Subgroups — #30745213

Order 17152.

798 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11761612	C₂	1	1	yes	maximal, characteristic
3	C₆₇⋊#1273	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₄	1	2	yes	—
6	#16863449	—	2	2	no	—
7	#16863449	—	2	2	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
9	C₂×C₂²⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
10	C₄×#11761612	—	2	2	no	—
11	C₄×#11761612	—	2	2	no	—
12	C₂×#14523954	C₄	1	2	yes	—
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₄	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	—	2	2	no	—
18	C₂×C₄×C₈×C₆₇	—	2	2	no	—
19	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×#11761612	—	2	4	no	—
22	C₂×#11761612	—	2	4	no	—
23	C₂×#11761612	C₈	1	8	yes	complemented
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
28	#14523954	—	2	4	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₄×C₆₇⋊C₈	—	2	2	no	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	—	2	4	no	—
37	C₂²×C₈×C₆₇	—	2	8	no	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	D₄	1	2	yes	—
44	C₂×C₄²×C₆₇	D₄	1	2	yes	—
45	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₆₇	—	2	4	no	—
48	C₄×C₈×C₆₇	—	2	2	no	—
49	C₄×C₈×C₆₇	—	2	2	no	—
50	#11761612	—	2	16	no	—
51	C₂×C₆₇⋊C₈	—	2	4	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₂×C₆₇⋊C₈	—	2	8	no	—
55	C₂×C₆₇⋊C₈	C₂×C₈	1	8	yes	complemented
56	C₂⁴×C₆₇	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
69	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₇	—	2	16	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	M₁₆	1	8	yes	complemented
77	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
87	C₂³×C₆₇	—	2	2	no	—
88	C₂³×C₆₇	—	2	2	no	—
89	C₂³×C₆₇	#95	1	2	yes	—
90	C₂³×C₆₇	C₄×C₈	1	2	yes	—
91	C₂³×C₆₇	#97	1	2	yes	—
92	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	#95	1	2	yes	—
100	C₂×C₄×C₆₇	#95	1	2	yes	—
101	C₂×C₄×C₆₇	#95	1	2	yes	—
102	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₆₇	#97	1	2	yes	—
104	C₂×C₄×C₆₇	C₂²⋊C₈	1	8	yes	—
105	C₂×C₄×C₆₇	C₄⋊C₈	1	8	yes	—
106	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₆₇	#131	1	4	yes	—
116	C₈×C₆₇	—	2	8	no	—
117	C₆₇⋊C₈	—	2	8	no	—
118	C₆₇⋊C₈	—	4	4	no	—
119	C₂²×C₆₇	—	2	4	no	—
120	C₂²×C₆₇	—	2	4	no	—
121	C₂²×C₆₇	—	2	2	no	—
122	C₂²×C₆₇	—	2	2	no	—
123	C₂²×C₆₇	#336	1	4	yes	—
124	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
127	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
128	C₂²×C₆₇	#405	1	2	yes	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	2	no	—
131	C₄×C₆₇	#336	1	4	yes	—
132	C₄×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
133	C₄×C₆₇	#407	1	2	yes	—
134	C₄×C₆₇	#408	1	2	yes	—
135	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
136	C₄×C₆₇	#423	1	2	yes	—
137	C₄×C₆₇	#424	1	2	yes	—
138	#7848	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	2	4	no	—
140	C₂×C₆₇	C₈×M₁₆	1	2	yes	—
141	C₂×C₆₇	#1451	1	2	yes	—
142	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
143	C₂×C₆₇	#1728	1	1	yes	characteristic
144	C₂×C₆₇	#1729	1	1	yes	characteristic
145	C₂×#431	—	67	1	no	—
146	C₂×C₄×M₁₆	—	67	1	no	—
147	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
148	#1273	—	67	4	no	—
149	C₆₇	#7848	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	—	67	2	no	—
151	C₂³×C₈	Dic₆₇	1	2	yes	—
152	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
153	#431	—	134	2	no	—
154	#431	—	134	2	no	—
155	C₂×C₄⋊C₈	—	67	2	no	—
156	C₂×C₂²⋊C₈	—	67	2	no	—
157	C₄×M₁₆	—	134	2	no	—
158	C₄×M₁₆	—	134	2	no	—
159	C₂×#97	—	67	2	no	—
160	C₂×C₄×C₈	—	2	2	no	—
161	C₂×C₄×C₈	—	2	2	no	—
162	C₂×C₄×C₈	—	67	2	no	—
163	C₂×C₄×C₈	—	67	2	no	—
164	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
165	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
166	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
