Subgroups — #30772003

Order 17152.

438 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863224	C₂	1	2	yes	maximal, complemented
2	#22560238	C₂	1	2	yes	maximal
3	#22558301	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523977	C₂²	1	4	yes	—
6	#16863256	—	2	2	no	—
7	#16863224	—	2	4	no	—
8	#16863159	—	2	2	no	—
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₆₇	—	2	4	no	—
11	C₂²×C₁₆×C₆₇	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₆₇	—	2	4	no	—
15	C₂×C₂²⋊Dic₆₇	—	4	2	no	—
16	#14523977	—	2	4	no	—
17	#14523977	—	2	4	no	—
18	C₆₇⋊#95	—	4	2	no	—
19	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₆₇	—	2	2	no	—
21	C₂²×C₈×C₆₇	—	2	2	no	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	—	2	2	no	—
27	C₂×C₁₆×C₆₇	—	2	8	no	—
28	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
31	C₂²×Dic₆₇	—	4	4	no	—
32	C₄⋊Dic₆₇	—	4	4	no	—
33	C₄⋊Dic₆₇	—	4	4	no	—
34	C₂²⋊Dic₆₇	—	8	4	no	—
35	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	—	2	2	no	—
37	C₂²×C₄×C₆₇	—	2	2	no	—
38	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	—	2	2	no	—
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂×C₈×C₆₇	—	2	2	no	—
45	C₂×C₈×C₆₇	—	2	4	no	—
46	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₆₇	#41	1	2	yes	—
51	C₁₆×C₆₇	—	2	2	no	—
52	C₁₆×C₆₇	C₂²⋊C₄	1	4	yes	complemented
53	C₂³×C₆₇	—	2	2	no	—
54	C₂³×C₆₇	—	2	2	no	—
55	C₂³×C₆₇	#107	1	2	yes	—
56	C₂³×C₆₇	SD₃₂	1	4	yes	—
57	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	2	no	—
61	C₂×C₄×C₆₇	—	2	4	no	—
62	C₂×C₄×C₆₇	#107	1	2	yes	—
63	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#121	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#135	1	2	yes	—
67	C₂×Dic₆₇	—	8	4	no	—
68	C₂×Dic₆₇	—	8	4	no	—
69	C₂×Dic₆₇	—	8	4	no	—
70	C₈×C₆₇	—	2	2	no	—
71	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈×C₆₇	#117	1	1	yes	characteristic
73	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
74	C₂²×C₆₇	—	2	2	no	—
75	C₂²×C₆₇	—	2	2	no	—
76	C₂²×C₆₇	—	2	8	no	—
77	C₂²×C₆₇	#367	1	4	yes	complemented
78	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
79	C₂²×C₆₇	#466	1	2	yes	—
80	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
81	C₂²×C₆₇	#508	1	2	yes	—
82	C₄×C₆₇	—	2	2	no	—
83	C₄×C₆₇	#380	1	1	yes	characteristic
84	C₄×C₆₇	#427	1	1	yes	characteristic
85	C₄×C₆₇	#467	1	2	yes	—
86	Dic₆₇	—	16	4	no	—
87	#12835	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	4	no	—
89	C₂×C₆₇	#1808	1	1	yes	characteristic
90	C₂×C₆₇	C₂×#367	1	1	yes	characteristic
91	C₂×C₆₇	#2158	1	1	yes	characteristic
92	C₂×C₆₇	#2217	1	4	yes	—
93	C₂³×C₁₆	D₆₇	1	1	yes	characteristic
94	C₂×#367	—	67	2	no	—
95	#1808	—	67	2	no	—
96	#1380	—	67	2	no	—
97	C₆₇	#12835	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
99	C₂²×C₁₆	—	2	4	no	—
100	C₂²×C₁₆	Dic₆₇	1	4	yes	complemented
101	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
103	C₂×#107	—	67	4	no	—
104	#380	—	134	2	no	—
105	#367	—	134	4	no	—
106	#340	—	134	2	no	—
107	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	Dic₁₃₄	1	2	yes	—
111	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
113	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	134	4	no	—
115	C₂×C₂²⋊C₄	—	268	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	8	no	—
118	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₆₇	1	4	yes	—
121	#107	—	134	4	no	—
122	#107	—	134	4	no	—
123	#95	—	268	2	no	—
124	C₂⁴	D₅₃₆	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	268	4	no	—
128	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
129	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
130	C₂²×C₄	Dic₂₆₈	1	2	yes	—
131	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
137	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
139	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
140	C₂×C₈	#11761628	1	2	yes	—
141	C₄⋊C₄	—	268	4	no	—
142	C₄⋊C₄	—	268	4	no	—
143	C₂²⋊C₄	—	536	4	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₂²⋊Dic₆₇	1	4	yes	complemented
146	C₂³	—	2	2	no	—
147	C₂³	—	2	2	no	—
148	C₂³	#14523977	1	2	yes	—
149	C₂³	#14523988	1	4	yes	—
150	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	536	4	no	—
156	C₂×C₄	—	536	4	no	—
157	C₂×C₄	—	536	4	no	—
158	C₂×C₄	#14523977	1	2	yes	—
159	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
160	C₂×C₄	#14524017	1	2	yes	—
161	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
162	C₂×C₄	#14524065	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
165	C₈	#14524002	1	1	yes	characteristic
166	C₈	#14524022	1	2	yes	—
167	C₂²	—	2	8	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	#16863224	1	4	yes	complemented
171	C₂²	C₂×#14523977	1	1	yes	characteristic
172	C₂²	#16863632	1	2	yes	—
173	C₂²	C₂×#14523988	1	2	yes	—
174	C₂²	#16863828	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	1072	4	no	—
177	C₄	#16863256	1	1	yes	characteristic
178	C₄	#16863434	1	1	yes	characteristic
179	C₄	#16863637	1	2	yes	—
180	C₂	—	2	4	no	—
181	C₂	#22560238	1	1	yes	characteristic
182	C₂	C₂×#16863224	1	1	yes	characteristic
183	C₂	#22561993	1	1	yes	characteristic
184	C₂	#22562305	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₃₆
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂³×C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂³×C₁₆	D₆₇
hypocenter	C₆₇	#12835
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16863224	C₂	semidirect
3. #22558301	C₂	semidirect
11. C₂²×C₁₆×C₆₇	C₄	semidirect
52. C₁₆×C₆₇	C₂²⋊C₄	semidirect
77. C₂²×C₆₇	#367	semidirect
97. C₆₇	#12835	semidirect
100. C₂²×C₁₆	Dic₆₇	semidirect
145. C₁₆	C₂²⋊Dic₆₇	semidirect
170. C₂²	#16863224	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863224	2
#22560238	2
#22558301	2
C₂³×C₁₆×C₆₇	2
#12835	67