Subgroups — #31010411

Order 17152.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863828	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863812	C₂	1	1	yes	maximal, characteristic, complemented
3	#22569930	C₂	1	2	yes	maximal, complemented
4	#22562340	C₂	1	4	yes	maximal, complemented
5	#22562335	C₂	1	2	yes	maximal, complemented
6	#22562330	C₂	1	2	yes	maximal, complemented
7	#22561956	C₂	1	2	yes	maximal, complemented
8	C₂×#368×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#16864196	—	2	2	no	—
10	C₂×#14524065	C₂²	1	2	yes	—
11	C₂×#14524049	C₂²	1	1	yes	characteristic
12	#16863828	—	2	1	no	—
13	#16863828	—	2	1	no	—
14	#16863828	—	2	1	no	—
15	#16863828	—	2	1	no	—
16	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic, complemented
17	C₂×#14523988	C₂²	1	2	yes	complemented
18	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic, complemented
19	#16863812	—	2	1	no	—
20	#16863812	—	2	1	no	—
21	#16863812	C₂²	1	2	yes	—
22	#16863812	C₂²	1	2	yes	—
23	C₂×#14523983	C₂²	1	1	yes	characteristic
24	C₂×#14523982	—	2	1	no	—
25	C₂×C₁₆×D₆₇	—	2	1	no	—
26	#16863672	C₂²	1	4	yes	—
27	#16863667	C₂²	1	4	yes	—
28	#16863660	—	2	4	no	—
29	#16863378	—	2	2	no	—
30	#16863211	C₂²	1	4	yes	complemented
31	#16863208	C₂²	1	4	yes	complemented
32	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#108×C₆₇	C₂²	1	2	yes	—
34	#368×C₆₇	C₂²	1	2	yes	complemented
35	#368×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#11761628	—	2	2	no	—
37	C₂×#11761628	D₄	1	1	yes	characteristic
38	#14524076	—	2	4	no	—
39	#14524076	—	4	2	no	—
40	#14524071	—	2	4	no	—
41	#14524071	—	4	2	no	—
42	#14524065	—	2	2	no	—
43	#14524065	—	2	2	no	—
44	#14524065	—	2	2	no	—
45	#14524065	—	2	2	no	—
46	#14524065	—	4	4	no	—
47	C₂×Dic₂₆₈	C₂³	1	2	yes	—
48	C₂×#11761617	—	2	2	no	—
49	C₂×D₅₃₆	C₂³	1	2	yes	—
50	#14524049	—	2	1	no	—
51	#14524049	—	2	1	no	—
52	#14524049	D₄	1	2	yes	—
53	#14524049	C₂³	1	2	yes	—
54	C₂×#11761612	D₄	1	1	yes	characteristic
55	C₂×#11761611	—	2	1	no	—
56	C₂×C₈×D₆₇	—	2	1	no	—
57	Dic₅₃₆	—	2	1	no	—
58	Dic₅₃₆	—	2	1	no	—
59	#14523988	—	2	2	no	—
60	#14523988	—	2	2	no	—
61	D₁₀₇₂	—	2	1	no	—
62	D₁₀₇₂	—	2	1	no	—
63	#14523983	—	2	1	no	—
64	#14523983	C₂³	1	2	yes	—
65	#14523982	—	2	2	no	—
66	#14523982	—	2	2	no	—
67	C₂×C₆₇⋊C₁₆	—	2	1	no	—
68	C₁₆×D₆₇	—	2	2	no	—
69	C₁₆×D₆₇	—	2	2	no	—
70	#14523961	—	2	4	no	—
71	#14523959	—	2	4	no	—
72	C₂×M₁₆×C₆₇	—	2	2	no	—
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
75	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
76	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
80	#108×C₆₇	—	2	2	no	—
81	#108×C₆₇	C₂³	1	4	yes	—
82	#11761628	—	2	1	no	—
83	#11761628	—	2	1	no	—
84	#11761628	—	4	2	no	—
85	#11761628	—	4	2	no	—
86	#11761628	—	4	2	no	—
87	#11761628	—	4	2	no	—
88	#11761628	D₈	1	2	yes	—
89	#11761628	C₂×D₄	1	2	yes	—
