Subgroups — #30914721

Order 17152.

674 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570737	C₂	1	1	yes	maximal, characteristic, complemented
2	#22570065	C₂	1	2	yes	maximal
3	#22568800	C₂	1	1	yes	maximal, characteristic
4	#22561715	C₂	1	2	yes	maximal, complemented
5	#22561559	C₂	1	2	yes	maximal, complemented
6	#22560987	C₂	1	1	yes	maximal, characteristic, complemented
7	#22560957	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560441	C₂	1	2	yes	maximal
9	#22560269	C₂	1	1	yes	maximal, characteristic
10	#22560046	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524049	C₂²	1	1	yes	characteristic
13	#16864027	C₂²	1	2	yes	—
14	#16863877	C₂²	1	2	yes	—
15	#16863803	—	2	2	no	—
16	#16863796	—	2	1	no	—
17	#16863793	—	2	1	no	—
18	#16863702	—	2	2	no	—
19	#16863699	—	2	2	no	—
20	#16863690	—	2	2	no	—
21	#16863675	—	2	2	no	—
22	C₂×#14523979	C₂²	1	1	yes	characteristic
23	#16863436	C₂²	1	2	yes	—
24	#16863436	C₂²	1	2	yes	—
25	#16863436	C₂²	1	2	yes	—
26	#16863435	—	2	2	no	—
27	#16863434	C₂²	1	1	yes	characteristic
28	#16863431	—	2	1	no	—
29	#106×D₆₇	—	2	1	no	—
30	#16863422	C₂²	1	2	yes	—
31	C₂×#14523971	C₂²	1	1	yes	characteristic
32	#16863395	C₂²	1	2	yes	—
33	#16863393	C₂²	1	1	yes	characteristic
34	#16863351	C₂²	1	1	yes	characteristic
35	#16863165	C₂²	1	2	yes	—
36	#16863165	C₂²	1	2	yes	—
37	#16863161	C₂²	1	1	yes	characteristic
38	#16863156	C₂²	1	1	yes	characteristic
39	#16863156	C₂²	1	2	yes	—
40	C₂×#107×C₆₇	C₂²	1	2	yes	—
41	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	#382×C₆₇	C₂²	1	2	yes	—
45	C₂×#11761628	C₂³	1	1	yes	characteristic
46	#14524049	—	2	2	no	—
47	#14524049	Q₈	1	4	yes	complemented
48	C₂×#11761612	C₂³	1	2	yes	—
49	C₂×#11761612	C₂³	1	1	yes	characteristic
50	C₂×#11761612	C₂³	1	1	yes	characteristic
51	C₂×#11761611	—	2	1	no	—
52	C₂×C₈×D₆₇	—	2	1	no	—
53	#14524042	—	2	2	no	—
54	#14524042	—	2	2	no	—
55	#14524034	—	2	2	no	—
56	#14524034	—	2	2	no	—
57	#14524022	—	4	2	no	—
58	#14524008	—	2	2	no	—
59	#14524005	—	2	2	no	—
60	C₄×C₂²⋊D₆₇	—	4	2	no	—
61	#14524002	C₂³	1	1	yes	characteristic
62	#14524002	C₂³	1	2	yes	—
63	#14524001	—	2	2	no	—
64	C₄⋊C₄×D₆₇	—	2	2	no	—
65	#14523979	—	2	2	no	—
66	#14523977	D₄	1	2	yes	—
67	#14523976	—	2	2	no	—
68	#14523976	—	2	2	no	—
69	#14523976	—	2	2	no	—
70	#14523975	D₄	1	1	yes	characteristic
71	#14523975	D₄	1	1	yes	characteristic
72	#14523974	—	2	2	no	—
73	#14523974	—	2	2	no	—
74	#14523974	—	2	2	no	—
75	Dic₆₇⋊C₈	—	2	2	no	—
76	Dic₆₇⋊C₈	—	2	2	no	—
77	#14523971	Q₈	1	4	yes	complemented
78	#14523968	D₄	1	2	yes	—
79	#14523966	—	2	2	no	—
80	#14523965	D₄	1	2	yes	—
81	#14523963	—	2	2	no	—
82	#14523953	—	2	1	no	—
83	C₈×Dic₆₇	—	2	1	no	—
84	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	#107×C₆₇	—	2	2	no	—
90	#107×C₆₇	—	2	2	no	—
91	#106×C₆₇	—	2	1	no	—
92	#106×C₆₇	—	2	1	no	—
93	#106×C₆₇	—	2	1	no	—
94	#106×C₆₇	—	2	1	no	—
95	C₄×C₈×C₆₇	—	2	2	no	—
96	#95×C₆₇	—	4	2	no	—
97	#11761628	—	2	1	no	—
98	#11761628	—	2	1	no	—
99	#11761628	C₂×Q₈	1	2	yes	—
100	#11761628	C₂×Q₈	1	2	yes	—
101	C₂×Dic₁₃₄	—	4	2	no	—
102	C₂×Dic₁₃₄	C₂×D₄	1	1	yes	characteristic
103	C₂×D₂₆₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₂²⋊D₆₇	—	2	1	no	—
105	C₂×C₄×D₆₇	—	2	1	no	—
106	C₂×C₄×D₆₇	—	4	2	no	—
107	#11761612	—	2	1	no	—
108	#11761612	—	2	1	no	—
109	#11761612	—	2	2	no	—
110	#11761612	—	2	2	no	—
111	#11761612	C₂×Q₈	1	2	yes	—
112	#11761612	C₂×Q₈	1	2	yes	—
