Subgroups — #30864462

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	C₂×#16863518	C₂	1	2	yes	maximal, complemented
2	C₂×#16863513	C₂	1	2	yes	maximal, complemented
3	#22568927	C₂	1	2	yes	maximal, complemented
4	#22561103	C₂	1	2	yes	maximal, complemented
5	#22561098	C₂	1	2	yes	maximal, complemented
6	#22561042	C₂	1	2	yes	maximal, complemented
7	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559895	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
10	C₂×#14524050	C₂²	1	4	yes	—
11	#16863518	—	2	8	no	—
12	#16863513	—	2	8	no	—
13	#16863468	—	2	8	no	—
14	C₂×#14523979	C₂²	1	2	yes	—
15	C₂×#14523979	C₂²	1	2	yes	—
16	#16863420	C₂²	1	2	yes	—
17	C₂×#14523970	C₂²	1	4	yes	—
18	C₂×#14523970	C₂²	1	4	yes	—
19	#16863363	C₂²	1	4	yes	—
20	C₂×#14523954	C₂²	1	1	yes	characteristic
21	C₂×#14523953	C₂²	1	1	yes	characteristic
22	C₂×#14523953	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
25	#16863242	C₂²	1	1	yes	characteristic
26	C₂×#135×C₆₇	C₂²	1	2	yes	—
27	C₂×#135×C₆₇	C₂²	1	2	yes	—
28	#532×C₆₇	C₂²	1	2	yes	—
29	#495×C₆₇	C₄	1	2	yes	complemented
30	#495×C₆₇	C₄	1	2	yes	complemented
31	#494×C₆₇	C₄	1	1	yes	characteristic, complemented
32	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
33	#492×C₆₇	C₄	1	2	yes	complemented
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#14524050	—	2	8	no	—
36	#14524050	—	2	8	no	—
37	C₂×#11761612	—	2	4	no	—
38	C₂×#11761612	C₂³	1	4	yes	—
39	C₂²×C₆₇⋊C₈	—	2	4	no	—
40	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
44	#14523979	—	2	4	no	—
45	#14523979	—	2	4	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
47	#14523970	—	2	8	no	—
48	#14523970	—	2	8	no	—
49	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
50	#14523954	D₄	1	4	yes	—
51	#14523953	—	2	2	no	—
52	#14523953	D₄	1	4	yes	complemented
53	C₈×Dic₆₇	—	2	2	no	—
54	C₈×Dic₆₇	D₄	1	4	yes	complemented
55	C₆₇⋊#95	—	2	2	no	—
56	C₂×#41×C₆₇	C₂³	1	4	yes	—
57	#135×C₆₇	—	2	4	no	—
58	#135×C₆₇	—	2	4	no	—
59	#135×C₆₇	—	2	4	no	—
60	#135×C₆₇	—	2	4	no	—
61	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
64	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
65	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
70	#127×C₆₇	C₂×C₄	1	2	yes	—
71	#124×C₆₇	C₂×C₄	1	4	yes	—
72	#121×C₆₇	—	2	4	no	—
73	C₄×D₄×C₆₇	—	2	4	no	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
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	C₂²×Dic₆₇	—	2	1	no	—
80	C₂²×Dic₆₇	—	2	1	no	—
81	#11761612	—	2	4	no	—
82	#11761612	—	2	4	no	—
83	#11761612	—	4	8	no	—
84	C₂×C₆₇⋊C₈	—	2	1	no	—
85	C₂×C₆₇⋊C₈	—	2	1	no	—
86	C₂×C₆₇⋊C₈	—	2	4	no	—
87	C₂×C₆₇⋊C₈	—	4	8	no	—
88	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
89	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
90	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
91	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
92	C₄×Dic₆₇	—	2	1	no	—
93	C₄×Dic₆₇	—	2	1	no	—
94	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
95	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
96	#41×C₆₇	—	2	4	no	—
97	#41×C₆₇	—	2	4	no	—
98	#41×C₆₇	—	2	4	no	—
99	#41×C₆₇	—	2	4	no	—
100	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	#41	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
107	Q₁₆×C₆₇	—	2	4	no	—
108	Q₁₆×C₆₇	—	2	4	no	—
109	SD₁₆×C₆₇	—	2	8	no	—
110	SD₁₆×C₆₇	—	2	8	no	—
111	D₈×C₆₇	—	2	4	no	—
112	D₈×C₆₇	—	2	4	no	—
