Subgroups — #30774440

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863798	C₂	1	1	yes	maximal, characteristic
2	#22560369	C₂	1	1	yes	maximal, characteristic
3	#22558301	C₂	1	4	yes	maximal
4	C₂×#346×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14524035	—	2	1	no	—
6	#16863798	—	2	2	no	—
7	#16863798	C₄	1	4	yes	complemented
8	C₂×#14523977	—	2	1	no	—
9	C₂×#14523977	C₂²	1	2	yes	—
10	C₂×#14523977	C₂²	1	2	yes	—
11	C₂×#14523975	—	2	1	no	—
12	#16863266	—	2	1	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#346×C₆₇	—	2	2	no	—
16	#14524035	—	2	4	no	—
17	#14524035	—	4	2	no	—
18	C₂×C₄⋊Dic₆₇	—	2	2	no	—
19	C₂×C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	—	4	2	no	—
21	C₂×Dic₆₇⋊C₄	—	4	2	no	—
22	#14523977	—	2	2	no	—
23	#14523977	—	2	4	no	—
24	#14523977	—	4	2	no	—
25	#14523977	C₂×C₄	1	4	yes	—
26	#14523975	—	2	4	no	—
27	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₆₇	—	2	2	no	—
31	C₂×C₁₆×C₆₇	—	2	2	no	—
32	C₂×C₁₆×C₆₇	Q₈	1	4	yes	—
33	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₆₇	D₄	1	2	yes	—
35	C₂²×Dic₆₇	—	4	2	no	—
36	C₂²×Dic₆₇	—	4	2	no	—
37	C₄⋊Dic₆₇	—	2	4	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	4	2	no	—
41	C₄⋊Dic₆₇	—	8	4	no	—
42	Dic₆₇⋊C₄	—	4	8	no	—
43	C₂²×C₄×C₆₇	D₈	1	2	yes	—
44	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
47	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₆₇	#41	1	2	yes	—
52	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₆₇	SD₁₆	1	2	yes	—
54	C₁₆×C₆₇	—	2	4	no	—
55	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₆₇	—	2	2	no	—
57	C₂×C₄×C₆₇	—	2	2	no	—
58	C₂×C₄×C₆₇	#102	1	4	yes	—
59	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×Dic₆₇	—	4	4	no	—
66	C₂×Dic₆₇	—	8	2	no	—
67	C₂×Dic₆₇	—	8	2	no	—
68	C₂×Dic₆₇	—	8	2	no	—
69	C₂×Dic₆₇	—	8	4	no	—
70	C₈×C₆₇	#103	1	4	yes	—
71	C₈×C₆₇	C₄×Q₈	1	2	yes	—
72	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
73	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
74	C₂²×C₆₇	#486	1	1	yes	characteristic
75	C₂²×C₆₇	#500	1	1	yes	characteristic
76	C₂²×C₆₇	#509	1	2	yes	—
77	C₂²×C₆₇	#510	1	2	yes	—
78	C₄×C₆₇	—	2	4	no	—
79	C₄×C₆₇	#391	1	1	yes	characteristic
80	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
81	C₄×C₆₇	#487	1	1	yes	characteristic
82	C₄×C₆₇	#498	1	1	yes	characteristic
83	Dic₆₇	—	8	4	no	—
84	Dic₆₇	—	16	2	no	—
85	#13284	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	#1990	1	1	yes	characteristic
87	C₂×C₆₇	#2139	1	2	yes	—
88	C₂×C₆₇	#2244	1	1	yes	characteristic
89	C₂×C₆₇	#2245	1	1	yes	characteristic
90	C₂×C₆₇	#2255	1	2	yes	—
91	C₂×#499	—	67	1	no	—
92	C₂×#346	D₆₇	1	1	yes	characteristic
93	#1845	—	67	1	no	—
94	#1380	—	67	4	no	—
95	C₆₇	#13284	1	1	yes	characteristic, complemented, Sylow
96	C₂×#125	—	134	1	no	—
97	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
98	#499	—	67	4	no	—
99	#499	—	134	2	no	—
100	C₂×#107	—	67	2	no	—
101	C₂×#107	—	67	2	no	—
102	C₂×#107	—	134	1	no	—
103	C₂×#106	—	134	1	no	—
104	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
105	#384	—	134	1	no	—
106	#346	—	2	2	no	—
107	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
109	#125	—	268	2	no	—
110	#125	—	134	4	no	—
111	C₂×C₄⋊C₄	—	268	2	no	—
112	C₂×C₄⋊C₄	—	268	2	no	—
113	C₂×C₄⋊C₄	—	134	2	no	—
114	C₂×C₄⋊C₄	—	134	2	no	—
115	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	Dic₁₃₄	1	4	yes	—
119	#107	—	268	2	no	—
120	#107	—	67	4	no	—
121	#107	—	134	4	no	—
122	#107	—	134	2	no	—
123	#106	—	134	4	no	—
124	C₄×C₈	C₄×D₆₇	1	2	yes	—
125	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
126	C₂²×C₄	—	268	2	no	—
127	C₂²×C₄	—	268	2	no	—
128	C₂²×C₄	D₅₃₆	1	2	yes	—
129	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
130	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
131	C₂×C₈	#11761617	1	2	yes	—
132	C₂×C₈	Dic₂₆₈	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
134	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
135	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
136	C₂×C₈	#11761628	1	2	yes	—
137	C₄⋊C₄	—	268	8	no	—
138	C₄⋊C₄	—	268	2	no	—
139	C₄⋊C₄	—	268	2	no	—
140	C₄⋊C₄	—	268	2	no	—
141	C₄⋊C₄	—	536	4	no	—
142	C₄⋊C₄	—	134	4	no	—
143	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
144	C₄²	#11761617	1	2	yes	—
145	C₁₆	—	2	4	no	—
146	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	268	4	no	—
150	C₂×C₄	—	536	2	no	—
151	C₂×C₄	—	536	2	no	—
152	C₂×C₄	—	536	4	no	—
153	C₂×C₄	—	536	2	no	—
154	C₂×C₄	#14523965	1	4	yes	—
155	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
156	C₂×C₄	#14524031	1	1	yes	characteristic
157	C₂×C₄	#14524043	1	1	yes	characteristic
158	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
159	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
161	C₈	#14523968	1	4	yes	—
162	C₈	C₄×Dic₁₃₄	1	2	yes	—
163	C₈	#14524022	1	2	yes	—
164	C₂²	C₂×#14523965	1	1	yes	characteristic
165	C₂²	#16863743	1	1	yes	characteristic
166	C₂²	#16863801	1	1	yes	characteristic
167	C₂²	#16863834	1	2	yes	—
168	C₂²	#16863839	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	536	4	no	—
171	C₄	—	1072	2	no	—
172	C₄	#16863302	1	1	yes	characteristic
173	C₄	C₂×#14523968	1	1	yes	characteristic
174	C₄	#16863747	1	1	yes	characteristic
175	C₄	#16863794	1	1	yes	characteristic
176	C₂	#22561170	1	1	yes	characteristic
177	C₂	#22561922	1	2	yes	—
178	C₂	#22562451	1	1	yes	characteristic
179	C₂	#22562455	1	1	yes	characteristic
180	C₂	#22562495	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#346×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#346	D₆₇
hypocenter	C₆₇	#13284
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
7. #16863798	C₄	semidirect
95. C₆₇	#13284	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863798	2
#22560369	2
#22558301	2
C₂×#346×C₆₇	2
#13284	67