Subgroups — #31015928

Order 17152.

2858 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863874	C₂	1	2	yes	maximal
2	#22563130	C₂	1	8	yes	maximal, complemented
3	Q₈×C₄⋊Dic₆₇	C₂	1	8	yes	maximal, complemented
4	#22562827	C₂	1	6	yes	maximal
5	#22562616	C₂	1	6	yes	maximal
6	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863874	—	2	16	no	—
8	C₂×Q₈×Dic₆₇	C₂²	1	16	yes	—
9	C₂×#14524002	C₂²	1	12	yes	—
10	#16863295	C₂²	1	48	yes	—
11	#16863266	C₂²	1	12	yes	—
12	#16863262	C₂²	1	12	yes	—
13	#16863256	C₂²	1	12	yes	—
14	#16863256	C₂²	1	4	yes	—
15	C₄×C₄⋊Dic₆₇	C₂²	1	12	yes	—
16	C₄×C₄⋊Dic₆₇	C₂²	1	12	yes	—
17	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	C₄	1	16	yes	complemented
19	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
23	Q₈×Dic₆₇	—	2	64	no	—
24	#14524002	—	2	48	no	—
25	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
26	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	—
27	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	—
28	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
29	C₂×C₄×Dic₆₇	C₂³	1	24	yes	—
30	C₆₇⋊#95	C₂³	1	8	yes	—
31	C₆₇⋊#95	C₂³	1	24	yes	—
32	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
33	C₂²×Q₈×C₆₇	D₄	1	8	yes	—
34	C₂²×Q₈×C₆₇	Q₈	1	8	yes	—
35	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
39	C₄×Q₈×C₆₇	C₂×C₄	1	32	yes	—
40	C₄×Q₈×C₆₇	C₂×C₄	1	32	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	24	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
45	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
46	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
47	C₄⋊Dic₆₇	—	2	48	no	—
48	C₄⋊Dic₆₇	—	2	16	no	—
49	C₄⋊Dic₆₇	—	4	24	no	—
50	C₂²⋊Dic₆₇	—	2	16	no	—
51	C₄×Dic₆₇	—	2	48	no	—
52	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₂×Q₈×C₆₇	C₄⋊C₄	1	64	yes	complemented
54	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
55	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	—
56	C₂×Q₈×C₆₇	C₂×Q₈	1	16	yes	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	4	6	no	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₆₇	C₂×Q₈	1	12	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₆₇	C₂²×C₄	1	48	yes	—
69	C₄²×C₆₇	C₂²×C₄	1	24	yes	—
70	C₄²×C₆₇	C₂²×C₄	1	24	yes	—
71	C₂³×C₆₇	—	2	2	no	—
72	C₂³×C₆₇	—	2	2	no	—
73	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
74	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
75	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
76	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
77	Q₈×C₆₇	C₂×C₄⋊C₄	1	64	yes	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	12	no	—
80	C₂×C₄×C₆₇	—	2	8	no	—
81	C₂×C₄×C₆₇	—	4	12	no	—
82	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	48	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
84	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
85	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
86	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
87	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
88	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
89	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
90	C₂×Dic₆₇	—	2	16	no	—
91	C₂×Dic₆₇	—	2	16	no	—
92	C₂×Dic₆₇	—	2	16	no	—
93	C₂²×C₆₇	—	2	2	no	—
94	C₂²×C₆₇	—	2	2	no	—
95	C₂²×C₆₇	—	2	4	no	—
96	C₂²×C₆₇	—	2	4	no	—
97	C₂²×C₆₇	C₂²×C₄⋊C₄	1	4	yes	—
98	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
99	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
100	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
101	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
102	C₂²×C₆₇	#585	1	2	yes	—
103	C₄×C₆₇	—	2	2	no	—
104	C₄×C₆₇	—	2	12	no	—
105	C₄×C₆₇	C₂²×C₄⋊C₄	1	24	yes	—
106	C₄×C₆₇	#519	1	4	yes	complemented
107	Dic₆₇	—	4	16	no	—
108	#34182	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	—	2	4	no	—
110	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#519	1	1	yes	characteristic
112	C₂×C₆₇	#3428	1	1	yes	characteristic
113	C₂×C₆₇	#3497	1	2	yes	—
114	C₂×C₆₇	#3502	1	2	yes	—
115	C₂×#519	—	67	2	no	—
116	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
117	#2351	—	67	8	no	—
118	Q₈×C₄⋊C₄	—	67	8	no	—
119	#2316	—	67	6	no	—
120	#2285	—	67	6	no	—
121	C₆₇	#34182	1	1	yes	characteristic, complemented, Sylow
122	C₂³×Q₈	C₂×D₆₇	1	1	yes	characteristic
123	#519	—	134	16	no	—