90	C₂×Dic₁₃₄	—	2	2	no	—
91	C₂×Dic₁₃₄	D₈	1	1	yes	characteristic
92	C₂×D₂₆₈	—	2	2	no	—
93	C₂×D₂₆₈	D₈	1	1	yes	characteristic
94	C₂×C₂²⋊D₆₇	—	2	1	no	—
95	C₂×C₂²⋊D₆₇	—	4	2	no	—
96	C₂×C₄×D₆₇	—	2	1	no	—
97	C₂×C₄×D₆₇	—	4	2	no	—
98	Dic₂₆₈	—	2	2	no	—
99	Dic₂₆₈	—	2	2	no	—
100	Dic₂₆₈	—	4	4	no	—
101	#11761617	—	4	4	no	—
102	#11761617	—	4	4	no	—
103	#11761617	—	4	2	no	—
104	#11761617	—	4	2	no	—
105	D₅₃₆	—	2	2	no	—
106	D₅₃₆	—	2	2	no	—
107	D₅₃₆	—	4	4	no	—
108	#11761612	—	2	1	no	—
109	#11761612	C₂×D₄	1	2	yes	—
110	#11761611	—	2	2	no	—
111	#11761611	—	2	2	no	—
112	C₂×C₆₇⋊C₈	—	2	1	no	—
113	C₈×D₆₇	—	2	2	no	—
114	C₈×D₆₇	—	2	2	no	—
115	C₆₇⋊C₁₆	—	2	2	no	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	M₁₆×C₆₇	—	2	4	no	—
118	M₁₆×C₆₇	—	4	2	no	—
119	C₂×C₈×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
124	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
125	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
126	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₆₇	#135	1	1	yes	characteristic
131	C₂²×D₆₇	—	2	1	no	—
132	C₂²×D₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#121	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	#135	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₆₇	—	2	2	no	—
140	C₂²⋊D₆₇	—	2	2	no	—
141	C₂²⋊D₆₇	—	8	2	no	—
142	C₂²⋊D₆₇	—	8	2	no	—
143	C₂×Dic₆₇	—	2	1	no	—
144	C₂×Dic₆₇	—	4	2	no	—
145	D₂₆₈	—	2	1	no	—
146	D₂₆₈	—	4	2	no	—
147	D₂₆₈	—	4	2	no	—
148	D₂₆₈	C₂×D₈	1	2	yes	—
149	C₄×D₆₇	—	2	2	no	—
150	C₄×D₆₇	—	2	2	no	—
151	C₄×D₆₇	—	8	2	no	—
152	C₄×D₆₇	—	8	2	no	—
153	Dic₁₃₄	—	2	1	no	—
154	Dic₁₃₄	—	4	2	no	—
155	Dic₁₃₄	—	4	2	no	—
156	Dic₁₃₄	C₂×D₈	1	2	yes	—
157	C₈×C₆₇	—	4	4	no	—
158	C₈×C₆₇	#121	1	1	yes	characteristic
159	C₈×C₆₇	#121	1	1	yes	characteristic
160	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
162	C₆₇⋊C₈	—	2	2	no	—
163	C₂²×C₆₇	—	4	1	no	—
164	C₂²×C₆₇	#466	1	1	yes	characteristic
165	C₂²×C₆₇	#466	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
167	C₂×D₆₇	—	2	2	no	—
168	C₂×D₆₇	—	4	2	no	—
169	C₂×D₆₇	—	8	2	no	—
170	C₂×D₆₇	—	16	2	no	—
171	C₄×C₆₇	#466	1	1	yes	characteristic
172	C₄×C₆₇	#466	1	1	yes	characteristic
173	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
174	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
175	Dic₆₇	—	2	2	no	—
176	Dic₆₇	—	8	2	no	—
177	#33617	—	67	1	no	maximal, Sylow
178	C₂×C₆₇	—	2	1	no	—
179	C₂×C₆₇	—	2	1	no	—
180	C₂×C₆₇	—	2	1	no	—
181	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
182	D₆₇	—	4	2	no	—
183	D₆₇	—	16	2	no	—
184	C₂×#508	—	67	1	no	—
185	C₂×#504	—	67	1	no	—
186	#3067	—	67	2	no	—
187	#2223	—	67	4	no	—
188	#2222	—	67	2	no	—
189	#2221	—	67	2	no	—
190	C₂×#368	D₆₇	1	1	yes	characteristic, complemented
191	#2148	—	67	2	no	—
192	C₆₇	#33617	1	1	yes	characteristic, complemented, Sylow
193	#575	—	134	2	no	—
