Subgroups — #31008932

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524049	C₂	1	1	yes	maximal, characteristic
2	#22562014	C₂	1	4	yes	maximal, complemented
3	#22562010	C₂	1	4	yes	maximal, complemented
4	#22561969	C₂	1	2	yes	maximal, complemented
5	#22561824	C₂	1	2	yes	maximal
6	C₂×#16863172	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14524049	C₄	1	1	yes	characteristic
9	C₂×#14524049	C₂²	1	1	yes	characteristic
10	C₄×#11761628	C₄	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
12	#16863351	C₄	1	1	yes	characteristic
13	#16863351	C₂²	1	1	yes	characteristic
14	#16863350	C₄	1	2	yes	—
15	#16863350	C₂²	1	2	yes	—
16	C₄×#11761612	C₄	1	1	yes	characteristic
17	C₄×#11761611	C₄	1	2	yes	—
18	C₄×C₈×D₆₇	C₂²	1	2	yes	—
19	Dic₆₇⋊C₁₆	C₂²	1	8	yes	—
20	(C₂×D₆₇)⋊C₁₆	C₂²	1	8	yes	—
21	#16863172	C₂²	1	2	yes	—
22	#16863172	C₂²	1	2	yes	—
23	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#346×C₆₇	C₂²	1	2	yes	—
26	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
27	C₂×#11761628	C₂×C₄	1	1	yes	characteristic
28	#14524049	—	2	4	no	—
29	#14524049	—	2	4	no	—
30	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
31	C₂×#11761612	C₂×C₄	1	1	yes	characteristic
32	C₂×#11761611	C₂×C₄	1	2	yes	—
33	C₂×#11761611	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₆₇	C₂³	1	2	yes	—
36	C₄×Dic₁₃₄	C₈	1	2	yes	—
37	C₄×D₂₆₈	C₈	1	2	yes	—
38	C₄×C₂²⋊D₆₇	C₈	1	4	yes	—
39	#14524002	C₂×C₄	1	1	yes	characteristic
40	#14524000	C₂×C₄	1	2	yes	—
41	C₄²×D₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₆₇⋊C₁₆	—	2	4	no	—
43	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
44	#14523954	C₂×C₄	1	2	yes	—
45	#14523953	C₂×C₄	1	2	yes	—
46	#14523953	C₂×C₄	1	2	yes	—
47	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
48	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
49	C₈×Dic₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₆₇	—	2	4	no	—
54	C₂×C₁₆×C₆₇	C₂³	1	4	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	#11761628	—	2	2	no	—
60	#11761628	—	2	2	no	—
61	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
62	C₂×D₂₆₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₂²⋊D₆₇	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₆₇	C₂×C₈	1	4	yes	—
65	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
66	#11761612	—	2	2	no	—
67	#11761612	—	2	2	no	—
68	#11761611	—	2	4	no	—
69	#11761611	—	2	4	no	—
70	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
72	C₈×D₆₇	—	2	4	no	—
73	C₈×D₆₇	—	2	4	no	—
74	C₄⋊Dic₆₇	C₂×C₈	1	2	yes	—
75	Dic₆₇⋊C₄	C₂×C₈	1	4	yes	—
76	C₂²⋊Dic₆₇	C₂×C₈	1	2	yes	—
77	(C₂×D₆₇)⋊C₄	C₂×C₈	1	4	yes	—
78	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
79	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
80	C₆₇⋊C₁₆	—	2	4	no	—
81	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	#41	1	4	yes	—
88	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₄²×C₆₇	C₂×C₈	1	2	yes	—
90	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₁₆×C₆₇	—	2	4	no	—
93	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
94	C₂²×D₆₇	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	#117	1	2	yes	—
97	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
102	C₂²⋊D₆₇	—	2	4	no	—
103	C₂×Dic₆₇	C₂²×C₈	1	2	yes	—
104	C₂×Dic₆₇	C₂²×C₈	1	4	yes	—
105	D₂₆₈	—	2	2	no	—
106	C₄×D₆₇	—	2	2	no	—
107	C₄×D₆₇	—	2	2	no	—
108	C₄×D₆₇	—	4	4	no	—
109	Dic₁₃₄	—	2	2	no	—
110	C₈×C₆₇	#117	1	2	yes	—
111	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
112	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
113	C₈×C₆₇	C₂×#41	1	2	yes	—
114	C₆₇⋊C₈	—	2	2	no	—
115	C₆₇⋊C₈	—	2	2	no	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	#431	1	2	yes	—
118	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
119	C₂×D₆₇	—	2	4	no	—
120	C₂×D₆₇	—	2	2	no	—
121	C₄×C₆₇	—	2	2	no	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	#431	1	2	yes	—
124	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
125	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
126	Dic₆₇	—	2	4	no	—
127	Dic₆₇	—	2	2	no	—
128	#33421	—	67	1	no	maximal, Sylow
129	C₂×C₆₇	—	2	2	no	—
130	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
131	C₂×C₆₇	#3407	1	2	yes	—
132	D₆₇	—	4	2	no	—
133	C₄×#131	—	67	1	no	—
134	#2164	—	67	4	no	—
