Subgroups — #31009071

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863814	C₂	1	2	yes	maximal, complemented
2	#22569113	C₂	1	1	yes	maximal, characteristic
3	#22562050	C₂	1	4	yes	maximal, complemented
4	#22562045	C₂	1	4	yes	maximal, complemented
5	#22561838	C₂	1	2	yes	maximal, complemented
6	#22561806	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864196	C₄	1	1	yes	characteristic
9	C₂×#14524049	C₂²	1	2	yes	—
10	#16863814	—	2	2	no	—
11	#16863814	—	2	2	no	—
12	#16863814	C₂²	1	4	yes	complemented
13	#16863814	C₂²	1	4	yes	complemented
14	C₂×#14523983	C₂²	1	1	yes	characteristic
15	C₂×#14523982	—	2	2	no	—
16	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₆₇	—	2	2	no	—
18	#16863519	C₄	1	4	yes	—
19	#16863514	C₂²	1	4	yes	—
20	#16863445	C₄	1	1	yes	characteristic
21	#16863414	C₄	1	2	yes	—
22	#16863181	C₂²	1	8	yes	complemented
23	#16863172	C₂²	1	2	yes	complemented
24	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	complemented
25	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
26	#405×C₆₇	C₂²	1	1	yes	characteristic
27	#364×C₆₇	C₂²	1	4	yes	complemented
28	C₂×#11761628	C₂×C₄	1	2	yes	—
29	#14524076	C₈	1	4	yes	—
30	#14524071	C₈	1	4	yes	—
31	#14524065	C₂×C₄	1	4	yes	—
32	#14524049	—	2	2	no	—
33	#14524049	—	2	2	no	—
34	#14524049	C₂×C₄	1	4	yes	—
35	#14524049	C₂³	1	4	yes	—
36	C₂×#11761612	C₂×C₄	1	2	yes	—
37	C₂×#11761611	—	2	2	no	—
38	C₂×C₈×D₆₇	—	2	2	no	—
39	#14523983	—	2	2	no	—
40	#14523982	—	2	4	no	—
41	#14523982	—	2	4	no	—
42	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
43	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
44	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
45	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
47	C₁₆×D₆₇	—	2	4	no	—
48	C₁₆×D₆₇	—	2	4	no	—
49	#14523979	C₂×C₄	1	4	yes	—
50	#14523971	C₂×C₄	1	8	yes	—
51	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₆₇	—	2	2	no	—
55	M₃₂×C₆₇	C₂³	1	4	yes	—
56	C₂×C₁₆×C₆₇	—	2	2	no	—
57	#97×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₆₇	C₂³	1	2	yes	—
59	#11761628	—	2	2	no	—
60	#11761628	—	2	2	no	—
61	#11761628	C₂×C₈	1	4	yes	—
62	#11761628	C₂²×C₄	1	4	yes	—
63	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
64	C₂×D₂₆₈	C₂×C₈	1	2	yes	—
65	C₂×C₂²⋊D₆₇	—	2	2	no	—
66	C₂×C₄×D₆₇	—	2	2	no	—
67	Dic₂₆₈	C₂×C₈	1	4	yes	—
68	#11761617	C₂×C₈	1	8	yes	—
69	D₅₃₆	C₂×C₈	1	4	yes	—
70	#11761612	—	2	2	no	—
71	#11761612	C₂²×C₄	1	4	yes	—
72	#11761611	—	2	4	no	—
73	#11761611	—	2	4	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₈×D₆₇	—	2	4	no	—
76	C₈×D₆₇	—	2	4	no	—
77	C₆₇⋊C₁₆	—	2	2	no	—
78	C₆₇⋊C₁₆	C₂×D₄	1	1	yes	characteristic
79	C₆₇⋊C₁₆	C₂×D₄	1	2	yes	—
80	C₆₇⋊C₁₆	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₆₇	#41	1	2	yes	—
87	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₆₇	—	2	1	no	—
89	C₂²⋊C₄×C₆₇	—	2	1	no	—
90	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
91	C₁₆×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	#131	1	1	yes	characteristic
93	C₂²×D₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
99	C₂²⋊D₆₇	—	2	4	no	—
100	C₂²⋊D₆₇	—	2	4	no	—
101	C₂×Dic₆₇	—	2	2	no	—
102	D₂₆₈	—	2	2	no	—
103	D₂₆₈	C₂²×C₈	1	4	yes	—
104	C₄×D₆₇	—	2	4	no	—
105	C₄×D₆₇	—	2	4	no	—
106	Dic₁₃₄	—	2	2	no	—
107	Dic₁₃₄	C₂²×C₈	1	4	yes	—
108	C₈×C₆₇	C₄×D₄	1	2	yes	—
109	C₈×C₆₇	C₂²×C₈	1	4	yes	—
110	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
112	C₆₇⋊C₈	—	2	4	no	—
113	C₂²×C₆₇	—	4	1	no	—
114	C₂²×C₆₇	C₈×D₄	1	2	yes	—
115	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
116	C₂×D₆₇	—	2	4	no	—
117	C₂×D₆₇	—	4	4	no	—
118	C₄×C₆₇	—	4	2	no	—
119	C₄×C₆₇	C₈×D₄	1	2	yes	—
120	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
122	Dic₆₇	—	2	4	no	—
123	#33441	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	2	2	no	—
126	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
127	D₆₇	—	4	4	no	—
128	C₂×#504	—	67	2	no	—
129	#2955	—	67	1	no	—
130	#2171	—	67	4	no	—
131	#2170	—	67	4	no	—
132	#2153	D₆₇	1	1	yes	characteristic, complemented
133	#2117	—	67	2	no	—
134	#2108	—	67	1	no	—
