Subgroups — #30774448

Order 17152.

342 conjugacy classes of subgroups.

The subgroup lattice has 196 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	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22558305	C₂	1	4	yes	maximal
4	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863953	—	2	1	no	—
6	#16863801	C₄	1	2	yes	—
7	#16863798	—	2	1	no	—
8	#16863798	C₄	1	2	yes	—
9	#16863794	—	2	1	no	—
10	C₂×#14523979	—	2	1	no	—
11	C₂×#14523979	C₂²	1	2	yes	—
12	#16863434	—	2	1	no	—
13	#16863434	C₂²	1	2	yes	—
14	#16863420	—	2	1	no	—
15	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#346×C₆₇	—	2	1	no	—
18	#346×C₆₇	—	2	1	no	—
19	C₂×#11761612	—	2	2	no	—
20	#14524043	—	2	2	no	—
21	#14524035	—	2	2	no	—
22	#14524022	—	4	2	no	—
23	C₄×Dic₁₃₄	—	4	2	no	—
24	#14524002	—	2	2	no	—
25	#14523979	—	2	4	no	—
26	#14523979	—	4	2	no	—
27	#14523977	—	2	2	no	—
28	#14523977	C₂×C₄	1	4	yes	—
29	#14523975	—	2	2	no	—
30	#14523975	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	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	C₂×C₁₆×C₆₇	—	2	2	no	—
36	C₂×C₁₆×C₆₇	—	2	2	no	—
37	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
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	#11761612	—	4	2	no	—
44	#11761612	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₄⋊Dic₆₇	—	2	4	no	—
47	C₄⋊Dic₆₇	—	4	2	no	—
48	Dic₆₇⋊C₄	—	4	4	no	—
49	C₂²⋊Dic₆₇	—	4	2	no	—
50	C₄×Dic₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₆₇	#41	1	2	yes	—
61	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₆₇	D₈	1	2	yes	—
64	C₁₆×C₆₇	—	2	4	no	—
65	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	#102	1	2	yes	—
68	C₂×C₄×C₆₇	#102	1	2	yes	—
69	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₆₇	—	4	4	no	—
76	Dic₁₃₄	—	8	4	no	—
77	C₈×C₆₇	#103	1	2	yes	—
78	C₈×C₆₇	#103	1	2	yes	—
79	C₈×C₆₇	C₄×Q₈	1	2	yes	—
80	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
81	C₆₇⋊C₈	—	8	2	no	—
82	C₂²×C₆₇	—	2	2	no	—
83	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
84	C₂²×C₆₇	#487	1	1	yes	characteristic
85	C₂²×C₆₇	#498	1	1	yes	characteristic
86	C₄×C₆₇	—	2	2	no	—
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	#391	1	1	yes	characteristic
89	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
90	C₄×C₆₇	#486	1	1	yes	characteristic
91	C₄×C₆₇	#500	1	1	yes	characteristic
92	Dic₆₇	—	8	4	no	—
93	#13286	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	—	2	1	no	—
95	C₂×C₆₇	—	2	1	no	—
96	C₂×C₆₇	#1990	1	1	yes	characteristic
97	C₂×C₆₇	#2143	1	2	yes	—
98	#3160	—	67	1	no	—
99	C₂×#346	D₆₇	1	1	yes	characteristic
100	#1848	—	67	1	no	—
101	#1382	—	67	4	no	—
102	C₆₇	#13286	1	1	yes	characteristic, complemented, Sylow
103	#533	—	134	1	no	—
104	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
105	#500	—	67	2	no	—
106	#499	—	67	2	no	—
107	#499	—	134	1	no	—
108	#498	—	134	1	no	—
109	C₂×#108	—	67	2	no	—
110	C₂×#108	—	134	1	no	—
111	#427	—	67	2	no	—
112	#427	—	134	1	no	—
113	#423	—	134	1	no	—
114	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
115	#346	—	2	1	no	—
116	#346	—	2	1	no	—
117	C₂×M₁₆	—	134	2	no	—
118	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
120	C₄⋊Q₈	—	134	2	no	—
121	#125	—	134	2	no	—
122	C₂²⋊Q₈	—	268	2	no	—
123	C₄×Q₈	—	268	2	no	—
124	#117	—	134	2	no	—
125	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
129	#108	—	268	2	no	—
130	#108	—	134	4	no	—
131	#107	—	67	4	no	—
132	#107	—	134	2	no	—
133	#106	—	134	2	no	—
134	#106	—	134	2	no	—
135	C₄⋊C₈	—	268	2	no	—
136	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
137	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×Q₈	—	268	2	no	—
141	C₂²×C₄	#11761617	1	2	yes	—
142	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
143	M₁₆	—	268	2	no	—
144	M₁₆	—	268	2	no	—
145	C₂×C₈	—	268	2	no	—
146	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	#11761617	1	2	yes	—
149	C₂×C₈	Dic₂₆₈	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
153	C₂×C₈	#11761628	1	2	yes	—
154	C₄⋊C₄	—	268	4	no	—
155	C₄⋊C₄	—	268	2	no	—
156	C₄⋊C₄	—	134	4	no	—
157	C₂²⋊C₄	—	268	2	no	—
158	C₄²	—	268	2	no	—
159	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
160	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
161	C₄²	D₅₃₆	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×#11761617	1	1	yes	characteristic
164	Q₈	—	536	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	268	4	no	—
167	C₂×C₄	#14523965	1	2	yes	—
168	C₂×C₄	#14523965	1	2	yes	—
169	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
170	C₂×C₄	#14524031	1	1	yes	characteristic
171	C₂×C₄	#14524043	1	1	yes	characteristic
172	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
173	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
175	C₈	—	536	2	no	—
176	C₈	#14523968	1	2	yes	—
177	C₈	#14523968	1	2	yes	—
178	C₈	C₄×Dic₁₃₄	1	2	yes	—
179	C₈	#14524022	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14523965	1	1	yes	characteristic
182	C₂²	#16863747	1	1	yes	characteristic
183	C₂²	#16863794	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	536	4	no	—
187	C₄	#16863302	1	1	yes	characteristic
188	C₄	C₂×#14523968	1	1	yes	characteristic
189	C₄	#16863743	1	1	yes	characteristic
190	C₄	#16863801	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#22561170	1	1	yes	characteristic
194	C₂	#22561938	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#346×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	#13286
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
37. C₂×C₁₆×C₆₇	Q₈	semidirect
102. C₆₇	#13286	semidirect
128. 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
#22570723	2
#22560381	2
#22558305	2
C₂×#346×C₆₇	2
#13286	67