Subgroups — C₂²×#486×C₆₇ #30739853

Order 17152.

3086 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⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#486×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#102×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₆₇	—	2	1	no	—
7	C₂³×D₄×C₆₇	—	2	1	no	—
8	C₂³×D₄×C₆₇	—	2	1	no	—
9	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
10	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	6	no	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	12	yes	complemented
13	C₂×#127×C₆₇	—	2	6	no	—
14	C₂×#127×C₆₇	C₂²	1	12	yes	—
15	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#486×C₆₇	C₂²	1	64	yes	direct product factor
19	C₂×#102×C₆₇	C₂²	1	48	yes	complemented
20	C₂×C₄×C₈×C₆₇	C₂²	1	12	yes	complemented
21	C₂⁵×C₆₇	—	2	2	no	—
22	C₂²×Q₈×C₆₇	—	2	12	no	—
23	C₂²×Q₈×C₆₇	—	4	8	no	—
24	C₂²×D₄×C₆₇	—	2	6	no	—
25	C₂²×D₄×C₆₇	—	2	12	no	—
26	C₂²×D₄×C₆₇	—	2	12	no	—
27	C₂²×D₄×C₆₇	—	2	16	no	—
28	C₂²×D₄×C₆₇	—	4	8	no	—
29	C₂²×D₄×C₆₇	—	4	8	no	—
30	C₂²×D₄×C₆₇	C₂³	1	12	yes	—
31	C₂³×C₄×C₆₇	—	2	2	no	—
32	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	—	2	8	no	—
36	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
37	C₄⋊Q₈×C₆₇	—	2	24	no	—
38	C₄⋊Q₈×C₆₇	C₂³	1	16	yes	complemented
39	#127×C₆₇	—	2	24	no	—
40	#127×C₆₇	C₂³	1	16	yes	—
41	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
44	C₂×C₄²×C₆₇	D₄	1	6	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
46	#102×C₆₇	C₂³	1	64	yes	complemented
47	C₄×C₈×C₆₇	C₂³	1	16	yes	complemented
48	C₂⁴×C₆₇	—	2	12	no	—
49	C₂⁴×C₆₇	—	4	8	no	—
50	C₂⁴×C₆₇	—	4	8	no	—
51	C₂⁴×C₆₇	—	4	8	no	—
52	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Q₈×C₆₇	—	2	16	no	—
54	C₂×Q₈×C₆₇	—	4	48	no	—
55	C₂×D₄×C₆₇	—	2	12	no	—
56	C₂×D₄×C₆₇	—	2	16	no	—
57	C₂×D₄×C₆₇	—	2	16	no	—
58	C₂×D₄×C₆₇	—	2	48	no	—
59	C₂×D₄×C₆₇	—	4	24	no	—
60	C₂×D₄×C₆₇	—	4	48	no	—
61	C₂×D₄×C₆₇	—	4	48	no	—
62	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
63	C₂²×C₄×C₆₇	—	2	12	no	—
64	C₂²×C₄×C₆₇	—	4	8	no	—
65	C₂²×C₄×C₆₇	D₈	1	8	yes	—
66	C₂²×C₄×C₆₇	SD₁₆	1	8	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₆₇	#41	1	8	yes	—
71	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
72	C₂×C₈×C₆₇	—	2	24	no	—
73	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
74	C₄⋊C₄×C₆₇	—	2	16	no	—
75	C₄⋊C₄×C₆₇	—	2	12	no	—
76	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
77	C₄²×C₆₇	C₂×D₄	1	12	yes	—
78	C₄²×C₆₇	C₂⁴	1	4	yes	—
79	C₂³×C₆₇	—	2	8	no	—
80	C₂³×C₆₇	—	4	24	no	—
81	C₂³×C₆₇	—	4	24	no	—
82	C₂³×C₆₇	—	4	24	no	—
83	C₂³×C₆₇	—	8	32	no	—
84	C₂³×C₆₇	#124	1	4	yes	—
85	C₂³×C₆₇	C₂×D₈	1	4	yes	—
86	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
87	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
88	Q₈×C₆₇	—	4	32	no	—
89	D₄×C₆₇	—	2	16	no	—
90	D₄×C₆₇	—	4	24	no	—
91	D₄×C₆₇	—	4	32	no	—
92	D₄×C₆₇	—	4	32	no	—
93	C₂×C₄×C₆₇	—	2	8	no	—
94	C₂×C₄×C₆₇	—	4	24	no	—
95	C₂×C₄×C₆₇	#124	1	12	yes	—
96	C₂×C₄×C₆₇	C₂×D₈	1	24	yes	—
97	C₂×C₄×C₆₇	C₂×SD₁₆	1	24	yes	—
98	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
99	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
100	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
101	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
