Subgroups — #31011341

Order 17152.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570749	C₂	1	2	yes	maximal
2	#22562494	C₂	1	4	yes	maximal, complemented
3	#22562486	C₂	1	2	yes	maximal, complemented
4	#22562483	C₂	1	2	yes	maximal, complemented
5	#22561998	C₂	1	2	yes	maximal
6	C₂×#16863222	C₂	1	1	yes	maximal, characteristic
7	#22561805	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic
9	#16864024	—	2	2	no	—
10	C₂×#14523983	C₂²	1	1	yes	characteristic
11	#16863803	—	2	4	no	—
12	#16863800	C₂²	1	4	yes	—
13	#16863797	C₂²	1	4	yes	—
14	#16863439	—	2	2	no	—
15	C₂×#14523977	C₂²	1	2	yes	—
16	#16863351	C₂²	1	1	yes	characteristic
17	#16863223	C₂²	1	4	yes	—
18	#16863222	C₂²	1	2	yes	—
19	#16863222	C₂²	1	2	yes	—
20	#16863221	C₂²	1	4	yes	—
21	#16863171	C₂²	1	2	yes	—
22	C₁₆×Dic₆₇	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	#427×C₆₇	C₂²	1	2	yes	—
25	#367×C₆₇	C₂²	1	2	yes	—
26	#366×C₆₇	C₂²	1	1	yes	characteristic
27	#366×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#11761612	D₄	1	1	yes	characteristic
29	#14524043	—	4	2	no	—
30	#14524042	—	2	4	no	—
31	#14524035	—	4	2	no	—
32	#14524034	—	2	4	no	—
33	#14524020	—	4	4	no	—
34	#14524002	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₆₇	—	2	2	no	—
36	#14523983	Q₈	1	4	yes	complemented
37	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
38	#14523977	—	2	2	no	—
39	#14523977	C₂³	1	4	yes	—
40	#14523976	—	2	4	no	—
41	#14523974	—	2	4	no	—
42	#14523953	D₄	1	2	yes	—
43	C₈×Dic₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	#117×C₆₇	—	2	2	no	—
46	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
49	#107×C₆₇	C₂³	1	2	yes	—
50	#107×C₆₇	C₂³	1	2	yes	—
51	#106×C₆₇	—	2	2	no	—
52	C₂×Dic₁₃₄	—	4	4	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	#11761612	Q₁₆	1	4	yes	—
55	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
56	C₄⋊Dic₆₇	—	2	4	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	4	no	—
59	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
60	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
61	Dic₆₇⋊C₄	—	4	4	no	—
62	Dic₆₇⋊C₄	—	4	4	no	—
63	C₂²⋊Dic₆₇	D₈	1	2	yes	—
64	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
65	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂²⋊C₄×C₆₇	—	4	2	no	—
74	C₄²×C₆₇	—	4	2	no	—
75	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
76	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
77	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	—	4	2	no	—
80	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₆₇	—	4	4	no	—
85	C₂×Dic₆₇	—	16	2	no	—
86	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
87	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
88	Dic₁₃₄	—	16	4	no	—
89	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
90	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
92	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	—
93	C₂²×C₆₇	—	2	1	no	—
94	C₂²×C₆₇	—	2	1	no	—
95	C₂²×C₆₇	#500	1	2	yes	—
96	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₆₇	—	8	2	no	—
98	C₄×C₆₇	—	8	2	no	—
99	C₄×C₆₇	#500	1	2	yes	—
100	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₆₇	—	2	2	no	—
103	Dic₆₇	—	2	2	no	—
104	Dic₆₇	—	8	4	no	—
105	#33729	—	67	1	no	maximal, Sylow
106	C₂×C₆₇	—	2	2	no	—
107	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
108	C₂×C₆₇	#3415	1	1	yes	characteristic
109	C₂×C₆₇	#3417	1	1	yes	characteristic
110	#3164	—	67	2	no	—
111	#2255	—	67	4	no	—
112	#2253	—	67	2	no	—
