Subgroups — C₂²×#488×C₆₇ #30739855

Order 17152.

2310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#488×C₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#103×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂²×#102×C₆₇	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#125×C₆₇	—	2	6	no	—
11	C₂×#125×C₆₇	C₂²	1	12	yes	complemented
12	C₂×#124×C₆₇	—	2	6	no	—
13	C₂×#124×C₆₇	C₂²	1	12	yes	—
14	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#488×C₆₇	C₂²	1	64	yes	direct product factor
19	C₂×#103×C₆₇	C₂²	1	24	yes	complemented
20	C₂×#102×C₆₇	C₂²	1	24	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	12	yes	complemented
22	C₂⁵×C₆₇	—	2	1	no	—
23	C₂²×Q₈×C₆₇	—	2	3	no	—
24	C₂²×Q₈×C₆₇	—	2	8	no	—
25	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
26	C₂²×D₄×C₆₇	—	2	3	no	—
27	C₂²×D₄×C₆₇	—	2	8	no	—
28	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
29	C₂³×C₄×C₆₇	—	2	1	no	—
30	C₂³×C₄×C₆₇	—	2	2	no	—
31	C₂³×C₄×C₆₇	D₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	—	2	8	no	—
34	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
35	#125×C₆₇	—	2	24	no	—
36	#125×C₆₇	C₂³	1	16	yes	complemented
37	#124×C₆₇	—	2	24	no	—
38	#124×C₆₇	C₂³	1	16	yes	—
39	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
42	C₂×C₂²⋊C₄×C₆₇	—	2	24	no	—
43	C₂×C₄²×C₆₇	D₄	1	6	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
45	#103×C₆₇	C₂³	1	32	yes	complemented
46	#102×C₆₇	C₂³	1	32	yes	—
47	C₄×C₈×C₆₇	C₂³	1	16	yes	complemented
48	C₂⁴×C₆₇	—	2	6	no	—
49	C₂⁴×C₆₇	—	4	4	no	—
50	C₂⁴×C₆₇	—	4	8	no	—
51	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Q₈×C₆₇	—	2	6	no	—
53	C₂×Q₈×C₆₇	—	2	24	no	—
54	C₂×Q₈×C₆₇	—	4	12	no	—
55	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
56	C₂×D₄×C₆₇	—	2	6	no	—
57	C₂×D₄×C₆₇	—	2	24	no	—
58	C₂×D₄×C₆₇	—	4	12	no	—
59	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
60	C₂²×C₄×C₆₇	—	2	8	no	—
61	C₂²×C₄×C₆₇	—	2	6	no	—
62	C₂²×C₄×C₆₇	—	2	12	no	—
63	C₂²×C₄×C₆₇	—	4	4	no	—
64	C₂²×C₄×C₆₇	—	4	8	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₆₇	#41	1	8	yes	—
68	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
69	C₂×C₈×C₆₇	—	2	24	no	—
70	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
71	C₄⋊C₄×C₆₇	—	2	32	no	—
72	C₄⋊C₄×C₆₇	—	2	12	no	—
73	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
74	C₂²⋊C₄×C₆₇	—	2	32	no	—
75	C₄²×C₆₇	C₂×D₄	1	12	yes	—
76	C₄²×C₆₇	C₂⁴	1	4	yes	—
77	C₂³×C₆₇	—	2	4	no	—
78	C₂³×C₆₇	—	4	12	no	—
79	C₂³×C₆₇	—	4	24	no	—
80	C₂³×C₆₇	—	8	16	no	—
81	C₂³×C₆₇	#124	1	4	yes	—
82	C₂³×C₆₇	#135	1	8	yes	—
83	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
84	Q₈×C₆₇	—	2	8	no	—
85	Q₈×C₆₇	—	4	12	no	—
86	D₄×C₆₇	—	2	8	no	—
87	D₄×C₆₇	—	4	12	no	—
88	C₂×C₄×C₆₇	—	2	24	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	8	no	—
91	C₂×C₄×C₆₇	—	4	12	no	—
92	C₂×C₄×C₆₇	—	4	24	no	—
93	C₂×C₄×C₆₇	#124	1	12	yes	—
94	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