102	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
103	C₈×C₆₇	—	2	8	no	—
104	C₂²×C₆₇	—	4	8	no	—
105	C₂²×C₆₇	—	4	8	no	—
106	C₂²×C₆₇	—	4	8	no	—
107	C₂²×C₆₇	—	8	48	no	—
108	C₂²×C₆₇	#486	1	16	yes	direct product factor
109	C₂²×C₆₇	C₂×#124	1	6	yes	—
110	C₂²×C₆₇	C₂²×D₈	1	6	yes	—
111	C₂²×C₆₇	C₂²×SD₁₆	1	6	yes	—
112	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
113	C₄×C₆₇	—	4	8	no	—
114	C₄×C₆₇	C₂×#124	1	6	yes	—
115	C₄×C₆₇	C₂²×D₈	1	8	yes	—
116	C₄×C₆₇	C₂²×SD₁₆	1	8	yes	—
117	C₄×C₆₇	C₂²×#41	1	2	yes	—
118	C₂²×#486	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₆₇	—	8	8	no	—
120	C₂×C₆₇	C₂×#486	1	12	yes	direct product factor
121	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
122	C₂×C₆₇	C₂³×D₈	1	1	yes	characteristic
123	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
124	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
125	C₂²×#127	C₂×C₆₇	1	1	yes	characteristic
126	C₂×#486	C₂×C₆₇	1	24	yes	direct product factor
127	C₂²×#102	C₂×C₆₇	1	4	yes	complemented
128	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
129	C₆₇	C₂²×#486	1	1	yes	characteristic, direct product factor, Sylow
130	C₂³×Q₈	—	2	1	no	—
131	C₂³×D₄	—	2	1	no	—
132	C₂³×D₄	—	2	1	no	—
133	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
134	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
135	C₂×C₄⋊Q₈	—	2	6	no	—
136	C₂×C₄⋊Q₈	C₂²×C₆₇	1	12	yes	complemented
137	C₂×#127	—	2	6	no	—
138	C₂×#127	C₂²×C₆₇	1	12	yes	—
139	C₂²×C₄⋊C₄	—	2	1	no	—
140	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
141	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
142	#486	C₂²×C₆₇	1	64	yes	direct product factor
143	C₂×#102	C₂²×C₆₇	1	48	yes	complemented
144	C₂×C₄×C₈	C₂²×C₆₇	1	12	yes	complemented
145	C₂⁵	—	2	2	no	—
146	C₂²×Q₈	—	2	12	no	—
147	C₂²×Q₈	—	4	8	no	—
148	C₂²×D₄	—	2	6	no	—
149	C₂²×D₄	—	2	12	no	—
150	C₂²×D₄	—	2	12	no	—
151	C₂²×D₄	—	2	16	no	—
152	C₂²×D₄	—	4	8	no	—
153	C₂²×D₄	—	4	8	no	—
154	C₂²×D₄	C₂³×C₆₇	1	12	yes	—
155	C₂³×C₄	—	2	2	no	—
156	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
157	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
158	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
159	C₂²×C₈	—	2	8	no	—
160	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
161	C₄⋊Q₈	—	2	24	no	—
162	C₄⋊Q₈	C₂³×C₆₇	1	16	yes	complemented
163	#127	—	2	24	no	—
164	#127	C₂³×C₆₇	1	16	yes	—
165	C₂×C₄⋊C₄	—	2	12	no	—
166	C₂×C₄⋊C₄	—	2	6	no	—
167	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
168	C₂×C₄²	D₄×C₆₇	1	6	yes	—
169	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
170	#102	C₂³×C₆₇	1	64	yes	complemented
171	C₄×C₈	C₂³×C₆₇	1	16	yes	complemented
172	C₂⁴	—	2	12	no	—
173	C₂⁴	—	4	8	no	—
174	C₂⁴	—	4	8	no	—
175	C₂⁴	—	4	8	no	—
176	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
177	C₂×Q₈	—	2	16	no	—
178	C₂×Q₈	—	4	48	no	—
179	C₂×D₄	—	2	12	no	—
180	C₂×D₄	—	2	16	no	—
181	C₂×D₄	—	2	16	no	—
182	C₂×D₄	—	2	48	no	—
183	C₂×D₄	—	4	24	no	—
184	C₂×D₄	—	4	48	no	—
185	C₂×D₄	—	4	48	no	—
186	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
187	C₂²×C₄	—	2	12	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	D₈×C₆₇	1	8	yes	—
190	C₂²×C₄	SD₁₆×C₆₇	1	8	yes	—
191	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