113	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₆₇	#41	1	2	yes	—
120	C₂×C₈×C₆₇	#41	1	2	yes	—
121	C₄⋊C₄×C₆₇	—	2	4	no	—
122	C₂²⋊C₄×C₆₇	—	2	8	no	—
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	—	2	4	no	—
127	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
128	Q₈×C₆₇	—	2	4	no	—
129	Q₈×C₆₇	—	2	4	no	—
130	D₄×C₆₇	—	2	4	no	—
131	D₄×C₆₇	—	2	4	no	—
132	D₄×C₆₇	—	4	4	no	—
133	D₄×C₆₇	—	4	4	no	—
134	D₄×C₆₇	—	8	4	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	4	no	—
138	C₂×C₄×C₆₇	—	4	4	no	—
139	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	2	2	no	—
149	C₂×Dic₆₇	—	4	2	no	—
150	C₂×Dic₆₇	—	4	2	no	—
151	C₈×C₆₇	C₄×D₄	1	2	yes	—
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	C₆₇⋊C₈	—	2	2	no	—
156	C₆₇⋊C₈	—	4	4	no	—
157	C₆₇⋊C₈	#127	1	4	yes	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	4	4	no	—
160	C₂²×C₆₇	—	4	8	no	—
161	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂²×C₆₇	#532	1	1	yes	characteristic
163	C₂²×C₆₇	#532	1	1	yes	characteristic
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	4	4	no	—
167	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
170	C₄×C₆₇	#532	1	1	yes	characteristic
171	Dic₆₇	—	4	2	no	—
172	Dic₆₇	—	4	2	no	—
173	#21707	—	67	1	no	maximal, Sylow
174	C₂×C₆₇	—	2	1	no	—
175	C₂×C₆₇	—	2	1	no	—
176	C₂×C₆₇	—	8	4	no	—
177	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
178	C₂×C₆₇	#2957	1	2	yes	—
179	#3147	D₆₇	1	1	yes	characteristic
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	—	67	1	no	—
187	#1726	—	67	1	no	—
188	C₆₇	#21707	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	C₂×D₆₇	1	2	yes	—
190	C₂×#135	C₂×D₆₇	1	2	yes	—
191	C₂×#131	—	67	4	no	—
192	#532	C₂×D₆₇	1	2	yes	—
193	#495	Dic₆₇	1	2	yes	complemented
194	#495	Dic₆₇	1	2	yes	complemented
195	#494	Dic₆₇	1	1	yes	characteristic, complemented
196	#493	Dic₆₇	1	1	yes	characteristic, complemented
197	#492	Dic₆₇	1	2	yes	complemented
198	#444	—	134	8	no	—
199	#443	—	134	8	no	—
200	#435	—	134	8	no	—
201	C₂×#108	—	67	2	no	—
202	C₂×#108	—	67	2	no	—
203	#423	—	67	2	no	—
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	C₂²×D₆₇	1	4	yes	—
215	#135	—	2	4	no	—
216	#135	—	2	4	no	—
217	#135	—	2	4	no	—
218	#135	—	2	4	no	—
219	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
220	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
221	C₂×SD₁₆	C₂×Dic₆₇	1	4	yes	—
222	C₂×SD₁₆	C₂×Dic₆₇	1	4	yes	—
223	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
224	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
225	#131	—	134	8	no	—
226	#131	—	134	8	no	—
227	C₂×M₁₆	—	67	4	no	—
228	C₂×M₁₆	—	134	4	no	—
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	C₂×Dic₆₇	1	2	yes	—
235	#127	C₂×Dic₆₇	1	2	yes	—
236	#124	C₂×Dic₆₇	1	4	yes	—
237	#121	—	2	4	no	—
238	C₄×D₄	—	2	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	—	134	4	no	—
243	#108	—	134	4	no	—
244	C₄⋊C₈	—	134	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	C₂×Dic₆₇	1	1	yes	characteristic
254	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
255	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
256	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
257	#95	—	134	2	no	—
258	#41	—	2	4	no	—
259	#41	—	2	4	no	—
260	#41	—	2	4	no	—
261	#41	—	2	4	no	—
262	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	C₂²×Dic₆₇	1	4	yes	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	134	1	no	—
268	C₂²×C₄	—	134	1	no	—