95	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
97	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
98	C₈×C₆₇	—	2	8	no	—
99	C₂²×C₆₇	—	4	4	no	—
100	C₂²×C₆₇	—	4	8	no	—
101	C₂²×C₆₇	—	8	24	no	—
102	C₂²×C₆₇	#488	1	16	yes	direct product factor
103	C₂²×C₆₇	C₂×#124	1	6	yes	—
104	C₂²×C₆₇	C₂×#135	1	12	yes	—
105	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
106	C₄×C₆₇	—	2	8	no	—
107	C₄×C₆₇	—	4	4	no	—
108	C₄×C₆₇	—	4	8	no	—
109	C₄×C₆₇	C₂×#124	1	6	yes	—
110	C₄×C₆₇	C₂²×#41	1	2	yes	—
111	C₂²×#488	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₆₇	—	8	4	no	—
113	C₂×C₆₇	C₂×#488	1	12	yes	direct product factor
114	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
115	C₂×C₆₇	C₂²×#135	1	2	yes	—
116	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
117	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic
118	C₂×#488	C₂×C₆₇	1	24	yes	direct product factor
119	C₂²×#103	C₂×C₆₇	1	2	yes	complemented
120	C₂²×#102	C₂×C₆₇	1	2	yes	—
121	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
122	C₆₇	C₂²×#488	1	1	yes	characteristic, direct product factor, Sylow
123	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
124	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
126	C₂×#125	—	2	6	no	—
127	C₂×#125	C₂²×C₆₇	1	12	yes	complemented
128	C₂×#124	—	2	6	no	—
129	C₂×#124	C₂²×C₆₇	1	12	yes	—
130	C₂²×C₄⋊C₄	—	2	2	no	—
131	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
132	C₂²×C₂²⋊C₄	—	2	2	no	—
133	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
134	#488	C₂²×C₆₇	1	64	yes	direct product factor
135	C₂×#103	C₂²×C₆₇	1	24	yes	complemented
136	C₂×#102	C₂²×C₆₇	1	24	yes	—
137	C₂×C₄×C₈	C₂²×C₆₇	1	12	yes	complemented
138	C₂⁵	—	2	1	no	—
139	C₂²×Q₈	—	2	3	no	—
140	C₂²×Q₈	—	2	8	no	—
141	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
142	C₂²×D₄	—	2	3	no	—
143	C₂²×D₄	—	2	8	no	—
144	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
145	C₂³×C₄	—	2	1	no	—
146	C₂³×C₄	—	2	2	no	—
147	C₂³×C₄	D₄×C₆₇	1	2	yes	—
148	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
149	C₂²×C₈	—	2	8	no	—
150	C₂²×C₈	C₂³×C₆₇	1	12	yes	—
151	#125	—	2	24	no	—
152	#125	C₂³×C₆₇	1	16	yes	complemented
153	#124	—	2	24	no	—
154	#124	C₂³×C₆₇	1	16	yes	—
155	C₂×C₄⋊C₄	—	2	24	no	—
156	C₂×C₄⋊C₄	—	2	6	no	—
157	C₂×C₄⋊C₄	C₂³×C₆₇	1	12	yes	—
158	C₂×C₂²⋊C₄	—	2	24	no	—
159	C₂×C₄²	D₄×C₆₇	1	6	yes	—
160	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
161	#103	C₂³×C₆₇	1	32	yes	complemented
162	#102	C₂³×C₆₇	1	32	yes	—
163	C₄×C₈	C₂³×C₆₇	1	16	yes	complemented
164	C₂⁴	—	2	6	no	—
165	C₂⁴	—	4	4	no	—
166	C₂⁴	—	4	8	no	—
167	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
168	C₂×Q₈	—	2	6	no	—
169	C₂×Q₈	—	2	24	no	—
170	C₂×Q₈	—	4	12	no	—
171	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
172	C₂×D₄	—	2	6	no	—
173	C₂×D₄	—	2	24	no	—
174	C₂×D₄	—	4	12	no	—
175	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	6	no	—