192	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
193	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
194	C₂²×C₄	#41×C₆₇	1	8	yes	—
195	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
196	C₂×C₈	—	2	24	no	—
197	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
198	C₄⋊C₄	—	2	16	no	—
199	C₄⋊C₄	—	2	12	no	—
200	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
201	C₄²	C₂×D₄×C₆₇	1	12	yes	—
202	C₄²	C₂⁴×C₆₇	1	4	yes	—
203	C₂³	—	2	8	no	—
204	C₂³	—	4	24	no	—
205	C₂³	—	4	24	no	—
206	C₂³	—	4	24	no	—
207	C₂³	—	8	32	no	—
208	C₂³	#124×C₆₇	1	4	yes	—
209	C₂³	C₂×D₈×C₆₇	1	4	yes	—
210	C₂³	C₂×SD₁₆×C₆₇	1	4	yes	—
211	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
212	Q₈	—	4	32	no	—
213	D₄	—	2	16	no	—
214	D₄	—	4	24	no	—
215	D₄	—	4	32	no	—
216	D₄	—	4	32	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	4	24	no	—
219	C₂×C₄	#124×C₆₇	1	12	yes	—
220	C₂×C₄	C₂×D₈×C₆₇	1	24	yes	—
221	C₂×C₄	C₂×SD₁₆×C₆₇	1	24	yes	—
222	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
223	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
224	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
225	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
226	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
227	C₈	—	2	8	no	—
228	C₂²	—	4	8	no	—
229	C₂²	—	4	8	no	—
230	C₂²	—	4	8	no	—
231	C₂²	—	8	48	no	—
232	C₂²	#486×C₆₇	1	16	yes	direct product factor
233	C₂²	C₂×#124×C₆₇	1	6	yes	—
234	C₂²	C₂²×D₈×C₆₇	1	6	yes	—
235	C₂²	C₂²×SD₁₆×C₆₇	1	6	yes	—
236	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
237	C₄	—	4	8	no	—
238	C₄	C₂×#124×C₆₇	1	6	yes	—
239	C₄	C₂²×D₈×C₆₇	1	8	yes	—
240	C₄	C₂²×SD₁₆×C₆₇	1	8	yes	—
241	C₄	C₂²×#41×C₆₇	1	2	yes	—
242	C₂	—	8	8	no	—
243	C₂	C₂×#486×C₆₇	1	12	yes	direct product factor
244	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
245	C₂	C₂³×D₈×C₆₇	1	1	yes	characteristic
246	C₂	C₂³×SD₁₆×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×#127×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₆₇	C₂	semidirect
3. C₂×#486×C₆₇	C₂	direct (N × H)
4. C₂²×#102×C₆₇	C₂	semidirect
5. C₂²×C₄×C₈×C₆₇	C₂	semidirect
12. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
18. #486×C₆₇	C₂²	direct (N × H)
19. C₂×#102×C₆₇	C₂²	semidirect
20. C₂×C₄×C₈×C₆₇	C₂²	semidirect
38. C₄⋊Q₈×C₆₇	C₂³	semidirect
46. #102×C₆₇	C₂³	semidirect
47. C₄×C₈×C₆₇	C₂³	semidirect
108. C₂²×C₆₇	#486	direct (N × H)
118. C₂²×#486	C₆₇	direct (N × H)
120. C₂×C₆₇	C₂×#486	direct (N × H)
124. C₂²×C₄⋊Q₈	C₂×C₆₇	semidirect
126. C₂×#486	C₂×C₆₇	direct (N × H)
127. C₂²×#102	C₂×C₆₇	semidirect
128. C₂²×C₄×C₈	C₂×C₆₇	semidirect
129. C₆₇	C₂²×#486	direct (N × H)
136. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
142. #486	C₂²×C₆₇	direct (N × H)
143. C₂×#102	C₂²×C₆₇	semidirect
144. C₂×C₄×C₈	C₂²×C₆₇	semidirect
162. C₄⋊Q₈	C₂³×C₆₇	semidirect
170. #102	C₂³×C₆₇	semidirect
171. C₄×C₈	C₂³×C₆₇	semidirect
232. C₂²	#486×C₆₇	direct (N × H)
243. C₂	C₂×#486×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⋊Q₈×C₆₇	2
C₂²×#127×C₆₇	2
C₂×#486×C₆₇	2
C₂²×#102×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂²×#486	67

Subgroups — C22×#486×C67 #30739853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#486×C₆₇ #30739853