135	C₆₇	#33441	1	1	yes	characteristic, complemented, Sylow
136	#575	—	67	1	no	—
137	C₂×#131	—	67	2	no	—
138	#504	—	67	4	no	—
139	#504	—	67	4	no	—
140	#504	—	134	2	no	—
141	#504	—	134	2	no	—
142	C₂×M₃₂	—	67	1	no	—
143	C₂×M₃₂	—	134	2	no	—
144	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
145	C₂²×C₁₆	—	67	1	no	—
146	C₂²×C₁₆	—	134	2	no	—
147	#444	—	67	4	no	—
148	#443	—	67	4	no	—
149	#430	—	67	1	no	—
150	#421	—	67	2	no	—
151	#405	C₂×D₆₇	1	1	yes	characteristic
152	#364	C₂×D₆₇	1	4	yes	complemented
153	#350	—	67	8	no	—
154	#346	—	67	2	no	—
155	C₄×C₁₆	—	67	2	no	—
156	C₂×#41	—	67	2	no	—
157	#137	—	67	4	no	—
158	AGL(1,ℤ/8)	—	67	4	no	—
159	#135	—	67	4	no	—
160	#131	—	67	4	no	—
161	#131	—	67	4	no	—
162	#131	—	134	2	no	—
163	#131	—	134	2	no	—
164	C₂×M₁₆	—	67	2	no	—
165	C₂×M₁₆	—	134	2	no	—
166	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
167	C₂²×C₈	—	134	2	no	—
168	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
169	#117	C₄×D₆₇	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	134	4	no	—
172	M₃₂	—	134	4	no	—
173	M₃₂	—	134	2	no	—
174	M₃₂	C₂²×D₆₇	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	67	2	no	—
177	C₂×C₁₆	—	67	2	no	—
178	C₂×C₁₆	—	67	1	no	—
179	C₂×C₁₆	—	67	1	no	—
180	C₂×C₁₆	—	67	2	no	—
181	C₂×C₁₆	—	134	4	no	—
182	C₂×C₁₆	—	134	4	no	—
183	#108	—	67	4	no	—
184	#104	—	67	8	no	—
185	#97	C₄×D₆₇	1	2	yes	—
186	C₄×C₈	C₂²×D₆₇	1	2	yes	—
187	#41	—	67	4	no	—
188	#41	—	67	4	no	—
189	#41	—	134	2	no	—
190	#41	—	134	2	no	—
191	C₂×Q₈	—	67	2	no	—
192	C₂×D₄	—	67	2	no	—
193	C₂×D₄	—	134	2	no	—
194	C₂²×C₄	—	134	2	no	—
195	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
196	Q₁₆	—	67	4	no	—
197	SD₁₆	—	67	8	no	—
198	D₈	—	67	4	no	—
199	M₁₆	—	67	4	no	—
200	M₁₆	—	134	2	no	—
201	M₁₆	—	134	4	no	—
202	M₁₆	—	134	4	no	—
203	M₁₆	C₈×D₆₇	1	4	yes	—
204	C₂×C₈	—	134	2	no	—
205	C₂×C₈	—	134	4	no	—
206	C₂×C₈	—	134	4	no	—
207	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
209	C₂×C₈	D₄×D₆₇	1	2	yes	—
210	C₂×C₈	#11761629	1	2	yes	—
211	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂×C₄×D₆₇	1	2	yes	—
215	C₁₆	—	2	4	no	—
216	C₁₆	—	67	1	no	—
217	C₁₆	—	67	2	no	—
218	C₁₆	—	67	1	no	—
219	C₁₆	—	134	2	no	—
220	C₂³	—	134	2	no	—
221	C₂³	#14524051	1	1	yes	characteristic
222	Q₈	—	67	4	no	—
223	Q₈	—	134	2	no	—
224	D₄	—	67	4	no	—
225	D₄	—	134	4	no	—
226	D₄	—	134	4	no	—
227	D₄	—	134	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	134	2	no	—
230	C₂×C₄	—	134	4	no	—
231	C₂×C₄	—	134	4	no	—
232	C₂×C₄	#14524005	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
234	C₂×C₄	#14524051	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
236	C₈	—	134	4	no	—
237	C₈	#14524005	1	2	yes	—
238	C₈	C₂×C₈×D₆₇	1	4	yes	—
239	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
240	C₈	C₂×#11761629	1	1	yes	characteristic
241	C₂²	—	4	1	no	—
242	C₂²	—	268	4	no	—
243	C₂²	—	134	4	no	—
244	C₂²	#16863461	1	2	yes	—
245	C₂²	C₂×#14524051	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	134	4	no	—
248	C₄	#16863461	1	2	yes	—
249	C₄	C₂×#14524005	1	1	yes	characteristic
250	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	268	4	no	—
254	C₂	C₂×#16863461	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2153	C₂
socle	C₂×C₆₇	C₂×C₈×D₄
prime radical	#16864196	C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2153	D₆₇
hypocenter	C₆₇	#33441
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

Complements

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

N	H ≅ G/N	type
1. C₂×#16863814	C₂	semidirect
3. #22562050	C₂	semidirect
4. #22562045	C₂	semidirect
5. #22561838	C₂	semidirect
6. #22561806	C₂	semidirect
7. C₆₇×#2153	C₂	semidirect
12. #16863814	C₂²	semidirect
13. #16863814	C₂²	semidirect
22. #16863181	C₂²	semidirect
23. #16863172	C₂²	semidirect
24. C₄×C₆₇⋊C₁₆	C₂²	semidirect
27. #364×C₆₇	C₂²	semidirect
132. #2153	D₆₇	semidirect
135. C₆₇	#33441	semidirect
152. #364	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863814	2
#22569113	2
#22562050	2
#22562045	2
#22561838	2
#22561806	2
C₆₇×#2153	2
#33441	67