Subgroups — #31008888

Order 17152.

622 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×#16863812	C₂	1	1	yes	maximal, characteristic
2	#22568817	C₂	1	2	yes	maximal
3	#22562003	C₂	1	2	yes	maximal, complemented
4	#22562002	C₂	1	4	yes	maximal, complemented
5	C₂×#16863226	C₂	1	1	yes	maximal, characteristic, complemented
6	#368×D₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#368×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864162	—	2	2	no	—
9	C₂×#14524049	C₂²	1	1	yes	characteristic
10	#16863812	C₄	1	4	yes	—
11	#16863812	C₄	1	4	yes	—
12	C₂×#14523983	C₂²	1	1	yes	characteristic
13	C₂×#14523982	C₂²	1	2	yes	—
14	C₂×C₁₆×D₆₇	C₂²	1	2	yes	—
15	#16863444	—	2	2	no	—
16	#16863443	—	2	4	no	—
17	C₂×#14523979	C₂²	1	2	yes	—
18	#108×D₆₇	C₂²	1	8	yes	—
19	#16863226	C₂²	1	2	yes	—
20	#16863226	C₂²	1	2	yes	—
21	#16863225	C₂²	1	8	yes	—
22	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#108×C₆₇	C₂²	1	2	yes	—
24	#368×C₆₇	C₂²	1	2	yes	—
25	#368×C₆₇	C₂²	1	2	yes	—
26	C₂×#11761628	D₄	1	1	yes	characteristic
27	#14524051	—	4	4	no	—
28	#14524049	—	4	4	no	—
29	#14524049	C₂×C₄	1	2	yes	—
30	#14524049	C₂×C₄	1	2	yes	—
31	#14524049	Q₈	1	2	yes	—
32	#14524049	Q₈	1	2	yes	—
33	C₂×#11761612	—	2	2	no	—
34	C₂×#11761612	D₄	1	1	yes	characteristic
35	C₂×#11761611	D₄	1	2	yes	—
36	M₁₆×D₆₇	—	2	8	no	—
37	C₂×C₈×D₆₇	C₂³	1	2	yes	—
38	#14523983	C₂×C₄	1	2	yes	—
39	#14523983	C₂×C₄	1	2	yes	—
40	#14523982	C₂×C₄	1	8	yes	—
41	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
42	C₁₆×D₆₇	C₂×C₄	1	8	yes	—
43	#14523979	—	2	2	no	—
44	#14523979	C₂³	1	4	yes	—
45	#14523978	—	2	8	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
52	#108×C₆₇	—	2	2	no	—
53	#108×C₆₇	C₂³	1	4	yes	—
54	#11761628	C₄⋊C₄	1	2	yes	—
55	#11761628	C₄⋊C₄	1	2	yes	—
56	#11761628	D₈	1	2	yes	—
57	#11761628	D₈	1	2	yes	—
58	C₂×Dic₁₃₄	Q₁₆	1	1	yes	characteristic
59	C₂×D₂₆₈	Q₁₆	1	1	yes	characteristic
60	C₂×C₂²⋊D₆₇	Q₁₆	1	2	yes	—
61	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
62	#11761612	—	2	4	no	—
63	#11761612	—	4	2	no	—
64	#11761612	C₄⋊C₄	1	2	yes	—
65	#11761612	C₄⋊C₄	1	2	yes	—
66	#11761611	—	4	8	no	—
67	#11761611	C₄⋊C₄	1	4	yes	—
68	#11761611	C₄⋊C₄	1	4	yes	—
69	C₂×C₆₇⋊C₈	—	4	2	no	—
70	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
71	C₈×D₆₇	—	4	8	no	—
72	C₈×D₆₇	C₂²×C₄	1	4	yes	—
73	C₈×D₆₇	C₂×Q₈	1	4	yes	—
74	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	M₁₆×C₆₇	—	2	4	no	—
77	M₁₆×C₆₇	—	4	2	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
86	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
87	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂²×D₆₇	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂²⋊D₆₇	#107	1	4	yes	—
96	C₂²⋊D₆₇	#107	1	4	yes	—
97	C₂×Dic₆₇	C₂×Q₁₆	1	2	yes	—
98	D₂₆₈	#107	1	2	yes	—
99	D₂₆₈	#107	1	2	yes	—
100	C₄×D₆₇	C₂×C₄⋊C₄	1	4	yes	—
101	C₄×D₆₇	C₂×D₈	1	4	yes	—
102	Dic₁₃₄	#107	1	2	yes	—
103	Dic₁₃₄	#107	1	2	yes	—
104	C₈×C₆₇	—	4	4	no	—
105	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₆₇⋊C₈	—	4	4	no	—
110	C₆₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
111	C₂²×C₆₇	—	4	1	no	—
112	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
114	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
115	C₂×D₆₇	—	4	2	no	—
116	C₂×D₆₇	C₂×#107	1	4	yes	—
117	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
118	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
119	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
121	Dic₆₇	C₂×#107	1	4	yes	—
122	#33414	—	67	1	no	maximal, Sylow
123	C₂×C₆₇	—	2	1	no	—
124	C₂×C₆₇	—	2	1	no	—
125	C₂×C₆₇	—	2	1	no	—
126	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
127	D₆₇	—	2	4	no	—
128	C₂×#504	—	67	1	no	—
129	#2916	—	67	2	no	—
130	#2161	—	67	2	no	—
131	#2161	—	67	4	no	—
132	C₂×#368	—	67	1	no	—
133	C₂×#368	—	67	4	no	—
134	C₂×#368	D₆₇	1	1	yes	characteristic, complemented
135	C₆₇	#33414	1	1	yes	characteristic, complemented, Sylow
136	#568	—	134	2	no	—
137	C₂×#131	—	67	1	no	—
138	#504	—	67	4	no	—
139	#504	—	67	4	no	—
