Subgroups — #30864454

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570637	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863517	C₂	1	2	yes	maximal, complemented
3	C₂×#16863512	C₂	1	2	yes	maximal, complemented
4	#22568926	C₂	1	2	yes	maximal, complemented
5	#22561100	C₂	1	2	yes	maximal, complemented
6	#22561095	C₂	1	2	yes	maximal, complemented
7	#22561040	C₂	1	2	yes	maximal, complemented
8	#22559895	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14524065	C₂²	1	2	yes	—
11	C₂×#14524065	C₂²	1	2	yes	—
12	C₂×#14524051	C₂²	1	4	yes	—
13	#16863946	C₂²	1	2	yes	—
14	#16863780	C₄	1	2	yes	complemented
15	#16863780	C₄	1	2	yes	complemented
16	#16863776	C₄	1	1	yes	characteristic, complemented
17	#16863773	C₄	1	1	yes	characteristic, complemented
18	#16863769	C₄	1	2	yes	complemented
19	#16863517	—	2	8	no	—
20	#16863512	—	2	8	no	—
21	#16863466	—	2	8	no	—
22	C₂×#14523969	C₂²	1	4	yes	—
23	C₂×#14523969	C₂²	1	4	yes	—
24	#16863364	C₂²	1	4	yes	—
25	C₂×#14523954	C₂²	1	1	yes	characteristic
26	C₂×#14523953	C₂²	1	1	yes	characteristic
27	C₂×#14523953	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
30	#16863242	C₂²	1	1	yes	characteristic
31	C₂×#108×C₆₇	C₂²	1	2	yes	—
32	C₂×#108×C₆₇	C₂²	1	2	yes	—
33	#423×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#11761628	C₂³	1	4	yes	—
36	#14524065	—	2	4	no	—
37	#14524065	—	2	4	no	—
38	#14524065	—	2	4	no	—
39	#14524065	—	2	4	no	—
40	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
41	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
42	C₂×#11761617	C₂×C₄	1	4	yes	—
43	C₂×#11761617	C₂×C₄	1	4	yes	—
44	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
45	C₂×D₅₃₆	C₂×C₄	1	2	yes	—
46	#14524051	—	2	8	no	—
47	#14524051	—	2	8	no	—
48	C₂×#11761611	—	2	4	no	—
49	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₆₇	—	2	4	no	—
52	#14524043	C₂×C₄	1	2	yes	—
53	#14524040	C₂×C₄	1	2	yes	—
54	#14524031	C₂×C₄	1	4	yes	—
55	#14524017	—	2	4	no	—
56	C₄×D₂₆₈	—	2	4	no	—
57	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
59	#14523969	—	2	8	no	—
60	#14523969	—	2	8	no	—
61	(C₂×D₆₇)⋊C₈	—	2	8	no	—
62	#14523954	D₄	1	4	yes	—
63	#14523953	—	2	2	no	—
64	#14523953	D₄	1	4	yes	complemented
65	C₈×Dic₆₇	—	2	2	no	—
66	C₈×Dic₆₇	D₄	1	4	yes	complemented
67	C₆₇⋊#95	—	2	2	no	—
68	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	#108×C₆₇	—	2	4	no	—
73	#108×C₆₇	—	2	4	no	—
74	C₄⋊C₈×C₆₇	—	2	4	no	—
75	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	#11761628	—	2	4	no	—
80	#11761628	—	2	4	no	—
81	#11761628	—	2	4	no	—
82	#11761628	—	2	4	no	—
83	C₂×Dic₁₃₄	C₂²×C₄	1	4	yes	—
84	C₂×D₂₆₈	—	2	4	no	—
85	C₂×D₂₆₈	C₂²×C₄	1	4	yes	—
86	C₂×C₂²⋊D₆₇	—	2	4	no	—
87	C₂²×Dic₆₇	—	2	1	no	—
88	C₂²×Dic₆₇	—	2	1	no	—
89	C₂×C₄×D₆₇	—	2	4	no	—
90	Dic₂₆₈	—	2	4	no	—
91	Dic₂₆₈	—	2	4	no	—
92	#11761617	—	2	8	no	—
93	#11761617	—	2	8	no	—
94	D₅₃₆	—	2	4	no	—
95	D₅₃₆	—	2	4	no	—
96	#11761611	—	4	8	no	—
97	C₂×C₆₇⋊C₈	—	2	1	no	—
98	C₂×C₆₇⋊C₈	—	2	1	no	—
99	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
100	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
101	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
103	C₈×D₆₇	—	4	8	no	—
104	C₄⋊Dic₆₇	—	2	4	no	—
105	(C₂×D₆₇)⋊C₄	—	2	8	no	—
106	C₄×Dic₆₇	—	2	1	no	—
107	C₄×Dic₆₇	—	2	1	no	—
108	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
109	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	#41	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
112	M₁₆×C₆₇	—	2	4	no	—