169	C₂×M₁₆	—	67	8	no	—
170	C₂×M₁₆	—	134	4	no	—
171	C₂×M₁₆	—	134	4	no	—
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	—	2	8	no	—
174	C₂²×C₈	—	67	4	no	—
175	C₂²×C₈	—	67	4	no	—
176	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
177	C₂×C₄²	Dic₁₃₄	1	2	yes	—
178	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
179	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
180	C₂×C₄²	D₂₆₈	1	2	yes	—
181	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
184	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
185	C₄⋊C₈	—	134	4	no	—
186	C₂²⋊C₈	—	134	4	no	—
187	#97	—	134	4	no	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	2	2	no	—
191	C₄×C₈	—	134	2	no	—
192	C₄×C₈	—	134	2	no	—
193	C₄×C₈	—	134	2	no	—
194	C₄×C₈	—	134	2	no	—
195	C₂⁴	C₄×Dic₆₇	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
201	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
202	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
203	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
204	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
205	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
206	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
207	C₂²×C₄	C₈×D₆₇	1	4	yes	—
208	C₂²×C₄	#11761611	1	4	yes	—
209	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
212	M₁₆	—	134	16	no	—
213	C₂×C₈	—	2	16	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	67	8	no	—
217	C₂×C₈	—	134	4	no	—
218	C₂×C₈	—	134	8	no	—
219	C₂×C₈	—	134	4	no	—
220	C₂×C₈	—	134	4	no	—
221	C₂×C₈	#11761612	1	8	yes	complemented
222	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
223	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
224	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
225	C₄²	Dic₆₇⋊C₄	1	2	yes	—
226	C₄²	Dic₆₇⋊C₄	1	2	yes	—
227	C₄²	C₄⋊Dic₆₇	1	2	yes	—
228	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
230	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
231	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₆₇⋊#95	1	2	yes	—
235	C₂³	C₈×Dic₆₇	1	2	yes	—
236	C₂³	#14523953	1	2	yes	—
237	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	C₆₇⋊#95	1	2	yes	—
245	C₂×C₄	C₆₇⋊#95	1	2	yes	—
246	C₂×C₄	C₆₇⋊#95	1	2	yes	—
247	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
248	C₂×C₄	#14523953	1	2	yes	—
249	C₂×C₄	(C₂×D₆₇)⋊C₈	1	8	yes	—
250	C₂×C₄	Dic₆₇⋊C₈	1	8	yes	—
251	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
258	C₂×C₄	C₂×#11761611	1	2	yes	—
259	C₂×C₄	C₂×#11761612	1	4	yes	—
260	C₂×C₄	#14524049	1	4	yes	—
261	C₈	—	2	8	no	—
262	C₈	—	268	4	no	—
263	C₈	—	134	8	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#16863150	1	4	yes	—
269	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
270	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
271	C₂²	C₂×#14523953	1	1	yes	characteristic
272	C₂²	C₄×#11761612	1	2	yes	—
273	C₂²	#16863351	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	#16863150	1	4	yes	—
277	C₄	C₂×(C₂×D₆₇)⋊C₈	1	2	yes	—
278	C₄	#16863360	1	2	yes	—
279	C₄	#16863362	1	2	yes	—
280	C₄	C₂×Dic₆₇⋊C₈	1	2	yes	—
281	C₄	#16863420	1	2	yes	—
282	C₄	#16863422	1	2	yes	—
283	C₂	—	2	4	no	—
284	C₂	C₈×#11761612	1	2	yes	—
285	C₂	#22558498	1	2	yes	—
286	C₂	C₂×#16863150	1	1	yes	characteristic
287	C₂	#22559903	1	1	yes	characteristic
288	C₂	#22559905	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄²
prime radical	C₂⁴×C₆₇	C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	#7848
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₆₇⋊#1273	C₂	semidirect
23. C₂×#11761612	C₈	semidirect
55. C₂×C₆₇⋊C₈	C₂×C₈	semidirect
76. C₂×C₈×C₆₇	M₁₆	semidirect
149. C₆₇	#7848	semidirect
221. C₂×C₈	#11761612	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
C₂×C₄×#11761612	2
C₆₇⋊#1273	2
C₂²×C₄×C₈×C₆₇	2
#7848	67