Subgroups — #30770865

Order 17152.

514 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863350	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₆₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×(C₂×D₆₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558287	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761611	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
10	C₂×#14524000	C₄	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16863350	—	2	2	no	—
14	#16863350	—	2	2	no	—
15	C₂×#14523954	C₄	1	1	yes	characteristic
16	C₂×#14523953	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
18	(C₂×D₆₇)⋊C₁₆	C₄	1	4	yes	complemented
19	(C₂×D₆₇)⋊C₁₆	C₄	1	4	yes	complemented
20	#16863171	—	2	2	no	—
21	C₁₆×Dic₆₇	—	2	2	no	—
22	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×#11761611	—	2	2	no	—
27	C₂×#11761611	—	2	2	no	—
28	C₂×#11761611	C₈	1	4	yes	—
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₆₇	—	2	1	no	—
32	C₂×C₈×D₆₇	—	2	1	no	—
33	C₂×C₈×D₆₇	—	2	1	no	—
34	C₂×C₈×D₆₇	—	2	1	no	—
35	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
37	#14524000	—	2	2	no	—
38	#14524000	—	2	2	no	—
39	C₂×Dic₆₇⋊C₄	C₈	1	2	yes	—
40	C₂×(C₂×D₆₇)⋊C₄	C₈	1	2	yes	—
41	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₆₇⋊C₁₆	—	2	1	no	—
43	C₂×C₆₇⋊C₁₆	—	2	1	no	—
44	C₂×C₆₇⋊C₁₆	—	2	1	no	—
45	C₂×C₆₇⋊C₁₆	—	2	1	no	—
46	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
48	#14523954	—	2	2	no	—
49	#14523953	—	2	2	no	—
50	C₈×Dic₆₇	D₄	1	2	yes	—
51	C₈×Dic₆₇	D₄	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₆₇	—	2	1	no	—
56	C₂×C₁₆×C₆₇	—	2	1	no	—
57	C₂×C₁₆×C₆₇	—	2	1	no	—
58	C₂×C₁₆×C₆₇	—	2	1	no	—
59	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	—	2	1	no	—
63	C₂³×D₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₆₇	—	2	1	no	—
67	C₂×C₄×D₆₇	—	2	1	no	—
68	C₂×C₄×D₆₇	—	2	1	no	—
69	C₂×C₄×D₆₇	—	2	1	no	—
70	C₂×C₄×D₆₇	C₄²	1	2	yes	—
71	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
72	#11761611	—	2	8	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	—	2	1	no	—
75	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
76	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
77	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
78	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
79	C₈×D₆₇	—	2	2	no	—
80	C₈×D₆₇	—	2	2	no	—
81	C₈×D₆₇	—	2	2	no	—
82	C₈×D₆₇	—	2	2	no	—
83	Dic₆₇⋊C₄	—	2	4	no	—
84	(C₂×D₆₇)⋊C₄	—	2	4	no	—
85	C₄×Dic₆₇	C₂²⋊C₄	1	2	yes	—
86	C₄×Dic₆₇	M₁₆	1	2	yes	—
87	C₆₇⋊C₁₆	—	2	2	no	—
88	C₆₇⋊C₁₆	—	2	2	no	—
89	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	—	2	1	no	—
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
94	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
99	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	—	2	1	no	—
102	C₁₆×C₆₇	—	2	2	no	—
103	C₁₆×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
105	C₂²×D₆₇	—	2	1	no	—
106	C₂²×D₆₇	—	2	1	no	—
107	C₂²×D₆₇	—	2	2	no	—
108	C₂²×D₆₇	—	2	2	no	—
109	C₂²×D₆₇	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₆₇	—	2	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
