Subgroups — #31009338

Order 17152.

498 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	#22569143	C₂	1	1	yes	maximal, characteristic
2	#22562117	C₂	1	2	yes	maximal, complemented
3	#22562115	C₂	1	4	yes	maximal, complemented
4	C₁₆×Dic₁₃₄	C₂	1	2	yes	maximal, complemented
5	#22561969	C₂	1	2	yes	maximal
6	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
7	#22561808	C₂	1	2	yes	maximal
8	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863953	C₄	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#16863526	C₄	1	2	yes	—
12	#16863526	C₄	1	2	yes	—
13	C₈×Dic₁₃₄	C₂²	1	2	yes	—
14	C₈×Dic₁₃₄	C₂²	1	2	yes	—
15	#16863422	C₄	1	2	yes	—
16	#16863420	C₄	1	1	yes	characteristic
17	#16863351	C₂²	1	2	yes	—
18	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
20	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
21	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
22	Dic₆₇⋊C₁₆	C₂²	1	4	yes	—
23	#16863171	C₂²	1	4	yes	—
24	C₁₆×Dic₆₇	C₂²	1	4	yes	—
25	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	#346×C₆₇	C₂²	1	2	yes	—
29	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
30	C₂×#11761612	C₂×C₄	1	2	yes	—
31	#14524043	C₈	1	1	yes	characteristic
32	#14524043	C₈	1	1	yes	characteristic
33	#14524035	C₈	1	2	yes	—
34	#14524022	C₈	1	2	yes	—
35	#14524022	C₈	1	2	yes	—
36	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
38	#14524002	C₂×C₄	1	2	yes	—
39	C₂×C₆₇⋊C₁₆	—	2	4	no	—
40	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
41	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
42	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
43	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
44	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
45	Dic₆₇⋊C₈	C₂×C₄	1	4	yes	—
46	#14523953	C₂×C₄	1	4	yes	—
47	C₈×Dic₆₇	C₂³	1	4	yes	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₆₇	—	2	2	no	—
52	M₃₂×C₆₇	Q₈	1	4	yes	complemented
53	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
54	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
55	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
60	C₂×Dic₁₃₄	C₂×C₈	1	2	yes	—
61	#11761612	—	2	4	no	—
62	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
63	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
64	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
65	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
66	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
67	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	—
68	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	—
69	C₂²⋊Dic₆₇	C₂×C₈	1	4	yes	—
70	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
71	C₆₇⋊C₁₆	—	2	2	no	—
72	C₆₇⋊C₁₆	—	2	2	no	—
73	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₆₇	#41	1	2	yes	—
82	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₁₆×C₆₇	—	2	2	no	—
88	C₁₆×C₆₇	C₂×Q₈	1	4	yes	—
89	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	4	1	no	—
92	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#131	1	2	yes	—
99	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×Dic₆₇	C₂²×C₈	1	4	yes	—
101	C₂×Dic₆₇	C₂²×C₈	1	4	yes	—
102	Dic₁₃₄	—	2	4	no	—
103	Dic₁₃₄	—	4	2	no	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	C₄×Q₈	1	2	yes	—
107	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
108	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₆₇⋊C₈	—	2	4	no	—
110	C₂²×C₆₇	—	2	1	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
113	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	1	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	C₈×Q₈	1	2	yes	—
118	C₄×C₆₇	#504	1	2	yes	—
119	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
121	Dic₆₇	—	2	4	no	—
122	Dic₆₇	—	2	4	no	—
123	#33476	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	—	2	2	no	—
125	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
126	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
127	C₂×C₆₇	#3407	1	1	yes	characteristic
128	#2960	—	67	1	no	—
129	#2183	—	67	4	no	—
130	#2183	—	67	2	no	—
131	Q₈×C₁₆	—	67	2	no	—
132	#2151	—	67	2	no	—
133	C₂×C₄⋊C₁₆	—	67	1	no	—
134	#2108	—	67	2	no	—