113	#2252	—	67	2	no	—
114	#2159	—	67	2	no	—
115	#2158	D₆₇	1	1	yes	characteristic
116	C₂×#366	—	67	1	no	—
117	#2108	—	67	1	no	—
118	C₆₇	#33729	1	1	yes	characteristic, complemented, Sylow
119	#544	—	134	2	no	—
120	C₂×M₃₂	—	67	1	no	—
121	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
122	#501	—	134	4	no	—
123	#500	—	67	4	no	—
124	#499	—	67	4	no	—
125	#428	—	134	2	no	—
126	#427	C₂×D₆₇	1	2	yes	—
127	C₂×#107	—	67	2	no	—
128	#405	—	67	1	no	—
129	#367	—	67	4	no	—
130	#367	C₂×D₆₇	1	2	yes	—
131	#366	—	67	2	no	—
132	#366	—	67	2	no	—
133	#366	—	67	4	no	—
134	#366	C₂×D₆₇	1	1	yes	characteristic
135	#366	C₂×D₆₇	1	1	yes	characteristic
136	#346	—	67	2	no	—
137	C₄×C₁₆	—	67	2	no	—
138	C₂×M₁₆	—	67	1	no	—
139	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
140	C₄⋊Q₈	—	268	2	no	—
141	C₄⋊Q₈	—	134	4	no	—
142	#125	—	268	2	no	—
143	#125	—	134	4	no	—
144	C₂²⋊Q₈	—	268	4	no	—
145	#117	—	2	2	no	—
146	#117	—	67	1	no	—
147	C₂×C₄⋊C₄	—	134	2	no	—
148	M₃₂	—	67	4	no	—
149	C₂×C₁₆	—	67	2	no	—
150	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
153	#107	—	67	4	no	—
154	#107	—	134	4	no	—
155	#107	—	134	2	no	—
156	#107	C₂²×D₆₇	1	2	yes	—
157	#107	C₂²×D₆₇	1	2	yes	—
158	#106	—	2	2	no	—
159	#106	—	134	4	no	—
160	#97	—	67	2	no	—
161	C₄×C₈	—	67	2	no	—
162	C₂×Q₈	—	268	4	no	—
163	C₂²×C₄	—	268	2	no	—
164	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
165	M₁₆	—	67	4	no	—
166	C₂×C₈	—	67	2	no	—
167	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
168	C₂×C₈	Q₈×D₆₇	1	2	yes	—
169	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
170	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	268	4	no	—
174	C₄⋊C₄	—	268	4	no	—
175	C₄⋊C₄	—	268	4	no	—
176	C₄⋊C₄	—	268	2	no	—
177	C₄⋊C₄	—	67	1	no	—
178	C₄⋊C₄	—	67	1	no	—
179	C₄⋊C₄	—	134	4	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	67	2	no	—
182	C₄²	—	4	2	no	—
183	C₄²	—	67	2	no	—
184	C₁₆	—	67	4	no	—
185	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
186	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
187	C₂³	D₈×D₆₇	1	1	yes	characteristic
188	Q₈	—	1072	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	268	4	no	—
192	C₂×C₄	—	1072	2	no	—
193	C₂×C₄	—	67	2	no	—
194	C₂×C₄	—	67	2	no	—
195	C₂×C₄	#14524042	1	2	yes	—
196	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
197	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
198	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
199	C₈	—	67	4	no	—
200	C₈	#14524042	1	2	yes	—
201	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#16863800	1	2	yes	—
206	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
207	C₄	—	8	2	no	—
208	C₄	—	8	2	no	—
209	C₄	—	536	4	no	—
210	C₄	—	134	2	no	—
211	C₄	—	134	2	no	—
212	C₄	#16863800	1	2	yes	—
213	C₄	C₂×#14524042	1	1	yes	characteristic
214	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#16863800	1	1	yes	characteristic
217	C₂	#22573347	1	1	yes	characteristic
218	C₂	#22573359	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2158	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#2158	D₆₇
hypocenter	C₆₇	#33729
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
2. #22562494	C₂	semidirect
3. #22562486	C₂	semidirect
4. #22562483	C₂	semidirect
36. #14523983	Q₈	semidirect
46. C₂×C₁₆×C₆₇	Q₈	semidirect
118. C₆₇	#33729	semidirect
150. C₂×C₁₆	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570749	2
#22562494	2
#22562486	2
#22562483	2
#22561998	2
C₂×#16863222	2
#22561805	2
C₆₇×#2158	2
#33729	67