119	C₂×Dic₆₇	—	2	1	no	—
120	C₂×Dic₆₇	—	2	1	no	—
121	C₂×Dic₆₇	—	2	2	no	—
122	C₂×Dic₆₇	—	2	2	no	—
123	C₂×Dic₆₇	C₄×C₈	1	2	yes	—
124	C₂×Dic₆₇	C₂²⋊C₈	1	4	yes	—
125	C₄×D₆₇	—	2	2	no	—
126	C₄×D₆₇	—	2	2	no	—
127	C₄×D₆₇	—	2	2	no	—
128	C₄×D₆₇	—	2	2	no	—
129	C₈×C₆₇	—	2	2	no	—
130	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
131	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
132	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₆₇⋊C₈	—	2	2	no	—
135	C₆₇⋊C₈	C₂²⋊C₈	1	4	yes	—
136	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₆₇	#431	1	1	yes	characteristic
139	C₂²×C₆₇	#504	1	2	yes	—
140	C₂²×C₆₇	#504	1	2	yes	—
141	C₂×D₆₇	—	2	2	no	—
142	C₂×D₆₇	—	2	2	no	—
143	C₂×D₆₇	—	2	2	no	—
144	C₂×D₆₇	—	4	2	no	—
145	C₂×D₆₇	—	4	2	no	—
146	C₄×C₆₇	—	4	2	no	—
147	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
149	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₆₇	#431	1	1	yes	characteristic
151	Dic₆₇	—	2	4	no	—
152	Dic₆₇	—	2	2	no	—
153	#12600	—	67	1	no	maximal, Sylow
154	C₂×C₆₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
155	C₂×C₆₇	#2108	1	1	yes	characteristic
156	C₂×C₆₇	#2108	1	1	yes	characteristic
157	C₂×C₆₇	#2113	1	2	yes	—
158	C₂×C₆₇	#2164	1	2	yes	—
159	D₆₇	—	4	2	no	—
160	C₂×#405	—	67	1	no	—
161	C₂×C₂²⋊C₁₆	—	67	1	no	—
162	C₂×C₂²⋊C₁₆	—	67	1	no	—
163	C₂×#346	—	67	1	no	—
164	C₂×C₄×C₁₆	—	67	1	no	—
165	#1374	—	67	1	no	—
166	#1374	D₆₇	1	1	yes	characteristic, complemented
167	C₆₇	#12600	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	67	1	no	—
169	C₂³×C₈	—	67	1	no	—
170	C₂×#117	—	67	1	no	—
171	C₂²×C₁₆	—	67	1	no	—
172	C₂²×C₁₆	—	67	1	no	—
173	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
174	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
175	#405	—	134	2	no	—
176	#405	—	134	2	no	—
177	C₂×#97	—	67	1	no	—
178	C₂×#97	—	67	1	no	—
179	C₂×C₄×C₈	—	67	1	no	—
180	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
181	C₂²⋊C₁₆	—	67	4	no	—
182	C₂²⋊C₁₆	—	67	4	no	—
183	#346	—	134	2	no	—
184	C₄×C₁₆	—	134	2	no	—
185	C₂³×C₄	—	67	1	no	—
186	C₂×M₁₆	—	67	4	no	—
187	C₂×M₁₆	—	134	2	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂²×C₈	—	67	1	no	—
190	C₂²×C₈	—	67	1	no	—
191	C₂²×C₈	—	67	2	no	—
192	C₂²×C₈	—	67	2	no	—
193	C₂²×C₈	—	134	1	no	—
194	C₂²×C₈	—	134	1	no	—
195	C₂²×C₈	—	134	1	no	—
196	C₂²×C₈	—	134	1	no	—
197	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
199	#117	—	134	2	no	—
200	#117	—	134	2	no	—
201	C₂×C₄⋊C₄	—	67	2	no	—
202	C₂×C₂²⋊C₄	—	67	2	no	—
203	C₂×C₄²	—	67	1	no	—
204	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	67	2	no	—
210	C₂×C₁₆	—	67	2	no	—
211	C₂×C₁₆	—	134	1	no	—
212	C₂×C₁₆	—	134	1	no	—
213	C₂×C₁₆	—	134	1	no	—
214	C₂×C₁₆	—	134	1	no	—
215	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
216	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
217	#97	—	134	2	no	—
218	#97	—	134	2	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	67	2	no	—
222	C₄×C₈	—	67	2	no	—
223	C₂⁴	—	67	1	no	—
224	C₂²×C₄	—	67	2	no	—
225	C₂²×C₄	—	67	1	no	—
226	C₂²×C₄	—	67	2	no	—
227	C₂²×C₄	—	67	2	no	—
228	C₂²×C₄	—	134	1	no	—
