Subgroups — #30765610

Order 17152.

386 conjugacy classes of subgroups.

The subgroup lattice has 242 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	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558179	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	—	2	1	no	—
6	#16863449	—	2	1	no	—
7	#16863172	C₂²	1	2	yes	—
8	C₄×C₆₇⋊C₁₆	—	2	1	no	—
9	#16863149	—	2	1	no	—
10	#16863147	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
13	C₄×Q₁₆×C₆₇	—	2	1	no	—
14	C₄×Q₁₆×C₆₇	—	2	1	no	—
15	C₄×Q₁₆×C₆₇	C₄	1	2	yes	—
16	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#103×C₆₇	—	2	1	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#14523983	—	2	4	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
24	C₄×C₆₇⋊C₈	—	4	1	no	—
25	C₂²×Q₈×C₆₇	—	2	1	no	—
26	C₂×Q₁₆×C₆₇	—	2	1	no	—
27	C₂×Q₁₆×C₆₇	—	2	2	no	—
28	C₂×Q₁₆×C₆₇	C₈	1	4	yes	complemented
29	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₇	—	2	2	no	—
33	C₄×Q₈×C₆₇	—	2	2	no	—
34	C₄×Q₈×C₆₇	—	4	1	no	—
35	C₄×Q₈×C₆₇	—	4	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
38	C₂×C₄²×C₆₇	—	4	1	no	—
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	#107×C₆₇	—	2	1	no	—
41	#107×C₆₇	C₂×C₄	1	2	yes	—
42	#103×C₆₇	—	2	2	no	—
43	#103×C₆₇	—	2	2	no	—
44	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₆₇⋊C₁₆	—	2	4	no	—
50	C₂×Q₈×C₆₇	—	2	4	no	—
51	C₂×Q₈×C₆₇	—	2	2	no	—
52	C₂×Q₈×C₆₇	—	4	1	no	—
53	C₂×Q₈×C₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	—	4	1	no	—
55	C₂²×C₄×C₆₇	—	4	1	no	—
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
59	Q₁₆×C₆₇	—	2	4	no	—
60	Q₁₆×C₆₇	—	2	2	no	—
61	Q₁₆×C₆₇	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₄⋊C₄×C₆₇	—	4	1	no	—
71	C₄⋊C₄×C₆₇	—	4	2	no	—
72	C₄⋊C₄×C₆₇	—	4	2	no	—
73	C₄²×C₆₇	—	4	2	no	—
74	C₄²×C₆₇	—	4	2	no	—
75	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₆₇	D₈	1	1	yes	characteristic
77	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₆₇	#102	1	1	yes	characteristic
80	Q₈×C₆₇	—	2	4	no	—
81	Q₈×C₆₇	—	4	4	no	—
82	Q₈×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	4	1	no	—
85	C₂×C₄×C₆₇	—	4	4	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	8	2	no	—
89	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
90	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#104	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₈×C₆₇	—	2	1	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
100	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
101	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	—	2	1	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	D₄⋊C₈	1	2	yes	—
106	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	4	4	no	—
111	C₄×C₆₇	—	8	2	no	—
112	C₄×C₆₇	D₄⋊C₈	1	2	yes	—
113	C₄×C₆₇	#362	1	2	yes	—
114	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
116	#11600	—	67	1	no	maximal, Sylow
117	C₂×C₆₇	—	2	2	no	—
118	C₂×C₆₇	C₂×D₄⋊C₈	1	1	yes	characteristic
119	C₂×C₆₇	#2145	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
121	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
122	C₄×M₃₂	—	67	1	no	—
123	#1564	—	67	1	no	—
124	#1334	—	67	4	no	—
125	C₆₇	#11600	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₁₆	Dic₆₇	1	1	yes	characteristic
127	C₂×C₄×Q₈	—	2	1	no	—
128	C₂×M₃₂	—	134	1	no	—
129	C₄×Q₁₆	—	2	1	no	—
130	C₄×Q₁₆	—	2	1	no	—
131	C₄×Q₁₆	Dic₆₇	1	2	yes	—
132	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
133	#431	—	134	1	no	—
134	C₂×#107	Dic₆₇	1	1	yes	characteristic
135	C₂×#103	—	2	1	no	—
136	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
137	#346	—	67	2	no	—
138	C₄×C₁₆	—	134	1	no	—
139	#335	—	134	1	no	—
140	#334	—	67	2	no	—
141	C₂²×Q₈	—	2	1	no	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	C₆₇⋊C₈	1	4	yes	complemented
145	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
146	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
147	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	4	1	no	—
151	C₄×Q₈	—	4	1	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	4	1	no	—
154	C₂×C₄²	—	4	1	no	—
155	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
156	M₃₂	—	134	4	no	—
157	C₂×C₁₆	—	134	2	no	—
158	#107	—	2	1	no	—
159	#107	C₂×Dic₆₇	1	2	yes	—
160	C₄⋊C₈	—	134	2	no	—
161	#103	—	2	2	no	—
162	#103	—	2	2	no	—
163	C₂²⋊C₈	—	268	1	no	—
164	C₄×C₈	—	268	1	no	—
165	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	—	4	1	no	—
174	C₂²×C₄	—	4	1	no	—
175	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
176	C₂²×C₄	#11761613	1	1	yes	characteristic
177	C₂²×C₄	#11761616	1	1	yes	characteristic
178	Q₁₆	—	2	4	no	—
179	Q₁₆	—	2	2	no	—
180	Q₁₆	C₂×C₆₇⋊C₈	1	4	yes	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	268	2	no	—
183	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
185	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
186	C₂×C₈	#11761612	1	2	yes	—
187	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
188	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
196	C₄²	#11761613	1	1	yes	characteristic
197	C₄²	#11761616	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
199	C₁₆	—	134	4	no	—
200	C₂³	#14523964	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
211	C₂×C₄	#14523964	1	1	yes	characteristic
212	C₂×C₄	#14523964	1	1	yes	characteristic
213	C₂×C₄	#14523964	1	1	yes	characteristic
214	C₂×C₄	#14523970	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
216	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
217	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	536	2	no	—
221	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
222	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
223	C₈	C₂×#11761612	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	D₄⋊(C₆₇⋊C₈)	1	2	yes	—
227	C₂²	C₂×#14523964	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	8	2	no	—
233	C₄	D₄⋊(C₆₇⋊C₈)	1	2	yes	—
234	C₄	#16863212	1	2	yes	—
235	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
236	C₄	C₂×#14523970	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	C₂×D₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
239	C₂	#22561945	1	1	yes	characteristic
240	C₂	C₂×#16863212	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523964
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#11600
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #22558179	C₂	semidirect
28. C₂×Q₁₆×C₆₇	C₈	semidirect
125. C₆₇	#11600	semidirect
144. C₂×Q₁₆	C₆₇⋊C₈	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
#22558957	2
#22558179	2
C₂×C₄×Q₁₆×C₆₇	2
#11600	67