Subgroups — #31011349

Order 17152.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 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	#22562493	C₂	1	4	yes	maximal, complemented
3	#22562490	C₂	1	2	yes	maximal, complemented
4	#22562487	C₂	1	2	yes	maximal, complemented
5	#22561998	C₂	1	2	yes	maximal
6	C₂×#16863224	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	#16863224	C₂²	1	2	yes	—
18	#16863224	C₂²	1	2	yes	—
19	#16863223	C₂²	1	4	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	1	yes	characteristic
26	#367×C₆₇	C₂²	1	1	yes	characteristic
27	#366×C₆₇	C₂²	1	2	yes	—
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	2	yes	—
60	Dic₆₇⋊C₄	—	4	4	no	—
61	Dic₆₇⋊C₄	—	4	4	no	—
62	C₂²⋊Dic₆₇	D₈	1	2	yes	—
63	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
64	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₂²⋊C₄×C₆₇	—	4	2	no	—
73	C₄²×C₆₇	—	4	2	no	—
74	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
75	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
76	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₆₇	—	4	4	no	—
84	C₂×Dic₆₇	—	16	2	no	—
85	C₂×Dic₆₇	C₂×D₈	1	4	yes	—
86	Dic₁₃₄	—	16	4	no	—
87	C₈×C₆₇	C₄⋊Q₈	1	2	yes	—
88	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
89	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
90	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	—
91	C₂²×C₆₇	—	2	2	no	—
92	C₂²×C₆₇	#500	1	2	yes	—
93	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₆₇	—	8	2	no	—
95	C₄×C₆₇	—	8	2	no	—
96	C₄×C₆₇	#500	1	2	yes	—
97	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
98	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
99	Dic₆₇	—	2	4	no	—
100	Dic₆₇	—	8	4	no	—
101	#33730	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	—	2	2	no	—
103	C₂×C₆₇	C₂×#500	1	1	yes	characteristic
104	C₂×C₆₇	#3416	1	2	yes	—
105	#3164	—	67	2	no	—
106	#2255	—	67	4	no	—
107	#2254	—	67	2	no	—
108	#2253	—	67	2	no	—
109	#2159	—	67	2	no	—
110	#2158	D₆₇	1	1	yes	characteristic
111	C₂×#367	—	67	1	no	—
112	#2108	—	67	1	no	—
113	C₆₇	#33730	1	1	yes	characteristic, complemented, Sylow
114	#544	—	134	2	no	—
115	C₂×M₃₂	—	67	1	no	—
116	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
117	#501	—	134	4	no	—
118	#500	—	67	4	no	—
119	#499	—	67	4	no	—
120	#428	—	134	2	no	—
121	#427	C₂×D₆₇	1	2	yes	—
122	C₂×#107	—	67	2	no	—
123	#405	—	67	1	no	—
124	#367	—	67	2	no	—
125	#367	—	67	2	no	—
126	#367	—	67	4	no	—
127	#367	C₂×D₆₇	1	1	yes	characteristic
128	#367	C₂×D₆₇	1	1	yes	characteristic
129	#366	—	67	4	no	—
130	#366	C₂×D₆₇	1	2	yes	—
131	#346	—	67	2	no	—
132	C₄×C₁₆	—	67	2	no	—
133	C₂×M₁₆	—	67	1	no	—
134	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	268	2	no	—
136	C₄⋊Q₈	—	134	4	no	—
137	#125	—	268	2	no	—
138	#125	—	134	4	no	—
139	C₂²⋊Q₈	—	268	4	no	—
140	#117	—	2	2	no	—
141	#117	—	67	1	no	—
142	C₂×C₄⋊C₄	—	134	2	no	—
143	M₃₂	—	67	4	no	—
144	C₂×C₁₆	—	67	2	no	—
145	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
146	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
148	#107	—	67	4	no	—
149	#107	—	134	4	no	—
150	#107	—	134	2	no	—
151	#107	C₂²×D₆₇	1	2	yes	—
152	#107	C₂²×D₆₇	1	2	yes	—
153	#106	—	2	2	no	—
154	#106	—	134	4	no	—
155	#97	—	67	2	no	—
156	C₄×C₈	—	67	2	no	—
157	C₂×Q₈	—	268	4	no	—
158	C₂²×C₄	—	268	2	no	—
159	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
160	M₁₆	—	67	4	no	—
161	C₂×C₈	—	67	2	no	—
162	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₆₇	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
165	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	268	4	no	—
169	C₄⋊C₄	—	268	4	no	—
170	C₄⋊C₄	—	268	4	no	—
171	C₄⋊C₄	—	268	2	no	—
172	C₄⋊C₄	—	67	2	no	—
173	C₄⋊C₄	—	134	4	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	—	67	2	no	—
176	C₄²	—	4	2	no	—
177	C₄²	—	67	2	no	—
178	C₁₆	—	67	4	no	—
179	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
180	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
181	C₂³	D₈×D₆₇	1	1	yes	characteristic
182	Q₈	—	1072	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	268	4	no	—
186	C₂×C₄	—	1072	2	no	—
187	C₂×C₄	—	67	4	no	—
188	C₂×C₄	#14524042	1	2	yes	—
189	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
191	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
192	C₈	—	67	4	no	—
193	C₈	#14524042	1	2	yes	—
194	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#16863800	1	2	yes	—
198	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
199	C₄	—	8	2	no	—
200	C₄	—	8	2	no	—
201	C₄	—	536	4	no	—
202	C₄	—	134	4	no	—
203	C₄	#16863800	1	2	yes	—
204	C₄	C₂×#14524042	1	1	yes	characteristic
205	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#16863800	1	1	yes	characteristic
208	C₂	#22573352	1	2	yes	—

	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	#33730
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. #22562493	C₂	semidirect
3. #22562490	C₂	semidirect
4. #22562487	C₂	semidirect
36. #14523983	Q₈	semidirect
46. C₂×C₁₆×C₆₇	Q₈	semidirect
113. C₆₇	#33730	semidirect
145. 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
#22562493	2
#22562490	2
#22562487	2
#22561998	2
C₂×#16863224	2
#22561805	2
C₆₇×#2158	2
#33730	67