194	C₂×#135	—	67	2	no	—
195	C₂×#131	—	67	1	no	—
196	#508	—	134	1	no	—
197	#508	—	134	1	no	—
198	#508	—	134	1	no	—
199	#508	—	134	1	no	—
200	C₂×Q₃₂	—	67	1	no	—
201	C₂×SD₃₂	—	67	2	no	—
202	C₂×D₁₆	—	67	1	no	—
203	#504	—	67	2	no	—
204	#504	—	67	2	no	—
205	#504	—	134	1	no	—
206	#504	—	134	1	no	—
207	C₂×M₃₂	—	67	1	no	—
208	C₂×M₃₂	—	134	1	no	—
209	C₂²×C₁₆	—	134	1	no	—
210	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
211	#473	—	67	4	no	—
212	#472	—	67	4	no	—
213	#471	—	134	4	no	—
214	C₂×#108	C₂×D₆₇	1	2	yes	—
215	#413	—	134	2	no	—
216	#368	C₂×D₆₇	1	2	yes	complemented
217	#368	C₂×D₆₇	1	2	yes	complemented
218	#362	—	67	4	no	—
219	#361	—	67	4	no	—
220	C₂×#41	—	67	1	no	—
221	C₂×#41	—	134	2	no	—
222	#137	—	268	2	no	—
223	#137	—	134	4	no	—
224	AGL(1,ℤ/8)	—	268	2	no	—
225	AGL(1,ℤ/8)	—	134	4	no	—
226	#135	—	268	4	no	—
227	#135	—	134	2	no	—
228	#135	—	134	2	no	—
229	#135	—	134	2	no	—
230	#135	—	134	2	no	—
231	C₂×Q₁₆	—	67	2	no	—
232	C₂×SD₁₆	—	134	2	no	—
233	C₂×D₈	—	67	2	no	—
234	#131	—	67	2	no	—
235	#131	—	67	2	no	—
236	#131	—	134	1	no	—
237	#131	—	134	1	no	—
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	67	1	no	—
240	C₂×M₁₆	—	134	1	no	—
241	C₂²×C₈	—	134	1	no	—
242	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
243	Q₃₂	—	134	1	no	—
244	Q₃₂	—	134	1	no	—
245	SD₃₂	—	134	2	no	—
246	SD₃₂	—	134	2	no	—
247	D₁₆	—	134	1	no	—
248	D₁₆	—	134	1	no	—
249	M₃₂	—	67	2	no	—
250	M₃₂	—	134	1	no	—
251	M₃₂	—	134	2	no	—
252	M₃₂	—	134	2	no	—
253	C₂×C₁₆	—	134	1	no	—
254	C₂×C₁₆	—	134	2	no	—
255	C₂×C₁₆	—	134	2	no	—
256	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
257	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
258	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
259	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
262	#108	—	2	2	no	—
263	#108	C₂²×D₆₇	1	4	yes	—
264	#101	—	134	4	no	—
265	#100	—	134	4	no	—
266	#41	—	268	2	no	—
267	#41	—	268	2	no	—
268	#41	—	268	2	no	—
269	#41	—	268	2	no	—
270	#41	—	67	2	no	—
271	#41	—	67	2	no	—
272	#41	—	134	1	no	—
273	#41	—	134	1	no	—
274	C₂×Q₈	—	67	1	no	—
275	C₂×Q₈	—	134	2	no	—
276	C₂×D₄	—	268	2	no	—
277	C₂×D₄	—	67	1	no	—
278	C₂×D₄	—	134	1	no	—
279	C₂×D₄	—	134	2	no	—
280	C₂²×C₄	—	268	2	no	—
281	C₂²×C₄	—	134	1	no	—
282	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
283	Q₁₆	—	268	4	no	—
284	Q₁₆	—	134	2	no	—
285	Q₁₆	—	134	2	no	—
286	SD₁₆	—	268	4	no	—
287	SD₁₆	—	268	4	no	—
288	SD₁₆	—	268	2	no	—
289	SD₁₆	—	268	2	no	—
290	D₈	—	268	4	no	—
291	D₈	—	134	2	no	—
292	D₈	—	134	2	no	—
293	M₁₆	—	2	4	no	—
294	M₁₆	—	4	2	no	—
295	M₁₆	—	67	2	no	—
296	M₁₆	—	134	1	no	—
297	M₁₆	—	134	2	no	—
298	M₁₆	—	134	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	134	1	no	—
301	C₂×C₈	—	134	2	no	—
302	C₂×C₈	—	134	2	no	—
303	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
