Subgroups — #31010751

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562387	C₂	1	2	yes	maximal, complemented
3	#22562343	C₂	1	2	yes	maximal, complemented
4	#22562241	C₂	1	2	yes	maximal, complemented
5	#22562120	C₂	1	2	yes	maximal, complemented
6	C₄×#14523984	C₂	1	4	yes	maximal, complemented
7	#22562008	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₆₈	—	2	1	no	—
11	C₂×#14523984	C₂²	1	4	yes	—
12	C₄×D₅₃₆	—	2	1	no	—
13	C₄×D₅₃₆	—	2	1	no	—
14	C₄×D₅₃₆	C₂²	1	2	yes	—
15	C₄×D₅₃₆	C₂²	1	2	yes	—
16	C₂×#14523977	C₂²	1	1	yes	characteristic
17	C₂×#14523965	—	2	1	no	—
18	#16863221	C₂²	1	4	yes	complemented
19	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	complemented
20	#16863197	C₂²	1	4	yes	complemented
21	#16863196	C₂²	1	4	yes	—
22	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
23	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	complemented
24	#532×C₆₇	—	2	1	no	—
25	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
27	#486×C₆₇	—	2	1	no	—
28	#466×C₆₇	—	2	1	no	—
29	#466×C₆₇	C₂²	1	2	yes	—
30	C₄×D₈×C₆₇	—	2	1	no	—
31	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×D₂₆₈	—	2	1	no	—
34	C₂²×C₄×D₆₇	—	4	1	no	—
35	C₂×D₅₃₆	—	2	4	no	—
36	C₂×D₅₃₆	—	2	1	no	—
37	C₂×D₅₃₆	C₂³	1	2	yes	—
38	C₄×D₂₆₈	—	2	2	no	—
39	C₄×D₂₆₈	—	2	2	no	—
40	C₄×D₂₆₈	—	4	1	no	—
41	C₄×D₂₆₈	—	4	1	no	—
42	C₂×C₄⋊Dic₆₇	—	2	1	no	—
43	C₂×(C₂×D₆₇)⋊C₄	—	4	1	no	—
44	#14523984	—	2	8	no	—
45	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
46	#14523977	—	2	1	no	—
47	#14523977	C₂³	1	2	yes	—
48	#14523965	—	2	2	no	—
49	#14523965	—	2	2	no	—
50	C₂×#41×C₆₇	—	4	1	no	—
51	C₂×D₈×C₆₇	—	2	1	no	—
52	C₂×D₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₆₇	—	2	1	no	—
56	#127×C₆₇	—	2	1	no	—
57	#124×C₆₇	—	4	1	no	—
58	#121×C₆₇	—	2	2	no	—
59	#121×C₆₇	—	4	1	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	4	1	no	—
62	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
63	#107×C₆₇	—	2	1	no	—
64	#107×C₆₇	C₂³	1	2	yes	—
65	#102×C₆₇	—	2	2	no	—
66	#102×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂³×D₆₇	—	4	1	no	—
72	C₂×D₂₆₈	—	2	4	no	—
73	C₂×D₂₆₈	—	2	2	no	—
74	C₂×D₂₆₈	—	4	2	no	—
75	C₂×D₂₆₈	—	4	1	no	—
76	C₂²×Dic₆₇	—	4	1	no	—
77	C₂×C₄×D₆₇	—	4	2	no	—
78	C₂×C₄×D₆₇	—	4	2	no	—
79	C₂×C₄×D₆₇	—	8	4	no	—
80	D₅₃₆	—	2	4	no	—
81	D₅₃₆	—	2	2	no	—
82	D₅₃₆	#41	1	4	yes	complemented
83	C₄⋊Dic₆₇	—	2	2	no	—
84	C₄⋊Dic₆₇	—	4	1	no	—
85	(C₂×D₆₇)⋊C₄	—	4	2	no	—
86	(C₂×D₆₇)⋊C₄	—	4	2	no	—
87	C₆₇⋊C₁₆	—	2	4	no	—
88	#41×C₆₇	—	8	4	no	—
89	C₂×Q₈×C₆₇	—	4	1	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	4	1	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂²×C₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	D₈×C₆₇	—	2	4	no	—
99	D₈×C₆₇	—	4	2	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	4	1	no	—
102	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	4	1	no	—
108	C₂²⋊C₄×C₆₇	—	4	2	no	—
109	C₂²⋊C₄×C₆₇	—	4	2	no	—
110	C₄²×C₆₇	D₈	1	1	yes	characteristic
111	C₄²×C₆₇	D₈	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₆₇	—	4	2	no	—
115	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
116	C₂²×D₆₇	—	4	4	no	—
117	C₂²×D₆₇	—	4	2	no	—
118	C₂²×D₆₇	—	8	2	no	—
119	C₂²×D₆₇	—	8	2	no	—
120	Q₈×C₆₇	—	8	2	no	—
121	D₄×C₆₇	—	4	2	no	—
122	D₄×C₆₇	—	4	2	no	—
123	D₄×C₆₇	—	8	2	no	—
124	D₄×C₆₇	—	8	4	no	—