140	C₂×M₃₂	—	67	1	no	—
141	C₂×M₃₂	—	67	2	no	—
142	C₂²×C₁₆	—	67	2	no	—
143	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
144	#430	—	134	4	no	—
145	#430	—	134	2	no	—
146	C₂×#108	—	67	2	no	—
147	C₂×#108	—	67	8	no	—
148	C₂×#108	C₂×D₆₇	1	2	yes	—
149	#368	—	67	2	no	—
150	#368	—	67	2	no	—
151	#368	—	67	8	no	—
152	#368	C₂×D₆₇	1	2	yes	—
153	#368	C₂×D₆₇	1	2	yes	—
154	C₂×#41	—	67	1	no	—
155	#131	—	268	4	no	—
156	#131	—	268	4	no	—
157	#131	—	67	2	no	—
158	#131	—	67	2	no	—
159	#131	—	67	2	no	—
160	#131	—	67	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	67	1	no	—
163	C₂×M₁₆	—	67	2	no	—
164	C₂×M₁₆	—	134	8	no	—
165	C₂×M₁₆	—	134	2	no	—
166	C₂²×C₈	—	67	2	no	—
167	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
168	M₃₂	—	67	2	no	—
169	M₃₂	—	67	2	no	—
170	M₃₂	—	67	8	no	—
171	C₂×C₁₆	—	67	2	no	—
172	C₂×C₁₆	—	67	8	no	—
173	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
174	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
175	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
176	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
177	#108	—	2	2	no	—
178	#108	—	67	4	no	—
179	#108	—	134	8	no	—
180	#108	—	134	2	no	—
181	#108	C₂²×D₆₇	1	4	yes	—
182	#41	—	67	2	no	—
183	#41	—	67	2	no	—
184	#41	—	67	2	no	—
185	#41	—	67	2	no	—
186	C₂×Q₈	—	67	1	no	—
187	C₂×D₄	—	67	2	no	—
188	C₂×D₄	—	67	1	no	—
189	C₂²×C₄	—	67	2	no	—
190	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	268	8	no	—
194	M₁₆	—	268	2	no	—
195	M₁₆	—	67	2	no	—
196	M₁₆	—	67	2	no	—
197	M₁₆	—	67	4	no	—
198	M₁₆	—	67	4	no	—
199	M₁₆	—	134	4	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	268	8	no	—
202	C₂×C₈	—	268	2	no	—
203	C₂×C₈	—	67	2	no	—
204	C₂×C₈	—	67	4	no	—
205	C₂×C₈	—	67	4	no	—
206	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
208	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
209	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
210	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
212	C₁₆	—	67	4	no	—
213	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
214	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
215	C₂³	—	67	2	no	—
216	C₂³	Q₁₆×D₆₇	1	1	yes	characteristic
217	Q₈	—	67	2	no	—
218	Q₈	—	67	2	no	—
219	D₄	—	67	4	no	—
220	D₄	—	67	4	no	—
221	D₄	—	67	2	no	—
222	D₄	—	67	2	no	—
223	C₂×C₄	—	67	2	no	—
224	C₂×C₄	—	67	4	no	—
225	C₂×C₄	—	67	4	no	—
226	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
227	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
228	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
229	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
230	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
232	C₈	—	4	4	no	—
233	C₈	—	268	4	no	—
234	C₈	—	67	4	no	—
235	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
236	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
237	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
238	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
239	C₂²	—	4	1	no	—
240	C₂²	—	268	2	no	—
241	C₂²	—	67	4	no	—
242	C₂²	#107×D₆₇	1	1	yes	characteristic
243	C₂²	#107×D₆₇	1	1	yes	characteristic
244	C₂²	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
245	C₄	—	67	4	no	—
246	C₄	#107×D₆₇	1	1	yes	characteristic
247	C₄	#107×D₆₇	1	1	yes	characteristic
248	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
249	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	134	4	no	—
254	C₂	C₂×#107×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#368×C₆₇	C₂
socle	C₂×C₆₇	C₂²×#107
prime radical	C₂×#11761628	D₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#368	D₆₇
hypocenter	C₆₇	#33414
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
3. #22562003	C₂	semidirect
4. #22562002	C₂	semidirect
5. C₂×#16863226	C₂	semidirect
6. #368×D₆₇	C₂	semidirect
7. C₂×#368×C₆₇	C₂	semidirect
134. C₂×#368	D₆₇	semidirect
135. C₆₇	#33414	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863812	2
#22568817	2
#22562003	2
#22562002	2
C₂×#16863226	2
#368×D₆₇	2
C₂×#368×C₆₇	2
#33414	67