Subgroups — #31010409

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	#22569929	C₂	1	2	yes	maximal, complemented
3	#22562339	C₂	1	4	yes	maximal, complemented
4	#22562334	C₂	1	2	yes	maximal, complemented
5	#22562329	C₂	1	2	yes	maximal, complemented
6	C₂×#16863226	C₂	1	1	yes	maximal, characteristic, complemented
7	#22561959	C₂	1	2	yes	maximal, complemented
8	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#16864194	—	2	2	no	—
10	C₂×#14524065	C₂²	1	2	yes	—
11	#16863828	—	2	1	no	—
12	#16863828	—	2	1	no	—
13	#16863828	—	2	1	no	—
14	#16863828	—	2	1	no	—
15	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×#14523988	C₂²	1	2	yes	complemented
17	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic, complemented
18	#16863671	C₂²	1	4	yes	—
19	#16863666	C₂²	1	4	yes	—
20	#16863659	—	2	4	no	—
21	C₂×#14523979	C₂²	1	2	yes	—
22	#16863378	—	2	2	no	—
23	#16863226	C₂²	1	2	yes	complemented
24	#16863226	C₂²	1	2	yes	complemented
25	#16863213	C₂²	1	4	yes	complemented
26	#16863210	C₂²	1	4	yes	complemented
27	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
28	#504×C₆₇	—	2	1	no	—
29	#504×C₆₇	—	2	1	no	—
30	#504×C₆₇	C₂²	1	2	yes	—
31	#504×C₆₇	C₂²	1	2	yes	—
32	C₂×M₃₂×C₆₇	—	2	1	no	—
33	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₁₆×C₆₇	—	2	1	no	—
35	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#11761628	—	2	2	no	—
37	#14524074	—	4	2	no	—
38	#14524073	—	2	4	no	—
39	#14524069	—	4	2	no	—
40	#14524068	—	2	4	no	—
41	#14524065	—	2	2	no	—
42	#14524065	—	2	2	no	—
43	#14524065	—	2	2	no	—
44	#14524065	—	2	2	no	—
45	#14524063	—	4	4	no	—
46	C₂×Dic₂₆₈	C₂³	1	2	yes	—
47	C₂×#11761617	—	2	2	no	—
48	C₂×D₅₃₆	C₂³	1	2	yes	—
49	C₂×#11761612	—	2	2	no	—
50	Dic₅₃₆	—	2	1	no	—
51	Dic₅₃₆	—	2	1	no	—
52	#14523988	—	2	2	no	—
53	#14523988	—	2	2	no	—
54	D₁₀₇₂	—	2	1	no	—
55	D₁₀₇₂	—	2	1	no	—
56	#14523979	—	2	2	no	—
57	#14523979	C₂³	1	4	yes	—
58	#14523961	—	2	4	no	—
59	#14523959	—	2	4	no	—
60	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
61	#131×C₆₇	—	2	1	no	—
62	#131×C₆₇	—	2	1	no	—
63	#131×C₆₇	D₄	1	2	yes	—
64	#131×C₆₇	C₂³	1	2	yes	—
65	C₂×M₁₆×C₆₇	—	2	1	no	—
66	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	—	2	1	no	—
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	M₃₂×C₆₇	—	2	1	no	—
70	M₃₂×C₆₇	—	2	2	no	—
71	M₃₂×C₆₇	—	2	2	no	—
72	M₃₂×C₆₇	C₂³	1	2	yes	—
73	C₂×C₁₆×C₆₇	—	2	1	no	—
74	C₂×C₁₆×C₆₇	—	2	2	no	—
75	C₂×C₁₆×C₆₇	—	2	2	no	—
76	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
82	#11761628	—	4	2	no	—
83	#11761628	—	4	2	no	—
84	#11761628	—	4	2	no	—
85	#11761628	—	4	2	no	—
86	C₂×Dic₁₃₄	—	2	2	no	—
87	C₂×D₂₆₈	—	2	2	no	—
88	C₂×C₂²⋊D₆₇	—	4	2	no	—
89	C₂×C₄×D₆₇	—	4	2	no	—
90	Dic₂₆₈	—	2	2	no	—
91	Dic₂₆₈	—	2	2	no	—
92	#11761618	—	4	4	no	—
93	#11761617	—	4	2	no	—
94	#11761617	—	4	2	no	—
95	#11761616	—	4	4	no	—
96	#11761615	—	4	4	no	—
97	D₅₃₆	—	2	2	no	—
98	D₅₃₆	—	2	2	no	—
99	#11761613	—	4	4	no	—
100	#11761612	—	2	4	no	—
101	#11761612	—	4	2	no	—
102	C₂×C₆₇⋊C₈	—	4	2	no	—
103	#41×C₆₇	—	2	1	no	—
104	#41×C₆₇	—	2	1	no	—
105	#41×C₆₇	D₈	1	2	yes	—
106	#41×C₆₇	C₂×D₄	1	2	yes	—
107	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
108	C₂×D₄×C₆₇	—	2	1	no	—
109	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	M₁₆×C₆₇	—	2	1	no	—
113	M₁₆×C₆₇	—	2	2	no	—
