Subgroups — #30770925

Order 17152.

474 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂²⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
5	#22558286	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523983	—	2	2	no	—
9	C₂×#14523983	—	2	2	no	—
10	C₂×#14523983	C₄	1	4	yes	complemented
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊Dic₆₇	C₄	1	1	yes	characteristic
17	#16863171	—	2	2	no	—
18	C₁₆×Dic₆₇	—	2	2	no	—
19	#16863150	C₄	1	1	yes	characteristic
20	#16863150	C₄	1	1	yes	characteristic
21	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²⋊C₁₆×C₆₇	C₄	1	4	yes	complemented
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₆₇	C₈	1	2	yes	—
28	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
30	#14523983	—	2	8	no	—
31	C₂×C₆₇⋊C₁₆	—	2	1	no	—
32	C₂×C₆₇⋊C₁₆	—	2	1	no	—
33	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
36	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
37	C₂²⋊(C₆₇⋊C₈)	C₈	1	4	yes	—
38	C₈×Dic₆₇	—	2	1	no	—
39	C₈×Dic₆₇	—	2	1	no	—
40	C₈×Dic₆₇	—	2	1	no	—
41	C₈×Dic₆₇	—	2	1	no	—
42	C₆₇⋊#95	C₈	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₂²×C₈×C₆₇	—	2	1	no	—
46	C₂²×C₈×C₆₇	—	2	1	no	—
47	C₂²×C₈×C₆₇	—	2	1	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₆₇	—	2	1	no	—
53	C₂×C₁₆×C₆₇	—	2	1	no	—
54	C₂×C₁₆×C₆₇	—	2	1	no	—
55	C₂×C₁₆×C₆₇	—	2	1	no	—
56	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
58	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
59	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
60	C₂×C₆₇⋊C₈	—	2	1	no	—
61	C₂×C₆₇⋊C₈	—	2	1	no	—
62	C₂×C₆₇⋊C₈	—	2	1	no	—
63	C₂×C₆₇⋊C₈	—	2	1	no	—
64	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
65	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
66	C₂²⋊Dic₆₇	—	2	4	no	—
67	C₄×Dic₆₇	—	2	1	no	—
68	C₄×Dic₆₇	—	2	1	no	—
69	C₄×Dic₆₇	—	2	1	no	—
70	C₄×Dic₆₇	—	2	1	no	—
71	C₆₇⋊C₁₆	—	2	2	no	—
72	C₆₇⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
73	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	—	2	1	no	—
83	C₂×C₈×C₆₇	—	2	1	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	C₄²	1	2	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
94	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
95	C₁₆×C₆₇	—	2	2	no	—
96	C₁₆×C₆₇	—	2	2	no	—
97	C₂³×C₆₇	—	2	1	no	—
98	C₂³×C₆₇	—	2	1	no	—
99	C₂³×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	C₄×C₈	1	2	yes	—
102	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
116	C₂×Dic₆₇	—	2	2	no	—
117	C₂×Dic₆₇	—	2	2	no	—
118	C₂×Dic₆₇	—	2	2	no	—
119	C₂×Dic₆₇	—	2	2	no	—
120	C₂×Dic₆₇	—	2	2	no	—
121	C₂×Dic₆₇	M₃₂	1	4	yes	complemented
122	C₈×C₆₇	—	2	2	no	—
123	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₈×C₆₇	#117	1	1	yes	characteristic
125	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
126	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
127	C₆₇⋊C₈	—	2	2	no	—
128	C₆₇⋊C₈	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	2	no	—
132	C₂²×C₆₇	—	4	2	no	—
133	C₂²×C₆₇	—	4	2	no	—
134	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
135	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
136	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
138	C₂²×C₆₇	#504	1	2	yes	—
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₆₇	#405	1	1	yes	characteristic
142	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
143	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
144	Dic₆₇	—	2	4	no	—
145	Dic₆₇	—	4	2	no	—
146	#12612	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	4	2	no	—
148	C₂×C₆₇	C₈×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₆₇	C₄×M₃₂	1	1	yes	characteristic
150	C₂×C₆₇	#2108	1	1	yes	characteristic
151	C₂×C₆₇	#2168	1	2	yes	—
152	C₂×C₆₇	#2169	1	2	yes	—
153	C₂²×M₃₂	—	67	1	no	—
154	C₂×C₂²⋊C₁₆	D₆₇	1	1	yes	characteristic
155	C₂×#346	—	67	1	no	—
156	C₂×C₄×C₁₆	—	67	1	no	—
157	C₈×C₂²⋊C₄	—	67	1	no	—
158	#1374	—	67	1	no	—
159	#1374	—	67	1	no	—
160	C₆₇	#12612	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
162	C₂×M₃₂	—	67	4	no	—
163	C₂×M₃₂	—	134	2	no	—
164	C₂×M₃₂	—	134	2	no	—
165	C₂²×C₁₆	—	67	1	no	—
166	C₂²×C₁₆	—	67	1	no	—