113	M₁₆×C₆₇	—	2	4	no	—
114	C₂×C₈×C₆₇	—	2	4	no	—
115	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₆₇	#41	1	2	yes	—
122	C₂×C₈×C₆₇	#41	1	2	yes	—
123	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂×D₄	1	2	yes	—
126	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₂²×D₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₂²⋊D₆₇	—	4	4	no	—
138	C₂²⋊D₆₇	—	4	4	no	—
139	C₂×Dic₆₇	—	2	4	no	—
140	C₂×Dic₆₇	—	2	2	no	—
141	C₂×Dic₆₇	—	2	2	no	—
142	C₂×Dic₆₇	—	4	2	no	—
143	C₂×Dic₆₇	—	4	2	no	—
144	D₂₆₈	—	2	4	no	—
145	D₂₆₈	—	2	4	no	—
146	D₂₆₈	—	8	4	no	—
147	C₄×D₆₇	—	4	4	no	—
148	C₄×D₆₇	—	4	4	no	—
149	Dic₁₃₄	—	2	4	no	—
150	Dic₁₃₄	—	2	4	no	—
151	C₈×C₆₇	—	4	4	no	—
152	C₈×C₆₇	C₄×D₄	1	2	yes	—
153	C₈×C₆₇	C₄×D₄	1	2	yes	—
154	C₈×C₆₇	C₄×D₄	1	2	yes	—
155	C₈×C₆₇	C₄×D₄	1	2	yes	—
156	C₆₇⋊C₈	—	2	2	no	—
157	C₆₇⋊C₈	#127	1	4	yes	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
160	C₂²×C₆₇	#532	1	1	yes	characteristic
161	C₂²×C₆₇	#532	1	1	yes	characteristic
162	C₂×D₆₇	—	4	4	no	—
163	C₂×D₆₇	—	4	8	no	—
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
169	C₄×C₆₇	#532	1	1	yes	characteristic
170	Dic₆₇	—	4	2	no	—
171	Dic₆₇	—	4	2	no	—
172	Dic₆₇	—	4	4	no	—
173	#21707	—	67	1	no	maximal, Sylow
174	C₂×C₆₇	—	2	1	no	—
175	C₂×C₆₇	—	2	1	no	—
176	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
177	C₂×C₆₇	#2957	1	2	yes	—
178	D₆₇	—	8	4	no	—
179	#3147	—	67	1	no	—
180	C₂×#444	—	67	2	no	—
181	C₂×#443	—	67	2	no	—
182	#2932	—	67	2	no	—
183	#1975	—	67	2	no	—
184	#1974	—	67	2	no	—
185	#1963	—	67	2	no	—
186	#1848	D₆₇	1	1	yes	characteristic, complemented
187	#1726	—	67	1	no	—
188	C₆₇	#21707	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	—	67	2	no	—
190	C₂×#135	—	67	2	no	—
191	C₂×#131	—	67	4	no	—
192	#532	—	67	2	no	—
193	#495	—	67	2	no	—
194	#495	—	67	2	no	—
195	#494	—	67	1	no	—
196	#493	—	67	1	no	—
197	#492	—	67	2	no	—
198	#444	—	134	8	no	—
199	#443	—	134	8	no	—
200	#435	—	134	8	no	—
201	C₂×#108	C₂×D₆₇	1	2	yes	—
202	C₂×#108	C₂×D₆₇	1	2	yes	—
203	#423	C₂×D₆₇	1	2	yes	—
204	C₂×#104	—	67	4	no	—
205	C₂×#104	—	67	4	no	—
206	#408	—	67	4	no	—
207	C₂×#97	—	67	1	no	—
208	C₂×#97	—	67	1	no	—
209	C₂×#97	—	67	1	no	—
210	C₂×C₄×C₈	—	67	1	no	—
211	C₂×C₄×C₈	—	67	1	no	—
212	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
213	#376	—	67	1	no	—
214	C₂×#41	—	67	4	no	—
215	#135	—	134	4	no	—
216	#135	—	134	4	no	—
217	#135	—	134	4	no	—
218	#135	—	134	4	no	—
219	C₂×Q₁₆	—	67	2	no	—
220	C₂×Q₁₆	—	67	2	no	—
221	C₂×SD₁₆	—	67	4	no	—
222	C₂×SD₁₆	—	67	4	no	—
223	C₂×D₈	—	67	2	no	—
224	C₂×D₈	—	67	2	no	—
225	#131	—	134	8	no	—
226	#131	—	134	8	no	—
227	C₂×M₁₆	—	134	4	no	—
228	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
229	C₂²×C₈	—	67	1	no	—
230	C₂²×C₈	—	67	1	no	—
231	C₂²×C₈	—	134	4	no	—
232	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
234	C₄⋊Q₈	—	67	2	no	—
235	#127	—	67	2	no	—
236	#124	—	67	4	no	—
237	#121	—	134	4	no	—
238	C₄×D₄	—	134	4	no	—
239	C₂×C₄²	—	67	1	no	—
240	C₂×C₄²	—	67	1	no	—
241	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
242	#108	—	2	4	no	—
243	#108	—	2	4	no	—
244	C₄⋊C₈	—	2	4	no	—
245	#104	—	134	8	no	—
246	#104	—	134	8	no	—
247	C₂²⋊C₈	—	134	8	no	—
248	#97	—	67	4	no	—
249	#97	—	67	4	no	—
250	#97	—	134	2	no	—
251	C₄×C₈	—	67	4	no	—
252	C₄×C₈	—	134	2	no	—
253	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
254	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
255	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
