Subgroups — #30900952

Order 17152.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569061	C₂	1	1	yes	maximal, characteristic, complemented
2	#22561669	C₂	1	2	yes	maximal, complemented
3	#22561479	C₂	1	2	yes	maximal, complemented
4	#22561120	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560997	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560698	C₂	1	2	yes	maximal, complemented
7	#22560334	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560318	C₂	1	2	yes	maximal, complemented
9	#22560263	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560009	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#472×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#16863871	C₂²	1	2	yes	—
13	#16863508	—	2	2	no	—
14	D₈×Dic₆₇	—	2	2	no	—
15	#16863444	C₂²	1	2	yes	—
16	C₂×#14523979	C₂²	1	1	yes	characteristic
17	#16863434	C₂²	1	1	yes	characteristic
18	#16863413	C₂²	1	2	yes	—
19	#16863401	C₂²	1	2	yes	—
20	#16863401	C₂²	1	2	yes	—
21	#16863380	C₂²	1	1	yes	characteristic
22	C₂×#14523960	C₂²	1	2	yes	—
23	#16863370	C₂²	1	1	yes	characteristic
24	#16863351	C₂²	1	1	yes	characteristic
25	#16863351	C₂²	1	1	yes	characteristic
26	#16863168	C₂²	1	2	yes	—
27	#16863168	C₂²	1	2	yes	—
28	#16863167	C₂²	1	1	yes	characteristic
29	#16863167	C₂²	1	1	yes	characteristic
30	#16863167	C₂²	1	2	yes	—
31	#16863153	C₂²	1	2	yes	—
32	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
33	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
35	#472×C₆₇	—	2	2	no	—
36	#472×C₆₇	—	2	2	no	—
37	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#100×C₆₇	C₂²	1	2	yes	—
39	C₂×#11761612	C₂³	1	1	yes	characteristic
40	C₂×#11761612	C₂³	1	1	yes	characteristic
41	C₂×#11761612	C₂³	1	2	yes	—
42	D₄×Dic₆₇	—	2	4	no	—
43	D₄×Dic₆₇	—	4	2	no	—
44	#14524002	C₂³	1	1	yes	characteristic
45	#14524002	C₂³	1	2	yes	—
46	#14524002	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
48	#14523979	—	2	1	no	—
49	#14523979	—	2	1	no	—
50	#14523978	—	4	2	no	—
51	#14523977	—	2	1	no	—
52	#14523975	—	2	1	no	—
53	#14523970	—	4	2	no	—
54	#14523964	—	2	4	no	—
55	#14523964	—	4	2	no	—
56	#14523962	—	2	1	no	—
57	#14523960	—	2	1	no	—
58	#14523960	—	4	2	no	—
59	#14523958	—	2	2	no	—
60	#14523953	—	2	1	no	—
61	#14523953	—	2	1	no	—
62	C₈×Dic₆₇	—	2	1	no	—
63	C₈×Dic₆₇	—	2	1	no	—
64	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
66	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
67	AGL(1,ℤ/8)×C₆₇	—	4	2	no	—
68	C₂×SD₁₆×C₆₇	—	2	2	no	—
69	C₂×D₈×C₆₇	—	2	2	no	—
70	C₂×D₈×C₆₇	—	2	1	no	—
71	C₂×D₈×C₆₇	—	2	1	no	—
72	C₂×D₈×C₆₇	—	2	1	no	—
73	C₂×D₈×C₆₇	—	2	1	no	—
74	C₂×D₈×C₆₇	D₄	1	2	yes	—
75	C₂×D₈×C₆₇	D₄	1	2	yes	—
76	#131×C₆₇	—	2	1	no	—
77	#131×C₆₇	—	2	1	no	—
78	#131×C₆₇	—	2	1	no	—
79	#131×C₆₇	—	2	1	no	—
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
83	C₂²×C₈×C₆₇	—	2	1	no	—
84	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	#108×C₆₇	—	2	1	no	—
86	#108×C₆₇	—	2	1	no	—
87	#100×C₆₇	—	2	4	no	—
88	C₂²×Dic₆₇	—	4	4	no	—
89	#11761612	—	2	1	no	—