269	C₂²×C₄	#11761627	1	2	yes	—
270	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
271	Q₁₆	—	2	4	no	—
272	Q₁₆	—	2	4	no	—
273	SD₁₆	—	2	8	no	—
274	SD₁₆	—	2	8	no	—
275	D₈	—	2	4	no	—
276	D₈	—	2	4	no	—
277	M₁₆	—	268	8	no	—
278	M₁₆	—	134	4	no	—
279	M₁₆	—	134	4	no	—
280	C₂×C₈	—	268	8	no	—
281	C₂×C₈	—	67	2	no	—
282	C₂×C₈	—	67	2	no	—
283	C₂×C₈	—	67	2	no	—
284	C₂×C₈	—	67	2	no	—
285	C₂×C₈	—	134	4	no	—
286	C₂×C₈	—	134	1	no	—
287	C₂×C₈	—	134	1	no	—
288	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
289	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
290	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
291	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
292	C₂×C₈	D₄×D₆₇	1	2	yes	—
293	C₂×C₈	D₄×D₆₇	1	2	yes	—
294	C₂×C₈	#11761627	1	2	yes	—
295	C₂×C₈	#11761627	1	2	yes	—
296	C₄⋊C₄	—	2	4	no	—
297	C₂²⋊C₄	—	2	8	no	—
298	C₄²	—	67	2	no	—
299	C₄²	—	67	2	no	—
300	C₄²	—	134	1	no	—
301	C₄²	—	134	1	no	—
302	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
303	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
304	C₄²	D₄×D₆₇	1	2	yes	—
305	C₂³	—	2	4	no	—
306	C₂³	C₂×#11761627	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	2	4	no	—
309	D₄	—	2	4	no	—
310	D₄	—	2	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	8	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	268	2	no	—
319	C₂×C₄	—	268	2	no	—
320	C₂×C₄	—	134	2	no	—
321	C₂×C₄	—	134	2	no	—
322	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
323	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
324	C₂×C₄	C₂³×Dic₆₇	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×#11761627	1	1	yes	characteristic
329	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
330	C₈	—	268	4	no	—
331	C₈	—	67	4	no	—
332	C₈	—	134	2	no	—
333	C₈	D₄×Dic₆₇	1	2	yes	—
334	C₈	D₄×Dic₆₇	1	2	yes	—
335	C₈	D₄×Dic₆₇	1	2	yes	—
336	C₈	D₄×Dic₆₇	1	2	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	8	no	—
339	C₂²	—	4	4	no	—
340	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
341	C₂²	#16863944	1	1	yes	characteristic
342	C₂²	#16863944	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	4	no	—
346	C₄	—	268	2	no	—
347	C₄	—	268	2	no	—
348	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
349	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
350	C₄	#127×D₆₇	1	1	yes	characteristic
351	C₄	#16863944	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	8	4	no	—
355	C₂	#127×Dic₆₇	1	1	yes	characteristic
356	C₂	#22569126	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3147	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3147	D₆₇
hypocenter	C₆₇	#21707
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16863518	C₂	semidirect
2. C₂×#16863513	C₂	semidirect
3. #22568927	C₂	semidirect
4. #22561103	C₂	semidirect
5. #22561098	C₂	semidirect
6. #22561042	C₂	semidirect
7. #22560381	C₂	semidirect
8. #22559895	C₂	semidirect
29. #495×C₆₇	C₄	semidirect
30. #495×C₆₇	C₄	semidirect
31. #494×C₆₇	C₄	semidirect
32. #493×C₆₇	C₄	semidirect
33. #492×C₆₇	C₄	semidirect
52. #14523953	D₄	semidirect
54. C₈×Dic₆₇	D₄	semidirect
188. C₆₇	#21707	semidirect
193. #495	Dic₆₇	semidirect
194. #495	Dic₆₇	semidirect
195. #494	Dic₆₇	semidirect
196. #493	Dic₆₇	semidirect
197. #492	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863518	2
C₂×#16863513	2
#22568927	2
#22561103	2
#22561098	2
#22561042	2
#22560381	2
#22559895	2
C₆₇×#3147	2
#21707	67