229	C₂²×C₄	—	134	1	no	—
230	C₂²×C₄	—	134	1	no	—
231	C₂²×C₄	—	134	1	no	—
232	C₂²×C₄	C₈×D₆₇	1	2	yes	—
233	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
234	M₁₆	—	134	8	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	67	2	no	—
238	C₂×C₈	—	67	2	no	—
239	C₂×C₈	—	67	2	no	—
240	C₂×C₈	—	67	2	no	—
241	C₂×C₈	—	134	1	no	—
242	C₂×C₈	—	134	1	no	—
243	C₂×C₈	—	134	2	no	—
244	C₂×C₈	—	134	2	no	—
245	C₂×C₈	—	134	2	no	—
246	C₂×C₈	—	134	2	no	—
247	C₂×C₈	C₈×D₆₇	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
250	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
251	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
252	C₂×C₈	#11761627	1	1	yes	characteristic
253	C₂×C₈	#11761629	1	1	yes	characteristic
254	C₄⋊C₄	—	134	4	no	—
255	C₂²⋊C₄	—	134	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	67	2	no	—
259	C₄²	—	67	2	no	—
260	C₁₆	—	2	2	no	—
261	C₁₆	—	2	2	no	—
262	C₁₆	—	134	2	no	—
263	C₁₆	—	134	2	no	—
264	C₂³	—	67	2	no	—
265	C₂³	—	134	2	no	—
266	C₂³	—	134	2	no	—
267	C₂³	—	134	1	no	—
268	C₂³	—	134	1	no	—
269	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	67	2	no	—
274	C₂×C₄	—	67	4	no	—
275	C₂×C₄	—	134	2	no	—
276	C₂×C₄	—	134	2	no	—
277	C₂×C₄	—	134	1	no	—
278	C₂×C₄	—	134	1	no	—
279	C₂×C₄	—	134	2	no	—
280	C₂×C₄	—	134	2	no	—
281	C₂×C₄	—	134	2	no	—
282	C₂×C₄	—	134	2	no	—
283	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
285	C₂×C₄	#14524001	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	67	4	no	—
291	C₈	—	134	2	no	—
292	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
293	C₈	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
294	C₈	#14524005	1	1	yes	characteristic
295	C₈	#14524005	1	1	yes	characteristic
296	C₂²	—	268	2	no	—
297	C₂²	—	268	2	no	—
298	C₂²	—	134	2	no	—
299	C₂²	—	134	2	no	—
300	C₂²	—	134	2	no	—
301	C₂²	C₄×C₈×D₆₇	1	1	yes	characteristic
302	C₂²	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
303	C₂²	#16863448	1	1	yes	characteristic
304	C₂²	#16863812	1	2	yes	—
305	C₂²	#16863814	1	2	yes	—
306	C₄	—	4	2	no	—
307	C₄	—	134	4	no	—
308	C₄	—	134	2	no	—
309	C₄	#16863243	1	1	yes	characteristic
310	C₄	#16863348	1	1	yes	characteristic
311	C₄	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
312	C₄	#16863448	1	1	yes	characteristic
313	C₂	—	268	2	no	—
314	C₂	#22559966	1	1	yes	characteristic
315	C₂	#22561804	1	1	yes	characteristic
316	C₂	#22561807	1	1	yes	characteristic
317	C₂	#22561821	1	2	yes	—
318	C₂	#22562015	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₆₇×#1374	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂³×D₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1374	D₆₇
hypocenter	C₆₇	#12600
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. C₂×#16863171	C₂	semidirect
5. C₂×C₁₆×Dic₆₇	C₂	semidirect
6. #22558287	C₂	semidirect
7. C₆₇×#1374	C₂	semidirect
18. (C₂×D₆₇)⋊C₁₆	C₄	semidirect
19. (C₂×D₆₇)⋊C₁₆	C₄	semidirect
166. #1374	D₆₇	semidirect
167. C₆₇	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863350	2
C₂×(C₂×D₆₇)⋊C₁₆	2
C₂×#16863171	2
C₂×C₁₆×Dic₆₇	2
#22558287	2
C₆₇×#1374	2
#12600	67