Subgroups — #30774465

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
2	#22560381	C₂	1	1	yes	maximal, characteristic
3	#22558317	C₂	1	4	yes	maximal
4	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523979	—	2	1	no	—
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863420	—	2	1	no	—
9	#16863172	—	2	1	no	—
10	C₄×C₆₇⋊C₁₆	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
12	#500×C₆₇	—	2	1	no	—
13	#500×C₆₇	—	2	1	no	—
14	#500×C₆₇	C₄	1	2	yes	—
15	#500×C₆₇	C₄	1	2	yes	—
16	C₂×#107×C₆₇	—	2	1	no	—
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	—	2	2	no	—
20	#14523983	—	2	2	no	—
21	#14523983	Q₈	1	4	yes	complemented
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	#14523979	—	2	4	no	—
24	#14523979	—	4	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
26	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
27	C₂²×Q₈×C₆₇	—	4	1	no	—
28	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₆₇	—	2	2	no	—
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	4	1	no	—
33	C₄⋊Q₈×C₆₇	—	4	1	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
36	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
37	#107×C₆₇	—	2	2	no	—
38	#107×C₆₇	—	2	2	no	—
39	#107×C₆₇	—	2	2	no	—
40	#107×C₆₇	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	#11761612	—	4	2	no	—
46	#11761612	—	4	2	no	—
47	C₂×C₆₇⋊C₈	—	4	2	no	—
48	C₆₇⋊C₁₆	—	2	4	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	4	2	no	—
51	C₂×Q₈×C₆₇	—	8	4	no	—
52	C₂²×C₄×C₆₇	—	4	2	no	—
53	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
59	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₆₇	#41	1	2	yes	—
64	C₄⋊C₄×C₆₇	—	2	4	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₇	D₈	1	1	yes	characteristic
71	C₄²×C₆₇	D₈	1	1	yes	characteristic
72	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
73	Q₈×C₆₇	—	8	4	no	—
74	Q₈×C₆₇	—	8	4	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	4	4	no	—
78	C₂×C₄×C₆₇	—	8	4	no	—
79	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
89	C₈×C₆₇	#103	1	2	yes	—
90	C₈×C₆₇	#103	1	2	yes	—
91	C₈×C₆₇	C₄×Q₈	1	2	yes	—
92	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
93	C₆₇⋊C₈	—	8	2	no	—
94	C₂²×C₆₇	—	2	1	no	—
95	C₂²×C₆₇	—	2	1	no	—
96	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
97	C₂²×C₆₇	#487	1	1	yes	characteristic
98	C₂²×C₆₇	#498	1	1	yes	characteristic
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	8	4	no	—
102	C₄×C₆₇	#361	1	2	yes	—
103	C₄×C₆₇	#362	1	2	yes	—
104	C₄×C₆₇	#391	1	1	yes	characteristic
105	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
106	C₄×C₆₇	#486	1	1	yes	characteristic
107	C₄×C₆₇	#500	1	1	yes	characteristic
108	#13290	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	—	2	1	no	—
110	C₂×C₆₇	—	2	1	no	—
111	C₂×C₆₇	#1990	1	1	yes	characteristic
112	C₂×C₆₇	C₂×#361	1	1	yes	characteristic
113	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
114	C₂×#500	D₆₇	1	1	yes	characteristic
115	C₄×M₃₂	—	67	1	no	—
116	#1848	—	67	1	no	—
117	#1387	—	67	4	no	—
118	C₆₇	#13290	1	1	yes	characteristic, complemented, Sylow
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	C₂×M₃₂	—	67	2	no	—
121	#500	—	2	1	no	—
122	#500	—	2	1	no	—
123	#500	Dic₆₇	1	2	yes	—
124	#500	Dic₆₇	1	2	yes	—
125	C₂×#108	—	67	2	no	—
126	C₂×#108	—	134	1	no	—
127	C₂×#107	—	2	1	no	—
128	C₂×#107	C₂×D₆₇	1	2	yes	—
129	#423	—	134	1	no	—
130	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
131	#346	—	134	1	no	—
132	C₄×C₁₆	—	134	1	no	—
133	C₂²×Q₈	—	4	1	no	—
134	C₂×M₁₆	—	134	2	no	—
135	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	4	1	no	—
140	C₄⋊Q₈	—	4	1	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
144	M₃₂	—	67	4	no	—
145	M₃₂	—	134	2	no	—
146	C₂×C₁₆	—	134	2	no	—
147	#108	—	268	2	no	—
148	#108	—	134	4	no	—
149	#107	—	2	2	no	—
150	#107	—	2	2	no	—
151	#107	—	2	2	no	—
152	#107	C₂×Dic₆₇	1	4	yes	—
153	C₄⋊C₈	—	268	1	no	—
154	C₄⋊C₈	—	268	1	no	—
155	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
156	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	8	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	#11761616	1	1	yes	characteristic
164	C₂²×C₄	#11761616	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
166	M₁₆	—	268	2	no	—
167	M₁₆	—	268	2	no	—
168	C₂×C₈	—	268	2	no	—
169	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
171	C₂×C₈	#11761615	1	2	yes	—
172	C₂×C₈	#11761618	1	2	yes	—
173	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
175	C₂×C₈	Q₈×D₆₇	1	2	yes	—
176	C₂×C₈	#11761629	1	2	yes	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
182	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
183	C₄²	#11761613	1	1	yes	characteristic
184	C₄²	#11761613	1	1	yes	characteristic
185	C₁₆	—	134	4	no	—
186	C₂³	C₂×#11761616	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	8	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#14523964	1	1	yes	characteristic
194	C₂×C₄	#14523964	1	1	yes	characteristic
195	C₂×C₄	#14523964	1	1	yes	characteristic
196	C₂×C₄	#14523964	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
198	C₂×C₄	#14524032	1	1	yes	characteristic
199	C₂×C₄	#14524041	1	1	yes	characteristic
200	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
201	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
202	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
203	C₈	—	536	2	no	—
204	C₈	#14523967	1	2	yes	—
205	C₈	#14523967	1	2	yes	—
206	C₈	Q₈×Dic₆₇	1	2	yes	—
207	C₈	#14524019	1	2	yes	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₂×#14523964	1	1	yes	characteristic
211	C₂²	#16863748	1	1	yes	characteristic
212	C₂²	#16863792	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	8	4	no	—
216	C₄	#16863209	1	2	yes	—
217	C₄	#16863212	1	2	yes	—
218	C₄	#16863303	1	1	yes	characteristic
219	C₄	C₂×#14523967	1	1	yes	characteristic
220	C₄	#16863744	1	1	yes	characteristic
221	C₄	#16863799	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#22561171	1	1	yes	characteristic
225	C₂	C₂×#16863209	1	1	yes	characteristic
226	C₂	C₂×#16863212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#13290
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
21. #14523983	Q₈	semidirect
118. C₆₇	#13290	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523983	2
#22560381	2
#22558317	2
C₂×#500×C₆₇	2
#13290	67