178	C₂²×C₄	—	2	12	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	—	4	8	no	—
181	C₂²×C₄	C₂×D₄×C₆₇	1	12	yes	—
182	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
183	C₂²×C₄	#41×C₆₇	1	8	yes	—
184	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
185	C₂×C₈	—	2	24	no	—
186	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	12	no	—
189	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
190	C₂²⋊C₄	—	2	32	no	—
191	C₄²	C₂×D₄×C₆₇	1	12	yes	—
192	C₄²	C₂⁴×C₆₇	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	4	24	no	—
196	C₂³	—	8	16	no	—
197	C₂³	#124×C₆₇	1	4	yes	—
198	C₂³	#135×C₆₇	1	8	yes	—
199	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
200	Q₈	—	2	8	no	—
201	Q₈	—	4	12	no	—
202	D₄	—	2	8	no	—
203	D₄	—	4	12	no	—
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	4	12	no	—
208	C₂×C₄	—	4	24	no	—
209	C₂×C₄	#124×C₆₇	1	12	yes	—
210	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
211	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
212	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
213	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
214	C₈	—	2	8	no	—
215	C₂²	—	4	8	no	—
216	C₂²	—	4	4	no	—
217	C₂²	—	8	24	no	—
218	C₂²	#488×C₆₇	1	16	yes	direct product factor
219	C₂²	C₂×#124×C₆₇	1	6	yes	—
220	C₂²	C₂×#135×C₆₇	1	12	yes	—
221	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
222	C₄	—	2	8	no	—
223	C₄	—	4	8	no	—
224	C₄	—	4	4	no	—
225	C₄	C₂×#124×C₆₇	1	6	yes	—
226	C₄	C₂²×#41×C₆₇	1	2	yes	—
227	C₂	—	8	4	no	—
228	C₂	C₂×#488×C₆₇	1	12	yes	direct product factor
229	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
230	C₂	C₂²×#135×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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×#125×C₆₇	C₂	semidirect
3. C₂×#488×C₆₇	C₂	direct (N × H)
4. C₂²×#103×C₆₇	C₂	semidirect
6. C₂²×C₄×C₈×C₆₇	C₂	semidirect
11. C₂×#125×C₆₇	C₂²	semidirect
18. #488×C₆₇	C₂²	direct (N × H)
19. C₂×#103×C₆₇	C₂²	semidirect
21. C₂×C₄×C₈×C₆₇	C₂²	semidirect
36. #125×C₆₇	C₂³	semidirect
45. #103×C₆₇	C₂³	semidirect
47. C₄×C₈×C₆₇	C₂³	semidirect
102. C₂²×C₆₇	#488	direct (N × H)
111. C₂²×#488	C₆₇	direct (N × H)
113. C₂×C₆₇	C₂×#488	direct (N × H)
116. C₂²×#125	C₂×C₆₇	semidirect
118. C₂×#488	C₂×C₆₇	direct (N × H)
119. C₂²×#103	C₂×C₆₇	semidirect
121. C₂²×C₄×C₈	C₂×C₆₇	semidirect
122. C₆₇	C₂²×#488	direct (N × H)
127. C₂×#125	C₂²×C₆₇	semidirect
134. #488	C₂²×C₆₇	direct (N × H)
135. C₂×#103	C₂²×C₆₇	semidirect
137. C₂×C₄×C₈	C₂²×C₆₇	semidirect
152. #125	C₂³×C₆₇	semidirect
161. #103	C₂³×C₆₇	semidirect
163. C₄×C₈	C₂³×C₆₇	semidirect
218. C₂²	#488×C₆₇	direct (N × H)
228. C₂	C₂×#488×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×#125×C₆₇	2
C₂²×#124×C₆₇	2
C₂×#488×C₆₇	2
C₂²×#103×C₆₇	2
C₂²×#102×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂²×#488	67

Subgroups — C22×#488×C67 #30739855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#488×C₆₇ #30739855