125	D₄×C₆₇	—	8	2	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	4	1	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	8	4	no	—
131	C₂×C₄×C₆₇	D₁₆	1	4	yes	complemented
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₆₇	—	4	2	no	—
139	C₂×Dic₆₇	—	8	2	no	—
140	D₂₆₈	—	2	4	no	—
141	D₂₆₈	—	4	4	no	—
142	D₂₆₈	—	4	2	no	—
143	C₄×D₆₇	—	8	4	no	—
144	C₄×D₆₇	—	8	4	no	—
145	C₈×C₆₇	—	2	1	no	—
146	C₈×C₆₇	—	2	1	no	—
147	C₈×C₆₇	—	2	1	no	—
148	C₈×C₆₇	C₂×#41	1	2	yes	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	—	2	1	no	—
151	C₂²×C₆₇	—	8	2	no	—
152	C₂²×C₆₇	—	8	2	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
155	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₆₇	—	4	4	no	—
157	C₂×D₆₇	—	8	8	no	—
158	C₂×D₆₇	—	8	2	no	—
159	C₂×D₆₇	—	8	2	no	—
160	C₄×C₆₇	—	2	1	no	—
161	C₄×C₆₇	—	2	1	no	—
162	C₄×C₆₇	—	2	1	no	—
163	C₄×C₆₇	—	8	2	no	—
164	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
165	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
166	C₄×C₆₇	#525	1	1	yes	characteristic
167	C₄×C₆₇	#575	1	1	yes	characteristic
168	Dic₆₇	—	8	2	no	—
169	#33660	—	67	1	no	maximal, Sylow
170	C₂×C₆₇	—	2	2	no	—
171	C₂×C₆₇	—	16	2	no	—
172	C₂×C₆₇	#3100	1	1	yes	characteristic
173	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
174	C₂×C₆₇	#3414	1	1	yes	characteristic
175	D₆₇	—	8	4	no	—
176	#3169	D₆₇	1	1	yes	characteristic, complemented
177	C₂×C₄×D₈	—	67	1	no	—
178	#2232	—	67	2	no	—
179	#2224	—	67	2	no	—
180	#2206	—	67	2	no	—
181	#2184	—	67	2	no	—
182	C₄×D₁₆	—	67	4	no	—
183	#2162	—	67	1	no	—
184	C₆₇	#33660	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₈	—	67	1	no	—
186	#532	—	2	1	no	—
187	C₂×C₄×D₄	—	134	1	no	—
188	C₂×D₁₆	—	67	4	no	—
189	#500	C₂×D₆₇	1	1	yes	characteristic, complemented
190	#493	C₂×D₆₇	1	1	yes	characteristic, complemented
191	#486	—	2	1	no	—
192	#466	—	2	1	no	—
193	#466	C₂×D₆₇	1	2	yes	—
194	C₄×D₈	—	2	1	no	—
195	C₄×D₈	—	67	2	no	—
196	C₄×D₈	—	67	2	no	—
197	C₄×D₈	—	134	1	no	—
198	C₄×D₈	—	134	1	no	—
199	C₄×D₈	C₂×D₆₇	1	2	yes	complemented
200	C₂×#107	—	67	1	no	—
201	C₂×#102	—	134	1	no	—
202	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
203	#366	—	67	4	no	—
204	C₄⋊C₁₆	—	67	2	no	—
205	#357	—	67	4	no	—
206	#357	—	67	4	no	—
207	C₂²⋊C₁₆	—	67	2	no	—
208	C₄×C₁₆	—	67	2	no	—
209	C₂×#41	—	4	1	no	—
210	C₂²×D₄	—	134	1	no	—
211	C₂³×C₄	—	268	1	no	—
212	C₂×D₈	—	2	1	no	—
213	C₂×D₈	—	67	2	no	—
214	C₂×D₈	—	134	4	no	—
215	C₂×D₈	—	134	1	no	—
216	C₂×D₈	C₂²×D₆₇	1	2	yes	—
217	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
219	C₄⋊Q₈	—	2	1	no	—
220	#127	—	2	1	no	—
221	#124	—	4	1	no	—
222	#121	—	2	2	no	—
223	#121	—	4	1	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	4	1	no	—
226	C₄×D₄	—	268	1	no	—
227	C₄×D₄	—	268	1	no	—
228	C₄×D₄	—	134	2	no	—
229	C₄×D₄	—	134	2	no	—
230	C₂×C₄⋊C₄	—	134	1	no	—
231	C₂×C₂²⋊C₄	—	268	1	no	—
232	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
233	D₁₆	—	134	8	no	—
234	C₂×C₁₆	—	67	4	no	—
235	#107	—	2	1	no	—
236	#107	—	67	2	no	—
237	#107	—	134	1	no	—
238	#107	C₂²×D₆₇	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	134	2	no	—
242	#102	—	134	2	no	—
243	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
244	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
247	C₂⁴	—	268	1	no	—
