Subgroups — #30774452

Order 17152.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22560366	C₂	1	1	yes	maximal, characteristic
3	#22558315	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863953	—	2	1	no	—
6	#16863801	C₄	1	2	yes	complemented
7	#16863798	—	2	1	no	—
8	#16863798	C₄	1	2	yes	complemented
9	#16863794	—	2	1	no	—
10	#16863434	—	2	1	no	—
11	#16863434	C₂²	1	2	yes	—
12	C₂×#14523975	—	2	1	no	—
13	C₂×#14523975	C₂²	1	2	yes	—
14	#16863266	—	2	1	no	—
15	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#346×C₆₇	—	2	1	no	—
18	C₄×C₁₆×C₆₇	—	2	1	no	—
19	#14524043	—	2	2	no	—
20	#14524035	—	2	2	no	—
21	#14524022	—	4	2	no	—
22	C₄×Dic₁₃₄	—	4	2	no	—
23	#14524002	—	2	2	no	—
24	C₂×C₄⋊Dic₆₇	—	2	2	no	—
25	C₂×C₄⋊Dic₆₇	—	4	2	no	—
26	#14523977	—	2	2	no	—
27	#14523977	C₂×C₄	1	4	yes	—
28	#14523975	—	2	2	no	—
29	#14523975	—	2	2	no	—
30	#14523975	—	2	4	no	—
31	#14523975	—	4	2	no	—
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	M₃₂×C₆₇	—	2	2	no	—
36	M₃₂×C₆₇	Q₈	1	4	yes	complemented
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂×Dic₁₃₄	—	4	2	no	—
43	C₂²×Dic₆₇	—	4	2	no	—
44	C₄⋊Dic₆₇	—	2	4	no	—
45	C₄⋊Dic₆₇	—	4	2	no	—
46	C₄⋊Dic₆₇	—	4	2	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	C₄⋊Dic₆₇	—	8	4	no	—
49	Dic₆₇⋊C₄	—	4	4	no	—
50	C₂²⋊Dic₆₇	—	4	2	no	—
51	C₄×Dic₆₇	—	4	2	no	—
52	C₂²×C₄×C₆₇	D₈	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
57	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₆₇	#41	1	2	yes	—
62	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₆₇	SD₁₆	1	2	yes	—
65	C₁₆×C₆₇	—	2	4	no	—
66	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	#102	1	2	yes	—
69	C₂×C₄×C₆₇	#102	1	2	yes	—
70	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₆₇	—	4	4	no	—
77	C₂×Dic₆₇	—	8	4	no	—
78	C₂×Dic₆₇	—	8	2	no	—
79	Dic₁₃₄	—	8	4	no	—
80	C₈×C₆₇	#103	1	2	yes	—
81	C₈×C₆₇	#103	1	2	yes	—
82	C₈×C₆₇	C₄×Q₈	1	2	yes	—
83	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
86	C₂²×C₆₇	#486	1	1	yes	characteristic
87	C₂²×C₆₇	#500	1	1	yes	characteristic
88	C₄×C₆₇	—	2	2	no	—
89	C₄×C₆₇	#360	1	4	yes	—
90	C₄×C₆₇	#391	1	1	yes	characteristic
91	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
92	C₄×C₆₇	#487	1	1	yes	characteristic
93	C₄×C₆₇	#498	1	1	yes	characteristic
94	Dic₆₇	—	8	4	no	—
95	Dic₆₇	—	16	2	no	—
96	#13287	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	—	2	1	no	—
98	C₂×C₆₇	—	2	1	no	—
99	C₂×C₆₇	#1990	1	1	yes	characteristic
100	C₂×C₆₇	C₂×#360	1	2	yes	—
101	#3160	—	67	1	no	—
102	C₄×M₃₂	D₆₇	1	1	yes	characteristic
103	#1844	—	67	1	no	—
104	#1386	—	67	4	no	—
105	C₆₇	#13287	1	1	yes	characteristic, complemented, Sylow
106	#533	—	134	1	no	—
107	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
108	#500	—	67	2	no	—
109	#499	—	67	2	no	—
110	#499	—	134	1	no	—
111	#498	—	134	1	no	—
112	#427	—	67	2	no	—
113	#427	—	134	1	no	—
114	C₂×#106	—	67	2	no	—
115	C₂×#106	—	134	1	no	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	#384	—	134	1	no	—
118	#346	—	2	1	no	—
119	C₄×C₁₆	—	2	1	no	—
120	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
122	C₄⋊Q₈	—	134	2	no	—
123	#125	—	134	2	no	—
124	C₂²⋊Q₈	—	268	2	no	—
125	C₄×Q₈	—	268	2	no	—
126	#117	—	134	2	no	—
127	C₂×C₄⋊C₄	—	268	2	no	—
128	C₂×C₄⋊C₄	—	134	2	no	—
129	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	Dic₁₃₄	1	4	yes	complemented
132	C₂×C₁₆	—	2	2	no	—
133	#107	—	67	4	no	—
134	#107	—	134	2	no	—
135	#106	—	268	2	no	—
136	#106	—	134	4	no	—
137	#106	—	134	2	no	—
138	#106	—	134	2	no	—
139	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
140	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
143	C₂×Q₈	—	268	2	no	—
144	C₂²×C₄	—	268	2	no	—
145	C₂²×C₄	D₅₃₆	1	2	yes	—
146	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
149	C₂×C₈	#11761617	1	2	yes	—
150	C₂×C₈	Dic₂₆₈	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
153	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
154	C₂×C₈	#11761628	1	2	yes	—
155	C₄⋊C₄	—	268	4	no	—
156	C₄⋊C₄	—	268	2	no	—
157	C₄⋊C₄	—	268	2	no	—
158	C₄⋊C₄	—	268	2	no	—
159	C₄⋊C₄	—	536	4	no	—
160	C₄⋊C₄	—	134	4	no	—
161	C₂²⋊C₄	—	268	2	no	—
162	C₄²	—	268	2	no	—
163	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
164	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
165	C₄²	#11761617	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
168	Q₈	—	536	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	268	4	no	—
171	C₂×C₄	—	536	4	no	—
172	C₂×C₄	—	536	2	no	—
173	C₂×C₄	#14523965	1	2	yes	—
174	C₂×C₄	#14523965	1	2	yes	—
175	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#14524031	1	1	yes	characteristic
177	C₂×C₄	#14524043	1	1	yes	characteristic
178	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
179	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
180	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
181	C₈	#14523968	1	2	yes	—
182	C₈	#14523968	1	2	yes	—
183	C₈	C₄×Dic₁₃₄	1	2	yes	—
184	C₈	#14524022	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×#14523965	1	1	yes	characteristic
187	C₂²	#16863743	1	1	yes	characteristic
188	C₂²	#16863801	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	536	4	no	—
191	C₄	—	1072	2	no	—
192	C₄	#16863207	1	4	yes	—
193	C₄	#16863302	1	1	yes	characteristic
194	C₄	C₂×#14523968	1	1	yes	characteristic
195	C₄	#16863747	1	1	yes	characteristic
196	C₄	#16863794	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#22561170	1	1	yes	characteristic
200	C₂	C₂×#16863207	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523965
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#13287
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
6. #16863801	C₄	semidirect
8. #16863798	C₄	semidirect
36. M₃₂×C₆₇	Q₈	semidirect
105. C₆₇	#13287	semidirect
131. M₃₂	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22560366	2
#22558315	2
C₄×M₃₂×C₆₇	2
#13287	67