256	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
257	#95	—	134	2	no	—
258	#41	—	134	4	no	—
259	#41	—	134	4	no	—
260	#41	—	134	4	no	—
261	#41	—	134	4	no	—
262	C₂×Q₈	—	67	4	no	—
263	C₂×D₄	—	67	4	no	—
264	C₂×D₄	—	134	4	no	—
265	C₂×D₄	—	134	4	no	—
266	C₂²×C₄	—	134	1	no	—
267	C₂²×C₄	—	134	1	no	—
268	C₂²×C₄	—	134	4	no	—
269	C₂²×C₄	#11761629	1	2	yes	—
270	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
271	Q₁₆	—	134	4	no	—
272	Q₁₆	—	134	4	no	—
273	SD₁₆	—	134	8	no	—
274	SD₁₆	—	134	8	no	—
275	D₈	—	134	4	no	—
276	D₈	—	134	4	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	268	8	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	268	8	no	—
282	C₂×C₈	—	67	2	no	—
283	C₂×C₈	—	67	2	no	—
284	C₂×C₈	—	67	2	no	—
285	C₂×C₈	—	67	2	no	—
286	C₂×C₈	—	134	1	no	—
287	C₂×C₈	—	134	1	no	—
288	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
292	C₂×C₈	D₄×D₆₇	1	2	yes	—
293	C₂×C₈	D₄×D₆₇	1	2	yes	—
294	C₂×C₈	#11761629	1	2	yes	—
295	C₂×C₈	#11761629	1	2	yes	—
296	C₄⋊C₄	—	134	4	no	—
297	C₂²⋊C₄	—	134	8	no	—
298	C₄²	—	67	2	no	—
299	C₄²	—	67	2	no	—
300	C₄²	—	134	1	no	—
301	C₄²	—	134	1	no	—
302	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
304	C₄²	D₄×D₆₇	1	2	yes	—
305	C₂³	—	134	4	no	—
306	C₂³	C₂×#11761629	1	1	yes	characteristic
307	Q₈	—	134	4	no	—
308	Q₈	—	134	4	no	—
309	D₄	—	268	4	no	—
310	D₄	—	268	4	no	—
311	D₄	—	536	4	no	—
312	D₄	—	134	4	no	—
313	D₄	—	134	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	268	2	no	—
316	C₂×C₄	—	268	2	no	—
317	C₂×C₄	—	268	4	no	—
318	C₂×C₄	—	268	4	no	—
319	C₂×C₄	—	134	4	no	—
320	C₂×C₄	—	134	2	no	—
321	C₂×C₄	—	134	2	no	—
322	C₂×C₄	#14524005	1	2	yes	—
323	C₂×C₄	#14524005	1	2	yes	—
324	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
326	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
329	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
330	C₈	—	4	4	no	—
331	C₈	—	67	4	no	—
332	C₈	—	134	2	no	—
333	C₈	#14524005	1	2	yes	—
334	C₈	#14524005	1	2	yes	—
335	C₈	#14524005	1	2	yes	—
336	C₈	#14524005	1	2	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	268	8	no	—
339	C₂²	—	268	4	no	—
340	C₂²	C₂×#14524005	1	1	yes	characteristic
341	C₂²	#16863947	1	1	yes	characteristic
342	C₂²	#16863947	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	268	2	no	—
346	C₄	—	268	2	no	—
347	C₄	—	268	4	no	—
348	C₄	C₂×#14524005	1	1	yes	characteristic
349	C₄	C₂×#14524005	1	1	yes	characteristic
350	C₄	#127×D₆₇	1	1	yes	characteristic
351	C₄	#16863947	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	536	4	no	—
355	C₂	#22562771	1	1	yes	characteristic
356	C₂	#22569125	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#22570637	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#21707
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. C₂×#16863517	C₂	semidirect
3. C₂×#16863512	C₂	semidirect
4. #22568926	C₂	semidirect
5. #22561100	C₂	semidirect
6. #22561095	C₂	semidirect
7. #22561040	C₂	semidirect
8. #22559895	C₂	semidirect
9. C₆₇×#1848	C₂	semidirect
14. #16863780	C₄	semidirect
15. #16863780	C₄	semidirect
16. #16863776	C₄	semidirect
17. #16863773	C₄	semidirect
18. #16863769	C₄	semidirect
64. #14523953	D₄	semidirect
66. C₈×Dic₆₇	D₄	semidirect
186. #1848	D₆₇	semidirect
188. C₆₇	#21707	semidirect

Maximal subgroups

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

subgroup	index
#22570637	2
C₂×#16863517	2
C₂×#16863512	2
#22568926	2
#22561100	2
#22561095	2
#22561040	2
#22559895	2
C₆₇×#1848	2
#21707	67