90	#11761612	—	2	1	no	—
91	#11761612	—	2	2	no	—
92	#11761612	—	2	1	no	—
93	#11761612	—	2	2	no	—
94	#11761612	—	2	1	no	—
95	C₂×C₆₇⋊C₈	—	2	1	no	—
96	C₂×C₆₇⋊C₈	—	2	1	no	—
97	C₂×C₆₇⋊C₈	—	2	2	no	—
98	C₄⋊Dic₆₇	—	2	1	no	—
99	C₄⋊Dic₆₇	—	2	2	no	—
100	C₄⋊Dic₆₇	—	2	1	no	—
101	C₂²⋊Dic₆₇	—	2	1	no	—
102	C₂²⋊Dic₆₇	—	2	2	no	—
103	C₂²⋊Dic₆₇	—	4	4	no	—
104	C₂²⋊Dic₆₇	#41	1	2	yes	—
105	C₄×Dic₆₇	—	2	1	no	—
106	C₄×Dic₆₇	—	2	1	no	—
107	C₄×Dic₆₇	—	2	2	no	—
108	C₂⁴×C₆₇	—	2	2	no	—
109	#41×C₆₇	—	2	1	no	—
110	#41×C₆₇	—	2	1	no	—
111	#41×C₆₇	—	2	1	no	—
112	#41×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
114	C₂×D₄×C₆₇	—	2	1	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
121	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
124	SD₁₆×C₆₇	—	4	4	no	—
125	D₈×C₆₇	—	2	2	no	—
126	D₈×C₆₇	—	2	2	no	—
127	D₈×C₆₇	—	2	2	no	—
128	D₈×C₆₇	—	2	2	no	—
129	D₈×C₆₇	—	4	4	no	—
130	M₁₆×C₆₇	—	2	1	no	—
131	M₁₆×C₆₇	—	2	2	no	—
132	M₁₆×C₆₇	—	2	1	no	—
133	M₁₆×C₆₇	—	2	2	no	—
134	M₁₆×C₆₇	—	4	1	no	—
135	M₁₆×C₆₇	—	4	1	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	—	4	1	no	—
139	C₂×C₈×C₆₇	—	4	1	no	—
140	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
143	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
144	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
145	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
146	C₂³×C₆₇	—	2	1	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	16	2	no	—
151	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	1	no	—
153	Q₈×C₆₇	—	2	1	no	—
154	D₄×C₆₇	—	2	4	no	—
155	D₄×C₆₇	—	2	1	no	—
156	D₄×C₆₇	—	2	4	no	—
157	D₄×C₆₇	—	2	1	no	—
158	D₄×C₆₇	—	4	1	no	—
159	D₄×C₆₇	—	4	1	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	4	1	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×Dic₆₇	—	2	2	no	—
172	C₂×Dic₆₇	—	4	2	no	—
173	C₂×Dic₆₇	—	4	1	no	—
174	C₂×Dic₆₇	—	16	4	no	—
175	C₈×C₆₇	—	4	1	no	—
176	C₈×C₆₇	—	4	2	no	—
177	C₈×C₆₇	#123	1	1	yes	characteristic
178	C₈×C₆₇	#123	1	1	yes	characteristic
179	C₈×C₆₇	#123	1	1	yes	characteristic
180	C₈×C₆₇	#123	1	1	yes	characteristic
181	C₆₇⋊C₈	—	4	2	no	—
182	C₆₇⋊C₈	—	4	1	no	—
183	C₆₇⋊C₈	—	4	1	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	1	no	—
186	C₂²×C₆₇	—	4	4	no	—
187	C₂²×C₆₇	—	8	4	no	—
188	C₂²×C₆₇	—	8	1	no	—
189	C₂²×C₆₇	—	16	2	no	—
190	C₂²×C₆₇	—	16	2	no	—
191	C₂²×C₆₇	#529	1	1	yes	characteristic
192	C₂²×C₆₇	#546	1	1	yes	characteristic
193	C₂²×C₆₇	#555	1	1	yes	characteristic
194	C₄×C₆₇	—	4	1	no	—
195	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
196	C₄×C₆₇	#540	1	1	yes	characteristic
197	C₄×C₆₇	#546	1	1	yes	characteristic
198	C₄×C₆₇	#553	1	1	yes	characteristic
199	Dic₆₇	—	4	2	no	—
200	Dic₆₇	—	8	1	no	—
201	Dic₆₇	—	8	2	no	—
202	#24988	—	67	1	no	maximal, Sylow
203	C₂×C₆₇	—	2	1	no	—
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	1	no	—
206	C₂×C₆₇	—	8	1	no	—
207	C₂×C₆₇	—	8	4	no	—
208	C₂×C₆₇	#2420	1	1	yes	characteristic
209	C₂×#472	D₆₇	1	1	yes	characteristic
210	#2948	—	67	1	no	—
211	#2081	—	67	2	no	—