306	C₂×C₈	#11761629	1	1	yes	characteristic
307	C₂×C₈	#11761629	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
309	C₁₆	—	134	2	no	—
310	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
311	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
312	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
313	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
314	C₂³	—	268	2	no	—
315	C₂³	—	134	1	no	—
316	C₂³	#14524066	1	1	yes	characteristic
317	Q₈	—	268	2	no	—
318	Q₈	—	268	2	no	—
319	Q₈	—	67	2	no	—
320	Q₈	—	134	1	no	—
321	D₄	—	268	2	no	—
322	D₄	—	268	2	no	—
323	D₄	—	536	2	no	—
324	D₄	—	536	2	no	—
325	D₄	—	67	2	no	—
326	D₄	—	134	2	no	—
327	D₄	—	134	2	no	—
328	D₄	—	134	1	no	—
329	C₂×C₄	—	268	2	no	—
330	C₂×C₄	—	536	2	no	—
331	C₂×C₄	—	536	2	no	—
332	C₂×C₄	—	134	1	no	—
333	C₂×C₄	—	134	2	no	—
334	C₂×C₄	—	134	2	no	—
335	C₂×C₄	#14524018	1	1	yes	characteristic
336	C₂×C₄	#14524018	1	1	yes	characteristic
337	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
338	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
339	C₂×C₄	#14524066	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
341	C₈	—	4	4	no	—
342	C₈	—	134	2	no	—
343	C₈	#14524018	1	1	yes	characteristic
344	C₈	#14524018	1	1	yes	characteristic
345	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
346	C₈	C₂×#11761629	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	268	2	no	—
349	C₂²	—	536	2	no	—
350	C₂²	—	1072	2	no	—
351	C₂²	—	134	2	no	—
352	C₂²	#16863633	1	1	yes	characteristic
353	C₂²	#16863633	1	1	yes	characteristic
354	C₂²	C₂×#14524066	1	1	yes	characteristic
355	C₄	—	536	2	no	—
356	C₄	—	134	2	no	—
357	C₄	#16863633	1	1	yes	characteristic
358	C₄	#16863633	1	1	yes	characteristic
359	C₄	C₂×#14524018	1	1	yes	characteristic
360	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	268	2	no	—
365	C₂	—	1072	2	no	—
366	C₂	C₂×#16863633	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863633
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#368×C₆₇	C₂
socle	C₂×C₆₇	C₂×#466
hypercenter	C₂×#368	D₆₇
hypocenter	C₆₇	#33617
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂×#16863828	C₂	semidirect
2. C₂×#16863812	C₂	semidirect
3. #22569930	C₂	semidirect
4. #22562340	C₂	semidirect
5. #22562335	C₂	semidirect
6. #22562330	C₂	semidirect
7. #22561956	C₂	semidirect
8. C₂×#368×C₆₇	C₂	semidirect
16. C₂×Dic₅₃₆	C₂²	semidirect
17. C₂×#14523988	C₂²	semidirect
18. C₂×D₁₀₇₂	C₂²	semidirect
30. #16863211	C₂²	semidirect
31. #16863208	C₂²	semidirect
34. #368×C₆₇	C₂²	semidirect
35. #368×C₆₇	C₂²	semidirect
190. C₂×#368	D₆₇	semidirect
192. C₆₇	#33617	semidirect
216. #368	C₂×D₆₇	semidirect
217. #368	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863828	2
C₂×#16863812	2
#22569930	2
#22562340	2
#22562335	2
#22562330	2
#22561956	2
C₂×#368×C₆₇	2
#33617	67