114	M₁₆×C₆₇	—	2	2	no	—
115	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
123	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
124	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₁₆×C₆₇	—	2	2	no	—
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	—	2	1	no	—
131	C₂³×C₆₇	#135	1	1	yes	characteristic
132	C₂²×D₆₇	—	4	2	no	—
133	Q₈×C₆₇	—	2	1	no	—
134	Q₈×C₆₇	C₂×D₈	1	2	yes	—
135	D₄×C₆₇	—	2	2	no	—
136	D₄×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	2	1	no	—
138	D₄×C₆₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₆₇	—	8	2	no	—
149	C₂²⋊D₆₇	—	8	2	no	—
150	C₂×Dic₆₇	—	4	2	no	—
151	D₂₆₈	—	4	2	no	—
152	D₂₆₈	—	4	2	no	—
153	C₄×D₆₇	—	8	2	no	—
154	C₄×D₆₇	—	8	2	no	—
155	Dic₁₃₄	—	4	2	no	—
156	Dic₁₃₄	—	4	2	no	—
157	C₈×C₆₇	—	2	2	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	—	4	4	no	—
163	C₂²×C₆₇	—	2	2	no	—
164	C₂²×C₆₇	—	4	1	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	#466	1	1	yes	characteristic
167	C₂²×C₆₇	#466	1	1	yes	characteristic
168	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
169	C₂×D₆₇	—	8	2	no	—
170	C₂×D₆₇	—	16	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	#466	1	1	yes	characteristic
173	C₄×C₆₇	#466	1	1	yes	characteristic
174	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
175	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
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	—	4	2	no	—
182	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
183	D₆₇	—	16	2	no	—
184	C₂×#508	—	67	1	no	—
185	C₂×#504	D₆₇	1	1	yes	characteristic, complemented
186	#3067	—	67	2	no	—
187	#2223	—	67	4	no	—
188	#2222	—	67	2	no	—
189	#2221	—	67	2	no	—
190	C₂×#368	—	67	1	no	—
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	C₂×D₆₇	1	1	yes	characteristic
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	—	2	1	no	—
204	#504	—	2	1	no	—
205	#504	C₂×D₆₇	1	2	yes	—
206	#504	C₂×D₆₇	1	2	yes	—
207	C₂×M₃₂	—	2	1	no	—
208	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
209	C₂²×C₁₆	—	2	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	—	67	2	no	—
215	#413	—	134	2	no	—
216	#368	—	67	2	no	—
217	#368	—	67	2	no	—
218	#362	—	67	4	no	—
219	#361	—	67	4	no	—
220	C₂×#41	—	134	2	no	—
221	C₂×#41	D₂₆₈	1	1	yes	characteristic
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	—	2	1	no	—
235	#131	—	2	1	no	—
236	#131	D₂₆₈	1	2	yes	—
237	#131	C₂²×D₆₇	1	2	yes	—
238	C₂×M₁₆	—	2	1	no	—
239	C₂×M₁₆	—	134	2	no	—
240	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
241	C₂²×C₈	—	2	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	—	2	1	no	—
250	M₃₂	—	2	2	no	—
251	M₃₂	—	2	2	no	—
252	M₃₂	C₂²×D₆₇	1	2	yes	—
253	C₂×C₁₆	—	2	1	no	—
254	C₂×C₁₆	—	2	2	no	—
255	C₂×C₁₆	—	2	2	no	—
256	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
257	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
258	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
259	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
262	#108	—	67	4	no	—
263	#108	—	134	2	no	—
264	#101	—	134	4	no	—
265	#100	—	134	4	no	—
266	#41	—	2	1	no	—
267	#41	—	2	1	no	—
268	#41	—	268	2	no	—
269	#41	—	268	2	no	—
270	#41	—	268	2	no	—
271	#41	—	268	2	no	—
272	#41	D₅₃₆	1	2	yes	—
273	#41	C₂×D₂₆₈	1	2	yes	—
274	C₂×Q₈	—	134	2	no	—
275	C₂×Q₈	D₅₃₆	1	1	yes	characteristic
