Subgroups — #30824980

Order 17152.

730 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	2	yes	maximal
2	#22558984	C₂	1	8	yes	maximal, complemented
3	#22558968	C₂	1	4	yes	maximal, complemented
4	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	2	no	—
6	#16863420	—	2	2	no	—
7	#16863420	—	2	2	no	—
8	#16863420	C₂²	1	4	yes	—
9	#16863420	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
12	#16863149	C₂²	1	16	yes	complemented
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#431×C₆₇	C₄	1	4	yes	—
15	#431×C₆₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
17	C₂×C₂²⋊C₈×C₆₇	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#11761612	—	2	4	no	—
20	C₂×#11761612	—	2	4	no	—
21	C₂×#11761612	—	4	8	no	—
22	C₂²×C₆₇⋊C₈	—	2	4	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
26	C₄⋊(C₆₇⋊C₈)	C₂³	1	8	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
40	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₆₇	C₂³	1	4	yes	—
43	#11761612	—	4	8	no	—
44	#11761612	—	4	8	no	—
45	C₂×C₆₇⋊C₈	—	2	8	no	—
46	C₂×C₆₇⋊C₈	—	4	8	no	—
47	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	—	4	2	no	—
49	C₂²×C₄×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₆₇	D₈	1	4	yes	—
57	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₆₇	—	4	4	no	—
65	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
66	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₂×D₄	1	2	yes	—
72	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
74	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
76	C₂³×C₆₇	—	4	1	no	—
77	C₂³×C₆₇	—	4	1	no	—
78	C₂³×C₆₇	#107	1	4	yes	—
79	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	4	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	4	2	no	—
86	C₂×C₄×C₆₇	—	4	1	no	—
87	C₂×C₄×C₆₇	—	4	1	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	#107	1	4	yes	—
90	C₂×C₄×C₆₇	#107	1	8	yes	—
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
98	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
100	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
101	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₆₇	—	2	8	no	—
105	C₆₇⋊C₈	—	4	8	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	2	no	—
108	C₂²×C₆₇	—	2	2	no	—
109	C₂²×C₆₇	—	2	1	no	—
110	C₂²×C₆₇	—	4	4	no	—
111	C₂²×C₆₇	C₂×#107	1	4	yes	—
112	C₂²×C₆₇	C₂×#107	1	2	yes	—
113	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	2	no	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	C₂×#107	1	4	yes	—
119	C₄×C₆₇	C₂×#107	1	2	yes	—
120	C₄×C₆₇	#430	1	2	yes	—
121	C₄×C₆₇	#568	1	2	yes	—
122	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
123	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
124	#17814	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	—	2	2	no	—
126	C₂×C₆₇	—	2	4	no	—
127	C₂×C₆₇	#2905	1	1	yes	characteristic
128	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
129	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
130	C₂×#431	D₆₇	1	1	yes	characteristic
131	C₂×#423	—	67	2	no	—
132	#1571	—	67	8	no	—
133	#1567	—	67	4	no	—
134	C₆₇	#17814	1	1	yes	characteristic, complemented, Sylow
135	C₂²×M₁₆	—	134	2	no	—
136	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
137	#431	Dic₆₇	1	4	yes	—
138	#431	C₂×D₆₇	1	4	yes	—
139	#423	—	67	4	no	—
140	#423	—	67	4	no	—
141	#423	—	134	2	no	—
142	#423	—	134	2	no	—
143	C₂×C₄⋊C₈	—	67	4	no	—
144	C₂×C₄⋊C₈	—	134	2	no	—
145	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
146	C₂×C₂²⋊C₈	Dic₆₇	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
148	#335	—	67	16	no	—
149	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
150	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
151	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
152	C₂×M₁₆	—	268	8	no	—
153	C₂×M₁₆	—	134	4	no	—
154	C₂×M₁₆	—	134	4	no	—
155	C₂²×C₈	—	134	4	no	—
156	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
157	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
158	C₂×C₄²	Dic₁₃₄	1	2	yes	—
159	C₂×C₄²	D₂₆₈	1	2	yes	—
160	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
161	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
163	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
165	C₄⋊C₈	—	67	8	no	—
166	C₄⋊C₈	—	134	4	no	—
167	C₄⋊C₈	—	134	4	no	—
168	C₄⋊C₈	—	134	4	no	—
169	C₄⋊C₈	C₂×Dic₆₇	1	8	yes	—
170	C₂²⋊C₈	C₂×Dic₆₇	1	8	yes	—
171	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
172	C₄×C₈	C₂²×D₆₇	1	4	yes	—
173	C₂⁴	C₄⋊Dic₆₇	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
177	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
178	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
182	C₂²×C₄	D₅₃₆	1	4	yes	—
183	C₂²×C₄	Dic₂₆₈	1	4	yes	—
184	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
185	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
187	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
189	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
190	M₁₆	—	268	8	no	—
191	M₁₆	—	268	8	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	—	268	8	no	—
194	C₂×C₈	—	134	8	no	—
195	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
196	C₄²	C₄⋊Dic₆₇	1	2	yes	—
197	C₄²	C₄⋊Dic₆₇	1	2	yes	—
198	C₄²	C₄⋊Dic₆₇	1	2	yes	—
199	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
200	C₄²	C₂²×Dic₆₇	1	2	yes	—
201	C₄²	C₂×D₂₆₈	1	2	yes	—
202	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
203	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
204	C₄²	C₂×Dic₁₃₄	1	2	yes	—
205	C₄²	C₂³×D₆₇	1	1	yes	characteristic
206	C₂³	—	4	1	no	—
207	C₂³	—	4	1	no	—
208	C₂³	#14523977	1	4	yes	—
209	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
210	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	#14523977	1	4	yes	—
220	C₂×C₄	#14523977	1	8	yes	—
221	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
227	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
228	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
229	C₂×C₄	C₂×Dic₂₆₈	1	4	yes	—
230	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
231	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
233	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
234	C₈	—	2	8	no	—
235	C₈	—	268	8	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	4	4	no	—
241	C₂²	C₂×#14523977	1	4	yes	—
242	C₂²	C₂×#14523977	1	2	yes	—
243	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	1	no	—
248	C₄	C₂×#14523977	1	4	yes	—
249	C₄	C₂×#14523977	1	2	yes	—
250	C₄	#16863445	1	2	yes	—
251	C₄	#16864163	1	2	yes	—
252	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
253	C₄	C₂²×Dic₂₆₈	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	2	4	no	—
256	C₂	#22568773	1	1	yes	characteristic
257	C₂	C₂²×#14523977	1	1	yes	characteristic
258	C₂	C₂×#16863445	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	#17814
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #22558984	C₂	semidirect
3. #22558968	C₂	semidirect
12. #16863149	C₂²	semidirect
134. C₆₇	#17814	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863420	2
#22558984	2
#22558968	2
C₂×#431×C₆₇	2
#17814	67