212	#2048	—	67	2	no	—
213	#1978	—	67	1	no	—
214	#1953	—	67	1	no	—
215	#1904	—	67	2	no	—
216	#1833	—	67	1	no	—
217	#1829	—	67	2	no	—
218	#1814	—	67	1	no	—
219	#1757	—	67	1	no	—
220	C₆₇	#24988	1	1	yes	characteristic, complemented, Sylow
221	C₂×AGL(1,ℤ/8)	C₂×D₆₇	1	2	yes	—
222	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
223	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
224	#518	—	67	2	no	—
225	#472	—	2	2	no	—
226	#472	—	2	2	no	—
227	#442	—	134	2	no	—
228	C₄×D₈	—	134	2	no	—
229	#430	—	67	2	no	—
230	C₂×#108	—	67	1	no	—
231	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
232	#427	—	67	1	no	—
233	#421	—	67	2	no	—
234	#418	—	67	2	no	—
235	#418	—	67	2	no	—
236	#413	—	67	1	no	—
237	C₂×#100	—	67	2	no	—
238	C₂×#100	C₂×D₆₇	1	2	yes	—
239	#410	—	67	1	no	—
240	#405	—	67	1	no	—
241	#405	—	67	1	no	—
242	#344	—	67	2	no	—
243	#344	—	67	2	no	—
244	#344	—	67	1	no	—
245	#344	—	67	1	no	—
246	#344	—	67	2	no	—
247	H(3,ℤ/4)	—	67	2	no	—
248	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
249	C₂²×D₄	C₂²×D₆₇	1	2	yes	—
250	AGL(1,ℤ/8)	—	2	4	no	—
251	AGL(1,ℤ/8)	—	4	2	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×D₈	—	2	2	no	—
254	C₂×D₈	—	2	1	no	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	2	1	no	—
258	C₂×D₈	C₂²⋊D₆₇	1	2	yes	—
259	C₂×D₈	C₂²⋊D₆₇	1	2	yes	—
260	#131	—	2	1	no	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	#131	—	2	1	no	—
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	—	67	1	no	—
266	C₂×M₁₆	—	67	2	no	—
267	C₂×M₁₆	—	67	1	no	—
268	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
269	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
272	C₄×D₄	—	268	2	no	—
273	C₄×D₄	—	134	4	no	—
274	#117	—	67	1	no	—
275	#117	—	67	2	no	—
276	#117	—	67	1	no	—
277	C₂×C₂²⋊C₄	—	134	4	no	—
278	#108	—	2	1	no	—
279	#108	—	2	1	no	—
280	#108	—	268	2	no	—
281	#108	—	134	1	no	—
282	#108	—	134	1	no	—
283	#107	—	134	1	no	—
284	#106	—	134	1	no	—
285	#104	—	268	2	no	—
286	#102	—	268	2	no	—
287	#102	—	134	4	no	—
288	#101	—	134	1	no	—
289	#100	—	2	4	no	—
290	#100	—	268	2	no	—
291	#100	—	134	1	no	—
292	#99	—	134	2	no	—
293	#97	—	134	1	no	—
294	#97	—	134	1	no	—
295	C₄×C₈	—	134	1	no	—
296	C₄×C₈	—	134	1	no	—
297	C₂⁴	—	2	2	no	—
298	#41	—	2	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	—	2	1	no	—
302	C₂×Q₈	#11761627	1	1	yes	characteristic
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	C₂×C₂²⋊D₆₇	1	4	yes	—
310	C₂×D₄	#11761627	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	268	4	no	—
313	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
314	SD₁₆	—	4	4	no	—
315	D₈	—	2	2	no	—
316	D₈	—	2	2	no	—
317	D₈	—	2	2	no	—
318	D₈	—	2	2	no	—
319	D₈	—	4	4	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	2	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	4	1	no	—
325	M₁₆	—	4	1	no	—
326	M₁₆	—	134	1	no	—
327	M₁₆	—	134	1	no	—
328	M₁₆	—	134	2	no	—
329	M₁₆	—	134	1	no	—
330	M₁₆	—	134	1	no	—
331	M₁₆	—	134	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	134	1	no	—