135	#2163	—	67	4	no	—
136	#2151	—	67	2	no	—
137	#2113	—	67	2	no	—
138	C₄×M₃₂	D₆₇	1	1	yes	characteristic, complemented
139	C₂×#346	—	67	1	no	—
140	C₆₇	#33421	1	1	yes	characteristic, complemented, Sylow
141	C₂×#131	—	67	1	no	—
142	C₂×#131	—	67	1	no	—
143	C₄×#41	—	67	1	no	—
144	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
145	C₂²×C₁₆	—	67	2	no	—
146	#405	—	67	1	no	—
147	#405	—	67	1	no	—
148	#405	—	67	2	no	—
149	#405	—	67	2	no	—
150	C₄×M₁₆	—	67	1	no	—
151	C₄×M₁₆	—	67	2	no	—
152	C₂×C₄×C₈	—	67	2	no	—
153	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
154	C₄⋊C₁₆	—	67	8	no	—
155	C₂²⋊C₁₆	—	67	8	no	—
156	#346	—	67	2	no	—
157	#346	—	67	2	no	—
158	#346	C₂×D₆₇	1	2	yes	—
159	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
160	C₂×#41	—	67	1	no	—
161	#131	—	134	4	no	—
162	#131	—	134	4	no	—
163	C₂×M₁₆	—	67	1	no	—
164	C₂×M₁₆	—	67	1	no	—
165	C₂×M₁₆	—	67	2	no	—
166	C₂×M₁₆	—	67	2	no	—
167	C₂²×C₈	—	67	2	no	—
168	C₂²×C₈	—	67	2	no	—
169	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
171	C₄×Q₈	—	67	2	no	—
172	C₄×D₄	—	67	4	no	—
173	C₄×D₄	—	67	2	no	—
174	#117	—	67	1	no	—
175	#117	—	67	2	no	—
176	C₂×C₄²	—	67	2	no	—
177	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
178	M₃₂	—	2	4	no	—
179	C₂×C₁₆	—	67	4	no	—
180	C₂×C₁₆	—	134	4	no	—
181	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
182	#97	—	67	2	no	—
183	#97	—	67	2	no	—
184	#97	—	67	2	no	—
185	C₄×C₈	—	67	2	no	—
186	C₄×C₈	—	67	2	no	—
187	C₄×C₈	—	67	2	no	—
188	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
189	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
192	#41	—	134	2	no	—
193	#41	—	134	2	no	—
194	C₂×Q₈	—	67	1	no	—
195	C₂×D₄	—	67	2	no	—
196	C₂×D₄	—	67	1	no	—
197	C₂²×C₄	—	67	4	no	—
198	C₂²×C₄	—	67	2	no	—
199	C₂²×C₄	C₈×D₆₇	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
201	M₁₆	—	134	2	no	—
202	M₁₆	—	134	2	no	—
203	M₁₆	—	134	4	no	—
204	M₁₆	—	134	4	no	—
205	C₂×C₈	—	67	2	no	—
206	C₂×C₈	—	67	2	no	—
207	C₂×C₈	—	134	4	no	—
208	C₂×C₈	—	134	4	no	—
209	C₂×C₈	#11761611	1	4	yes	—
210	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
213	C₂×C₈	#11761628	1	4	yes	—
214	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
215	C₄⋊C₄	—	67	4	no	—
216	C₄⋊C₄	—	67	2	no	—
217	C₂²⋊C₄	—	67	2	no	—
218	C₂²⋊C₄	—	67	4	no	—
219	C₄²	—	67	4	no	—
220	C₄²	—	67	2	no	—
221	C₄²	C₈×D₆₇	1	2	yes	—
222	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
223	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
224	C₁₆	—	2	4	no	—
225	C₁₆	—	134	4	no	—
226	C₂³	—	67	2	no	—
227	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
228	Q₈	—	134	2	no	—
229	D₄	—	134	4	no	—
230	D₄	—	134	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	268	4	no	—
233	C₂×C₄	—	67	4	no	—
234	C₂×C₄	—	67	2	no	—
235	C₂×C₄	—	134	2	no	—
236	C₂×C₄	—	134	2	no	—
237	C₂×C₄	#14524000	1	2	yes	—
238	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
239	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
240	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
241	C₂×C₄	C₂×#11761611	1	2	yes	—
242	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
243	C₈	—	134	2	no	—
244	C₈	—	134	2	no	—
245	C₈	#14524000	1	2	yes	—
246	C₈	C₂×#11761611	1	2	yes	—
247	C₈	C₂×#11761611	1	2	yes	—
248	C₈	C₂×#11761628	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	134	2	no	—
251	C₂²	—	134	4	no	—
252	C₂²	#16863447	1	2	yes	—
253	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	134	2	no	—
257	C₄	—	134	4	no	—
258	C₄	#16863447	1	2	yes	—
259	C₄	C₂×#14524000	1	1	yes	characteristic
260	C₄	C₂²×#11761611	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	268	2	no	—
263	C₂	C₂×#16863447	1	1	yes	characteristic
264	C₂	#22573311	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₄×M₃₂×C₆₇	C₂
prime radical	C₂×#11761628	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#33421
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. #22562014	C₂	semidirect
3. #22562010	C₂	semidirect
4. #22561969	C₂	semidirect
6. C₂×#16863172	C₂	semidirect
7. C₄×M₃₂×C₆₇	C₂	semidirect
138. C₄×M₃₂	D₆₇	semidirect
140. C₆₇	#33421	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524049	2
#22562014	2
#22562010	2
#22561969	2
#22561824	2
C₂×#16863172	2
C₄×M₃₂×C₆₇	2
#33421	67