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	—	268	2	no	—
278	C₂×D₄	—	134	2	no	—
279	C₂×D₄	D₅₃₆	1	1	yes	characteristic
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	268	2	no	—
282	C₂²×C₄	C₂×D₂₆₈	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	1	no	—
294	M₁₆	—	2	2	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	268	2	no	—
297	M₁₆	—	134	4	no	—
298	M₁₆	C₂×D₂₆₈	1	2	yes	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	268	2	no	—
303	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
304	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
306	C₂×C₈	#11761628	1	1	yes	characteristic
307	C₂×C₈	#11761628	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
309	C₁₆	—	2	2	no	—
310	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
311	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
312	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
313	C₁₆	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
314	C₂³	—	2	1	no	—
315	C₂³	—	268	2	no	—
316	C₂³	#14524065	1	1	yes	characteristic
317	Q₈	—	2	1	no	—
318	Q₈	—	268	2	no	—
319	Q₈	—	268	2	no	—
320	Q₈	C₂×D₅₃₆	1	2	yes	—
321	D₄	—	2	2	no	—
322	D₄	—	2	2	no	—
323	D₄	—	2	1	no	—
324	D₄	—	268	2	no	—
325	D₄	—	268	2	no	—
326	D₄	—	536	2	no	—
327	D₄	—	536	2	no	—
328	D₄	C₂×D₅₃₆	1	2	yes	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	268	2	no	—
333	C₂×C₄	—	536	2	no	—
334	C₂×C₄	—	536	2	no	—
335	C₂×C₄	#14524017	1	1	yes	characteristic
336	C₂×C₄	#14524017	1	1	yes	characteristic
337	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
338	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
339	C₂×C₄	#14524065	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	268	4	no	—
343	C₈	#14524017	1	1	yes	characteristic
344	C₈	#14524017	1	1	yes	characteristic
345	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
346	C₈	C₂×#11761628	1	1	yes	characteristic
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	536	2	no	—
351	C₂²	—	1072	2	no	—
352	C₂²	#16863632	1	1	yes	characteristic
353	C₂²	#16863632	1	1	yes	characteristic
354	C₂²	C₂×#14524065	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	536	2	no	—
357	C₄	#16863632	1	1	yes	characteristic
358	C₄	#16863632	1	1	yes	characteristic
359	C₄	C₂×#14524017	1	1	yes	characteristic
360	C₄	C₂²×D₅₃₆	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	2	no	—
365	C₂	—	1072	2	no	—
366	C₂	C₂×#16863632	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863632
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂×C₆₇	C₂×#466
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#33617
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16863828	C₂	semidirect
2. #22569929	C₂	semidirect
3. #22562339	C₂	semidirect
4. #22562334	C₂	semidirect
5. #22562329	C₂	semidirect
6. C₂×#16863226	C₂	semidirect
7. #22561959	C₂	semidirect
8. C₂×#504×C₆₇	C₂	semidirect
15. C₂×Dic₅₃₆	C₂²	semidirect
16. C₂×#14523988	C₂²	semidirect
17. C₂×D₁₀₇₂	C₂²	semidirect
23. #16863226	C₂²	semidirect
24. #16863226	C₂²	semidirect
25. #16863213	C₂²	semidirect
26. #16863210	C₂²	semidirect
185. C₂×#504	D₆₇	semidirect
192. C₆₇	#33617	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
#22569929	2
#22562339	2
#22562334	2
#22562329	2
C₂×#16863226	2
#22561959	2
C₂×#504×C₆₇	2
#33617	67