124	C₂×C₄×Q₈	—	67	16	no	—
125	C₂×C₄×Q₈	Dic₆₇	1	16	yes	complemented
126	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
127	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
128	C₂×#117	—	67	12	no	—
129	C₂²×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
130	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
131	#389	—	67	48	no	—
132	#384	—	67	12	no	—
133	#382	—	67	12	no	—
134	#380	—	67	12	no	—
135	#380	—	67	4	no	—
136	C₄×C₄⋊C₄	—	67	12	no	—
137	C₄×C₄⋊C₄	—	67	12	no	—
138	C₂²×Q₈	Dic₁₃₄	1	8	yes	—
139	C₂²×Q₈	D₂₆₈	1	8	yes	—
140	C₂²×Q₈	C₂×Dic₆₇	1	8	yes	—
141	C₂²×Q₈	C₂²×D₆₇	1	4	yes	—
142	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
143	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
144	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
145	C₄×Q₈	—	134	64	no	—
146	C₄×Q₈	C₂×Dic₆₇	1	32	yes	—
147	C₄×Q₈	C₂×Dic₆₇	1	32	yes	—
148	#117	—	134	48	no	—
149	C₂×C₄⋊C₄	—	67	24	no	—
150	C₂×C₄⋊C₄	—	67	24	no	—
151	C₂×C₄⋊C₄	—	67	8	no	—
152	C₂×C₄⋊C₄	C₂×Dic₆₇	1	24	yes	—
153	C₂×C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
154	C₂×C₂²⋊C₄	—	67	8	no	—
155	C₂×C₄²	—	67	24	no	—
156	C₂×C₄²	C₂×Dic₆₇	1	24	yes	—
157	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
158	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
159	#95	—	67	8	no	—
160	#95	—	67	24	no	—
161	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
162	C₂×Q₈	C₄⋊Dic₆₇	1	64	yes	complemented
163	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
164	C₂×Q₈	C₂×D₂₆₈	1	16	yes	—
165	C₂×Q₈	C₂×Dic₁₃₄	1	16	yes	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	6	no	—
168	C₂²×C₄	—	67	16	no	—
169	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
170	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
171	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
172	C₂²×C₄	C₂×D₂₆₈	1	12	yes	—
173	C₂²×C₄	C₂×Dic₁₃₄	1	12	yes	—
174	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
175	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
176	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
177	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
178	C₄⋊C₄	—	268	24	no	—
179	C₄⋊C₄	—	134	48	no	—
180	C₄⋊C₄	—	134	16	no	—
181	C₄⋊C₄	C₂²×Dic₆₇	1	48	yes	—
182	C₂²⋊C₄	—	134	16	no	—
183	C₄²	—	134	48	no	—
184	C₄²	C₂²×Dic₆₇	1	24	yes	—
185	C₄²	C₂²×Dic₆₇	1	24	yes	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	C₂³×Dic₆₇	1	2	yes	—
189	C₂³	C₂²×D₂₆₈	1	2	yes	—
190	C₂³	C₂²×Dic₁₃₄	1	2	yes	—
191	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
192	Q₈	C₂×C₄⋊Dic₆₇	1	64	yes	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	4	12	no	—
197	C₂×C₄	—	134	16	no	—
198	C₂×C₄	—	134	16	no	—
199	C₂×C₄	—	134	16	no	—
200	C₂×C₄	C₂×C₄⋊Dic₆₇	1	48	yes	—
201	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
202	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
203	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
204	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
205	C₂×C₄	C₂²×D₂₆₈	1	12	yes	—
206	C₂×C₄	C₂²×Dic₁₃₄	1	12	yes	—
207	C₂×C₄	#14524091	1	4	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	C₂²×C₄⋊Dic₆₇	1	4	yes	—
213	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
214	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
215	C₂²	C₂³×Dic₁₃₄	1	1	yes	characteristic
216	C₂²	C₂×#14524091	1	2	yes	—
217	C₂²	#16864232	1	2	yes	—
218	C₄	—	2	2	no	—
219	C₄	—	2	12	no	—
220	C₄	—	268	16	no	—
221	C₄	C₂²×C₄⋊Dic₆₇	1	24	yes	—
222	C₄	#16863874	1	4	yes	complemented
223	C₂	—	2	4	no	—
224	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic
225	C₂	C₂×#16863874	1	1	yes	characteristic
226	C₂	#22573412	1	1	yes	characteristic
227	C₂	#22573980	1	2	yes	—
228	C₂	#22574019	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle, cosocle	C₂³×C₆₇	C₂⁵
prime radical	C₂⁴×C₆₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	#34182
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #22563130	C₂	semidirect
3. Q₈×C₄⋊Dic₆₇	C₂	semidirect
18. C₂×C₄×Q₈×C₆₇	C₄	semidirect
53. C₂×Q₈×C₆₇	C₄⋊C₄	semidirect
106. C₄×C₆₇	#519	semidirect
121. C₆₇	#34182	semidirect
125. C₂×C₄×Q₈	Dic₆₇	semidirect
162. C₂×Q₈	C₄⋊Dic₆₇	semidirect
222. C₄	#16863874	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16863874	2
#22563130	2
Q₈×C₄⋊Dic₆₇	2
#22562827	2
#22562616	2
C₂²×C₄×Q₈×C₆₇	2
#34182	67