167	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
168	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₈	—	67	1	no	—
170	C₂×C₄×C₈	—	67	1	no	—
171	C₂×C₄×C₈	—	67	1	no	—
172	C₄×C₂²⋊C₄	—	67	1	no	—
173	C₂²⋊C₁₆	Dic₆₇	1	4	yes	complemented
174	#346	—	134	2	no	—
175	C₄×C₁₆	—	134	2	no	—
176	#336	—	67	1	no	—
177	#336	—	67	1	no	—
178	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	67	1	no	—
184	C₂²×C₈	—	67	1	no	—
185	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₆₇	1	2	yes	—
187	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
188	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	—	67	2	no	—
190	C₂×C₄²	—	67	1	no	—
191	C₂×C₄²	—	67	1	no	—
192	M₃₂	—	134	8	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	67	2	no	—
198	C₂×C₁₆	—	67	2	no	—
199	C₂×C₁₆	—	67	2	no	—
200	C₂×C₁₆	—	67	2	no	—
201	C₂×C₁₆	—	134	1	no	—
202	C₂×C₁₆	—	134	1	no	—
203	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
204	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
205	C₂²⋊C₈	—	67	4	no	—
206	C₄×C₈	—	134	1	no	—
207	C₄×C₈	—	134	1	no	—
208	C₄×C₈	—	134	1	no	—
209	C₄×C₈	—	134	1	no	—
210	#95	—	67	2	no	—
211	C₂⁴	C₈×D₆₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	67	2	no	—
217	C₂²×C₄	—	67	2	no	—
218	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
219	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₈×D₆₇	1	2	yes	—
221	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	67	2	no	—
229	C₂×C₈	—	67	2	no	—
230	C₂×C₈	—	134	1	no	—
231	C₂×C₈	—	134	1	no	—
232	C₂×C₈	—	134	1	no	—
233	C₂×C₈	—	134	1	no	—
234	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
237	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
238	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
239	C₂×C₈	#11761627	1	1	yes	characteristic
240	C₂×C₈	#11761627	1	1	yes	characteristic
241	C₂²⋊C₄	—	134	4	no	—
242	C₄²	—	134	1	no	—
243	C₄²	—	134	1	no	—
244	C₄²	—	134	1	no	—
245	C₄²	—	134	1	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	—	2	2	no	—
248	C₁₆	—	67	4	no	—
249	C₁₆	—	134	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	C₈×Dic₆₇	1	2	yes	—
255	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	67	4	no	—
263	C₂×C₄	—	134	2	no	—
264	C₂×C₄	—	134	2	no	—
265	C₂×C₄	—	134	2	no	—
266	C₂×C₄	—	134	2	no	—
267	C₂×C₄	—	134	2	no	—
268	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
269	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
270	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
273	C₂×C₄	#14524051	1	1	yes	characteristic
274	C₂×C₄	#14524051	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	134	2	no	—
277	C₈	—	134	2	no	—
278	C₈	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
279	C₈	#14523999	1	1	yes	characteristic
280	C₈	D₄×Dic₆₇	1	1	yes	characteristic
281	C₈	D₄×Dic₆₇	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
288	C₂²	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
289	C₂²	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
290	C₂²	M₃₂×D₆₇	1	2	yes	—
291	C₂²	#16863812	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	268	2	no	—
294	C₄	—	134	4	no	—
295	C₄	C₂²⋊C₄×Dic₆₇	1	1	yes	characteristic
296	C₄	#16863354	1	1	yes	characteristic
297	C₄	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
298	C₄	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
299	C₂	—	4	2	no	—
300	C₂	#22559983	1	1	yes	characteristic
301	C₂	M₃₂×Dic₆₇	1	1	yes	characteristic
302	C₂	#22561805	1	1	yes	characteristic
303	C₂	#22562037	1	2	yes	—
304	C₂	#22562038	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₂×C₂²⋊C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₆₇
hypocenter	C₆₇	#12612
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16863171	C₂	semidirect
3. C₂×C₁₆×Dic₆₇	C₂	semidirect
5. #22558286	C₂	semidirect
6. #22558286	C₂	semidirect
10. C₂×#14523983	C₄	semidirect
24. C₂²⋊C₁₆×C₆₇	C₄	semidirect
72. C₆₇⋊C₁₆	C₂²⋊C₄	semidirect
121. C₂×Dic₆₇	M₃₂	semidirect
160. C₆₇	#12612	semidirect
173. C₂²⋊C₁₆	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂×#16863171	2
C₂×C₁₆×Dic₆₇	2
C₈×C₂²⋊Dic₆₇	2
#22558286	2
C₂×C₂²⋊C₁₆×C₆₇	2
#12612	67