Subgroups — #30774445

Order 17152.

350 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	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22558304	C₂	1	4	yes	maximal
4	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14523979	—	2	1	no	—
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863420	—	2	1	no	—
9	C₄×C₆₇⋊C₁₆	—	2	1	no	—
10	C₄×C₆₇⋊C₁₆	—	2	1	no	—
11	#533×C₆₇	—	2	1	no	—
12	#500×C₆₇	C₄	1	2	yes	—
13	#499×C₆₇	—	2	1	no	—
14	#499×C₆₇	C₄	1	2	yes	—
15	#498×C₆₇	—	2	1	no	—
16	#427×C₆₇	—	2	1	no	—
17	#427×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
23	#14523979	—	2	4	no	—
24	#14523979	—	4	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₆₇	—	2	2	no	—
29	#125×C₆₇	—	2	2	no	—
30	C₂²⋊Q₈×C₆₇	—	4	2	no	—
31	C₄×Q₈×C₆₇	—	4	2	no	—
32	#117×C₆₇	—	2	2	no	—
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	#107×C₆₇	—	2	2	no	—
35	#107×C₆₇	C₂×C₄	1	4	yes	—
36	#106×C₆₇	—	2	2	no	—
37	#106×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	#11761612	—	4	2	no	—
43	#11761612	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₆₇⋊C₁₆	—	2	4	no	—
46	C₂×Q₈×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
48	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
52	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₆₇	#41	1	2	yes	—
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	4	4	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₂²⋊C₄×C₆₇	—	4	2	no	—
61	C₄²×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₆₇	D₈	1	2	yes	—
65	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
66	Q₈×C₆₇	—	8	4	no	—
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	#102	1	2	yes	—
70	C₂×C₄×C₆₇	#102	1	2	yes	—
71	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
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	—	8	4	no	—
88	C₄×C₆₇	#359	1	4	yes	—
89	C₄×C₆₇	#391	1	1	yes	characteristic
90	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
91	C₄×C₆₇	#486	1	1	yes	characteristic
92	C₄×C₆₇	#500	1	1	yes	characteristic
93	#13285	—	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	C₂×#359	1	2	yes	—
98	#3160	D₆₇	1	1	yes	characteristic
99	C₂×C₄×C₁₆	—	67	1	no	—
100	#1848	—	67	1	no	—
101	#1382	—	67	4	no	—
102	C₆₇	#13285	1	1	yes	characteristic, complemented, Sylow
103	#533	—	2	1	no	—
104	C₂²×C₁₆	—	67	2	no	—
105	#500	Dic₆₇	1	2	yes	—
106	#499	—	2	1	no	—
107	#499	Dic₆₇	1	2	yes	—
108	#498	—	2	1	no	—
109	C₂×#108	—	67	2	no	—
110	C₂×#108	—	134	1	no	—
111	#427	—	2	1	no	—
112	#427	C₂×D₆₇	1	2	yes	—
113	#423	—	134	1	no	—
114	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
115	C₄×C₁₆	—	134	1	no	—
116	C₄×C₁₆	—	134	1	no	—
117	C₂×M₁₆	—	134	2	no	—
118	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	2	no	—
121	#125	—	2	2	no	—
122	C₂²⋊Q₈	—	4	2	no	—
123	C₄×Q₈	—	4	2	no	—
124	#117	—	2	2	no	—
125	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
126	C₂×C₁₆	—	67	4	no	—
127	C₂×C₁₆	—	134	2	no	—
128	C₂×C₁₆	—	134	2	no	—
129	#108	—	268	2	no	—
130	#108	—	134	4	no	—
131	#107	—	2	2	no	—
132	#107	C₂×Dic₆₇	1	4	yes	—
133	#106	—	2	2	no	—
134	#106	—	2	2	no	—
135	C₄⋊C₈	—	268	2	no	—
136	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₆₇	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	—	4	2	no	—
141	C₂²×C₄	#11761616	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
143	M₁₆	—	268	2	no	—
144	M₁₆	—	268	2	no	—
145	C₂×C₈	—	268	2	no	—
146	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
148	C₂×C₈	#11761615	1	2	yes	—
149	C₂×C₈	#11761618	1	2	yes	—
150	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₆₇	1	2	yes	—
153	C₂×C₈	#11761629	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
160	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
161	C₄²	#11761613	1	2	yes	—
162	C₁₆	—	134	4	no	—
163	C₂³	C₂×#11761616	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	#14523964	1	2	yes	—
168	C₂×C₄	#14523964	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
170	C₂×C₄	#14524032	1	1	yes	characteristic
171	C₂×C₄	#14524041	1	1	yes	characteristic
172	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
173	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
174	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
175	C₈	—	536	2	no	—
176	C₈	#14523967	1	2	yes	—
177	C₈	#14523967	1	2	yes	—
178	C₈	Q₈×Dic₆₇	1	2	yes	—
179	C₈	#14524019	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14523964	1	1	yes	characteristic
182	C₂²	#16863748	1	1	yes	characteristic
183	C₂²	#16863792	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	#16863203	1	4	yes	—
187	C₄	#16863303	1	1	yes	characteristic
188	C₄	C₂×#14523967	1	1	yes	characteristic
189	C₄	#16863744	1	1	yes	characteristic
190	C₄	#16863799	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#22561171	1	1	yes	characteristic
194	C₂	C₂×#16863203	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#13285
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
22. C₂×C₆₇⋊C₁₆	Q₈	semidirect
102. C₆₇	#13285	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₆₇⋊C₁₆	2
#22560381	2
#22558304	2
C₆₇×#3160	2
#13285	67