337	C₂×C₈	—	134	1	no	—
338	C₂×C₈	—	134	2	no	—
339	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
340	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
341	C₂×C₈	#11761627	1	1	yes	characteristic
342	C₂×C₈	#11761627	1	1	yes	characteristic
343	C₂×C₈	#11761627	1	1	yes	characteristic
344	C₂×C₈	#11761627	1	1	yes	characteristic
345	C₄⋊C₄	—	134	2	no	—
346	C₄⋊C₄	—	134	1	no	—
347	C₄⋊C₄	—	134	1	no	—
348	C₂²⋊C₄	—	268	4	no	—
349	C₂²⋊C₄	—	67	2	no	—
350	C₂²⋊C₄	—	134	1	no	—
351	C₂²⋊C₄	—	134	2	no	—
352	C₄²	—	134	1	no	—
353	C₄²	—	134	2	no	—
354	C₄²	—	134	1	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	2	1	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	16	2	no	—
360	C₂³	#41×D₆₇	1	1	yes	characteristic
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	D₄	—	2	1	no	—
364	D₄	—	2	4	no	—
365	D₄	—	2	1	no	—
366	D₄	—	2	4	no	—
367	D₄	—	4	1	no	—
368	D₄	—	4	1	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	268	2	no	—
375	C₂×C₄	—	268	1	no	—
376	C₂×C₄	—	1072	4	no	—
377	C₂×C₄	—	134	2	no	—
378	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
380	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
381	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
382	C₂×C₄	#14524088	1	1	yes	characteristic
383	C₂×C₄	#14524089	1	1	yes	characteristic
384	C₈	—	4	2	no	—
385	C₈	—	4	1	no	—
386	C₈	—	268	1	no	—
387	C₈	—	268	1	no	—
388	C₈	—	268	2	no	—
389	C₈	#14524024	1	1	yes	characteristic
390	C₈	#14524024	1	1	yes	characteristic
391	C₈	#14524024	1	1	yes	characteristic
392	C₈	#14524024	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	8	4	no	—
397	C₂²	—	8	1	no	—
398	C₂²	—	16	2	no	—
399	C₂²	—	16	2	no	—
400	C₂²	#16863927	1	1	yes	characteristic
401	C₂²	#16864034	1	1	yes	characteristic
402	C₂²	#16864110	1	1	yes	characteristic
403	C₄	—	4	1	no	—
404	C₄	—	268	2	no	—
405	C₄	—	536	2	no	—
406	C₄	—	536	1	no	—
407	C₄	C₂×#14524024	1	1	yes	characteristic
408	C₄	#16863999	1	1	yes	characteristic
409	C₄	#16864040	1	1	yes	characteristic
410	C₄	#16864098	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	4	no	—
415	C₂	—	8	1	no	—
416	C₂	#22563697	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22563697
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₂×#472×C₆₇	C₂
socle	C₂×C₆₇	#2420
hypercenter	C₂×#472	D₆₇
hypocenter	C₆₇	#24988
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#472

Complements

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

N	H ≅ G/N	type
1. #22569061	C₂	semidirect
2. #22561669	C₂	semidirect
3. #22561479	C₂	semidirect
4. #22561120	C₂	semidirect
5. #22560997	C₂	semidirect
6. #22560698	C₂	semidirect
7. #22560334	C₂	semidirect
8. #22560318	C₂	semidirect
9. #22560263	C₂	semidirect
10. #22560009	C₂	semidirect
220. C₆₇	#24988	semidirect

Maximal subgroups

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

subgroup	index
#22569061	2
#22561669	2
#22561479	2
#22561120	2
#22560997	2
#22560698	2
#22560334	2
#22560318	2
#22560263	2
#22560009	2
C₂×#472×C₆₇	2
#24988	67