135	C₄×M₃₂	D₆₇	1	1	yes	characteristic
136	C₆₇	#33476	1	1	yes	characteristic, complemented, Sylow
137	#533	—	67	1	no	—
138	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
139	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
140	C₂²×C₁₆	—	67	2	no	—
141	#446	—	67	2	no	—
142	#446	—	67	2	no	—
143	C₈×Q₈	—	67	2	no	—
144	C₈×Q₈	—	67	2	no	—
145	#424	—	67	2	no	—
146	#423	—	67	1	no	—
147	#405	—	67	2	no	—
148	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
149	C₄⋊C₁₆	—	67	4	no	—
150	C₄⋊C₁₆	—	67	4	no	—
151	C₄⋊C₁₆	—	67	1	no	—
152	C₄⋊C₁₆	—	67	1	no	—
153	C₄⋊C₁₆	—	67	2	no	—
154	#346	—	67	4	no	—
155	#346	C₂×D₆₇	1	2	yes	—
156	C₄×C₁₆	—	67	4	no	—
157	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
158	C₂×M₁₆	—	67	2	no	—
159	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	C₄⋊Q₈	—	67	1	no	—
162	C₄⋊Q₈	—	67	1	no	—
163	#125	—	67	2	no	—
164	C₂²⋊Q₈	—	67	2	no	—
165	C₂²⋊Q₈	—	67	2	no	—
166	C₄×Q₈	—	67	2	no	—
167	C₄×Q₈	—	67	2	no	—
168	#117	—	67	2	no	—
169	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	Dic₁₃₄	1	4	yes	complemented
172	C₂×C₁₆	—	67	2	no	—
173	C₂×C₁₆	—	67	2	no	—
174	C₂×C₁₆	—	134	4	no	—
175	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
176	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
177	C₄⋊C₈	—	67	4	no	—
178	C₄⋊C₈	—	67	4	no	—
179	C₄⋊C₈	—	67	2	no	—
180	C₄⋊C₈	—	67	2	no	—
181	#97	—	67	4	no	—
182	C₄×C₈	—	67	4	no	—
183	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
184	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
186	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
187	C₂×Q₈	—	67	2	no	—
188	C₂×Q₈	—	67	2	no	—
189	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
192	M₁₆	—	134	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	67	4	no	—
195	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
197	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
198	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
199	C₂×C₈	#11761628	1	2	yes	—
200	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
201	C₄⋊C₄	—	67	2	no	—
202	C₄⋊C₄	—	67	2	no	—
203	C₄⋊C₄	—	67	4	no	—
204	C₄⋊C₄	—	67	4	no	—
205	C₄⋊C₄	—	67	2	no	—
206	C₄⋊C₄	—	67	2	no	—
207	C₂²⋊C₄	—	67	4	no	—
208	C₄²	—	67	4	no	—
209	C₄²	C₈×D₆₇	1	1	yes	characteristic
210	C₄²	C₈×D₆₇	1	1	yes	characteristic
211	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
212	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
213	C₁₆	—	2	2	no	—
214	C₁₆	—	134	2	no	—
215	C₁₆	—	134	2	no	—
216	C₁₆	C₂×Dic₁₃₄	1	4	yes	—
217	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
218	Q₈	—	268	2	no	—
219	Q₈	—	134	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	67	4	no	—
223	C₂×C₄	—	67	4	no	—
224	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
225	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
228	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
229	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
230	C₂×C₄	#14524049	1	2	yes	—
231	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	134	4	no	—
235	C₈	C₄×Dic₁₃₄	1	2	yes	—
236	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
237	C₈	C₂×#11761628	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₈×Dic₁₃₄	1	2	yes	—
241	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	134	4	no	—
246	C₄	—	134	4	no	—
247	C₄	C₈×Dic₁₃₄	1	2	yes	—
248	C₄	#16863814	1	2	yes	—
249	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
250	C₄	C₂×#14524049	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×C₈×Dic₁₃₄	1	1	yes	characteristic
253	C₂	C₂×#16863814	1	1	yes	characteristic
254	C₂	#22573311	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₄×M₃₂×C₆₇	C₂
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#33476
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #22562117	C₂	semidirect
3. #22562115	C₂	semidirect
4. C₁₆×Dic₁₃₄	C₂	semidirect
52. M₃₂×C₆₇	Q₈	semidirect
136. C₆₇	#33476	semidirect
171. M₃₂	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22569143	2
#22562117	2
#22562115	2
C₁₆×Dic₁₃₄	2
#22561969	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22561808	2
C₄×M₃₂×C₆₇	2
#33476	67