248	#41	—	8	4	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	268	2	no	—
255	C₂×D₄	—	268	1	no	—
256	C₂×D₄	—	134	4	no	—
257	C₂×D₄	—	134	2	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	268	1	no	—
260	C₂²×C₄	—	268	2	no	—
261	C₂²×C₄	—	268	2	no	—
262	C₂²×C₄	—	536	4	no	—
263	C₂²×C₄	#11761613	1	1	yes	characteristic
264	C₂²×C₄	#11761613	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
266	D₈	—	2	4	no	—
267	D₈	—	4	2	no	—
268	D₈	—	67	4	no	—
269	D₈	—	134	4	no	—
270	D₈	—	134	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
274	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
275	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	268	1	no	—
280	C₄⋊C₄	—	134	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	268	2	no	—
284	C₂²⋊C₄	—	268	2	no	—
285	C₄²	#11761613	1	1	yes	characteristic
286	C₄²	#11761613	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
289	C₁₆	—	134	4	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	268	4	no	—
292	C₂³	—	268	2	no	—
293	C₂³	—	536	2	no	—
294	C₂³	—	536	2	no	—
295	C₂³	C₂×#11761613	1	1	yes	characteristic
296	Q₈	—	8	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	4	no	—
301	D₄	—	8	2	no	—
302	D₄	—	268	4	no	—
303	D₄	—	268	2	no	—
304	D₄	—	134	4	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	268	2	no	—
311	C₂×C₄	—	536	2	no	—
312	C₂×C₄	—	536	4	no	—
313	C₂×C₄	—	536	4	no	—
314	C₂×C₄	#14523984	1	4	yes	complemented
315	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
316	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
317	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
318	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
319	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	#41×D₆₇	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	268	4	no	—
331	C₂²	—	536	8	no	—
332	C₂²	—	536	2	no	—
333	C₂²	—	536	2	no	—
334	C₂²	C₂×#14523984	1	2	yes	—
335	C₂²	C₂²×#11761613	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	536	2	no	—
341	C₄	C₂×#14523984	1	2	yes	—
342	C₄	C₂×#14523984	1	2	yes	—
343	C₄	#16863903	1	1	yes	characteristic
344	C₄	#16864194	1	1	yes	characteristic
345	C₂	—	2	2	no	—
346	C₂	—	16	2	no	—
347	C₂	—	536	4	no	—
348	C₂	#22570205	1	1	yes	characteristic
349	C₂	C₂²×#14523984	1	1	yes	characteristic
350	C₂	#22573343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3169	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	#3169	D₆₇
hypocenter	C₆₇	#33660
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₅₃₆	C₂	semidirect
2. #22562387	C₂	semidirect
3. #22562343	C₂	semidirect
4. #22562241	C₂	semidirect
5. #22562120	C₂	semidirect
6. C₄×#14523984	C₂	semidirect
7. #22562008	C₂	semidirect
8. C₆₇×#3169	C₂	semidirect
18. #16863221	C₂²	semidirect
19. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
20. #16863197	C₂²	semidirect
23. C₄×C₆₇⋊C₁₆	C₂²	semidirect
25. #500×C₆₇	C₂²	semidirect
26. #493×C₆₇	C₂²	semidirect
31. C₄×D₈×C₆₇	C₂²	semidirect
82. D₅₃₆	#41	semidirect
131. C₂×C₄×C₆₇	D₁₆	semidirect
176. #3169	D₆₇	semidirect
184. C₆₇	#33660	semidirect
189. #500	C₂×D₆₇	semidirect
190. #493	C₂×D₆₇	semidirect
199. C₄×D₈	C₂×D₆₇	semidirect
314. C₂×C₄	#14523984	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₃₆	2
#22562387	2
#22562343	2
#22562241	2
#22562120	2
C₄×#14523984	2
#22562008	2
C₆₇×#3169	2
#33660	67