented
168	C₂²×C₈	—	2	8	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	D₄×C₆₇	1	8	yes	complemented
171	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
172	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
173	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
174	C₄×D₄	C₂×C₄×C₆₇	1	48	yes	—
175	C₄×D₄	C₂³×C₆₇	1	16	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	12	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	24	yes	—
178	C₂×C₄²	C₂×C₄×C₆₇	1	6	yes	—
179	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
180	C₄⋊C₈	C₂³×C₆₇	1	16	yes	complemented
181	C₂²⋊C₈	C₂³×C₆₇	1	32	yes	complemented
182	C₄×C₈	C₂³×C₆₇	1	16	yes	complemented
183	C₂⁴	—	2	8	no	—
184	C₂⁴	—	2	8	no	—
185	C₂⁴	—	2	8	no	—
186	C₂⁴	C₂²×C₄×C₆₇	1	12	yes	—
187	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
188	C₂×D₄	—	2	48	no	—
189	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	48	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	2	24	no	—
195	C₂²×C₄	M₁₆×C₆₇	1	8	yes	complemented
196	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
197	C₂²×C₄	C₂²×C₄×C₆₇	1	12	yes	—
198	C₂²×C₄	C₂²×C₄×C₆₇	1	24	yes	—
199	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
200	C₂²×C₄	C₂²×C₄×C₆₇	1	6	yes	—
201	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
202	C₂²×C₄	#41×C₆₇	1	4	yes	—
203	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
204	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
205	M₁₆	—	2	64	no	—
206	C₂×C₈	—	2	24	no	—
207	C₂×C₈	—	2	12	no	—
208	C₂×C₈	C₂×D₄×C₆₇	1	24	yes	complemented
209	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
210	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
211	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
212	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
213	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
214	C₄²	C₂²×C₄×C₆₇	1	12	yes	—
215	C₄²	C₂⁴×C₆₇	1	4	yes	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	24	no	—
218	C₂³	—	2	24	no	—
219	C₂³	—	4	32	no	—
220	C₂³	C₄×D₄×C₆₇	1	4	yes	—
221	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
222	C₂³	#131×C₆₇	1	4	yes	—
223	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
224	C₂³	C₂³×C₄×C₆₇	1	3	yes	—
225	D₄	—	2	32	no	—
226	C₂×C₄	—	2	24	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	48	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	C₄×D₄×C₆₇	1	12	yes	—
232	C₂×C₄	C₂×M₁₆×C₆₇	1	24	yes	complemented
233	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
234	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
235	C₂×C₄	C₂³×C₄×C₆₇	1	3	yes	—
236	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
237	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
238	C₂×C₄	C₂×#41×C₆₇	1	6	yes	—
239	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
240	C₈	—	2	8	no	—
241	C₈	—	2	4	no	—
242	C₈	C₂²×D₄×C₆₇	1	8	yes	complemented
243	C₂²	—	2	8	no	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	8	no	—
246	C₂²	—	4	48	no	—
247	C₂²	#436×C₆₇	1	16	yes	direct product factor
248	C₂²	C₂×C₄×D₄×C₆₇	1	6	yes	—
249	C₂²	C₂²×M₁₆×C₆₇	1	6	yes	—
250	C₂²	C₂×#131×C₆₇	1	6	yes	—
251	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	2	8	no	—
254	C₄	C₂×C₄×D₄×C₆₇	1	6	yes	—
255	C₄	C₂²×M₁₆×C₆₇	1	8	yes	complemented
256	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
257	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
258	C₂	—	4	8	no	—
259	C₂	C₂×#436×C₆₇	1	12	yes	direct product factor
260	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
261	C₂	C₂³×M₁₆×C₆₇	1	1	yes	characteristic
262	C₂	C₂²×#131×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	C₂³×D₄×C₆₇	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
1. C₂³×M₁₆×C₆₇	C₂	semidirect
3. C₂×#436×C₆₇	C₂	direct (N × H)
4. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
5. C₂²×C₂²⋊C₈×C₆₇	C₂	semidirect
6. C₂²×C₄×C₈×C₆₇	C₂	semidirect
10. C₂²×M₁₆×C₆₇	C₂²	semidirect
19. #436×C₆₇	C₂²	direct (N × H)
20. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
21. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
22. C₂×C₄×C₈×C₆₇	C₂²	semidirect
35. C₂×M₁₆×C₆₇	C₂³	semidirect
38. C₂²×C₈×C₆₇	D₄	semidirect
48. C₄⋊C₈×C₆₇	C₂³	semidirect
49. C₂²⋊C₈×C₆₇	C₂³	semidirect
50. C₄×C₈×C₆₇	C₂³	semidirect
63. C₂²×C₄×C₆₇	M₁₆	semidirect
76. C₂×C₈×C₆₇	C₂×D₄	semidirect
100. C₂×C₄×C₆₇	C₂×M₁₆	semidirect
110. C₈×C₆₇	C₂²×D₄	semidirect
115. C₂²×C₆₇	#436	direct (N × H)
123. C₄×C₆₇	C₂²×M₁₆	semidirect
126. C₂²×#436	C₆₇	direct (N × H)
128. C₂×C₆₇	C₂×#436	direct (N × H)
132. C₂³×M₁₆	C₂×C₆₇	semidirect
134. C₂×#436	C₂×C₆₇	direct (N × H)
135. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
136. C₂²×C₂²⋊C₈	C₂×C₆₇	semidirect
137. C₂²×C₄×C₈	C₂×C₆₇	semidirect
138. C₆₇	C₂²×#436	direct (N × H)
142. C₂²×M₁₆	C₂²×C₆₇	semidirect
151. #436	C₂²×C₆₇	direct (N × H)
152. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
153. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
154. C₂×C₄×C₈	C₂²×C₆₇	semidirect
167. C₂×M₁₆	C₂³×C₆₇	semidirect
170. C₂²×C₈	D₄×C₆₇	semidirect
180. C₄⋊C₈	C₂³×C₆₇	semidirect
181. C₂²⋊C₈	C₂³×C₆₇	semidirect
182. C₄×C₈	C₂³×C₆₇	semidirect
195. C₂²×C₄	M₁₆×C₆₇	semidirect
208. C₂×C₈	C₂×D₄×C₆₇	semidirect
232. C₂×C₄	C₂×M₁₆×C₆₇	semidirect
242. C₈	C₂²×D₄×C₆₇	semidirect
247. C₂²	#436×C₆₇	direct (N × H)
255. C₄	C₂²×M₁₆×C₆₇	semidirect
259. C₂	C₂×#436×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×M₁₆×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₂×#436×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂²×#436	67

Subgroups — C22×#436×C67 #30739383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#436×C₆₇ #30739383