113	#11761611	—	2	4	no	—
114	C₂×C₆₇⋊C₈	—	2	1	no	—
115	C₂×C₆₇⋊C₈	—	2	1	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₈×D₆₇	—	2	4	no	—
118	C₄⋊Dic₆₇	—	4	2	no	—
119	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
120	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
121	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
122	Dic₆₇⋊C₄	—	4	2	no	—
123	Dic₆₇⋊C₄	—	4	2	no	—
124	Dic₆₇⋊C₄	—	4	2	no	—
125	C₂²⋊Dic₆₇	—	2	2	no	—
126	C₂²⋊Dic₆₇	—	2	1	no	—
127	(C₂×D₆₇)⋊C₄	—	4	2	no	—
128	C₄×Dic₆₇	—	2	1	no	—
129	C₄×Dic₆₇	—	2	2	no	—
130	C₄×Dic₆₇	—	4	2	no	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
141	C₂×C₈×C₆₇	#41	1	2	yes	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
148	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂²×D₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	8	2	no	—
155	C₂×C₄×C₆₇	—	8	2	no	—
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂²⋊D₆₇	—	2	2	no	—
163	C₂²⋊D₆₇	—	2	2	no	—
164	C₂×Dic₆₇	—	2	1	no	—
165	C₂×Dic₆₇	—	2	2	no	—
166	C₂×Dic₆₇	—	2	2	no	—
167	C₂×Dic₆₇	—	2	2	no	—
168	D₂₆₈	—	2	1	no	—
169	D₂₆₈	C₄⋊Q₈	1	2	yes	—
170	C₄×D₆₇	—	2	2	no	—
171	C₄×D₆₇	—	2	2	no	—
172	C₄×D₆₇	—	16	2	no	—
173	Dic₁₃₄	—	2	1	no	—
174	Dic₁₃₄	—	16	2	no	—
175	Dic₁₃₄	C₄⋊Q₈	1	2	yes	—
176	C₈×C₆₇	—	2	2	no	—
177	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
178	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
179	C₆₇⋊C₈	—	2	2	no	—
180	C₆₇⋊C₈	—	2	2	no	—
181	C₆₇⋊C₈	—	4	2	no	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	#544	1	1	yes	characteristic
184	C₂²×C₆₇	D₄²	1	1	yes	characteristic
185	C₂²×C₆₇	#554	1	1	yes	characteristic
186	C₂×D₆₇	—	2	2	no	—
187	C₂×D₆₇	—	4	2	no	—
188	C₄×C₆₇	—	4	2	no	—
189	C₄×C₆₇	—	8	2	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
192	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
193	C₄×C₆₇	#547	1	1	yes	characteristic
194	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
195	Dic₆₇	—	2	2	no	—
196	Dic₆₇	—	4	2	no	—
197	Dic₆₇	—	4	2	no	—
198	Dic₆₇	—	4	2	no	—
199	#26080	—	67	1	no	maximal, Sylow
200	C₂×C₆₇	—	2	1	no	—
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	#2440	1	1	yes	characteristic
203	C₂×C₆₇	#3290	1	2	yes	—
204	D₆₇	—	4	2	no	—
205	#3163	—	67	1	no	—
206	#3083	—	67	2	no	—
207	#2912	—	67	1	no	—
208	#2089	—	67	2	no	—
209	#2061	—	67	2	no	—
210	#1951	—	67	1	no	—
211	#1945	—	67	1	no	—
212	#1860	—	67	2	no	—
213	#1846	D₆₇	1	1	yes	characteristic, complemented
214	#1815	—	67	1	no	—
215	#1764	—	67	1	no	—
216	C₆₇	#26080	1	1	yes	characteristic, complemented, Sylow
217	C₂×#131	—	67	1	no	—
218	#544	—	67	2	no	—
219	#520	—	67	2	no	—
220	#501	—	134	2	no	—
221	#499	—	134	1	no	—
222	#498	—	134	1	no	—
223	#479	—	134	2	no	—
224	#478	—	134	2	no	—
225	#477	—	134	2	no	—
226	#474	—	134	2	no	—
227	C₂×#108	—	67	1	no	—
228	#428	—	67	2	no	—
229	#428	—	67	2	no	—
230	#428	—	67	2	no	—
231	#428	—	134	2	no	—
232	#427	—	67	1	no	—
233	#427	—	134	1	no	—
234	C₂×#107	C₂×D₆₇	1	2	yes	—
235	C₂×#106	—	134	1	no	—
236	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
237	C₂×#106	C₂×D₆₇	1	1	yes	characteristic
238	#424	—	67	2	no	—
239	C₂×#104	—	67	1	no	—
240	#417	—	67	2	no	—
241	#416	—	67	1	no	—
242	#405	—	67	1	no	—
243	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
244	#382	C₂×D₆₇	1	2	yes	—
245	#343	—	67	2	no	—
246	#343	—	67	2	no	—
247	#341	—	67	1	no	—
248	#339	—	67	1	no	—
249	#339	—	67	2	no	—
250	C₂×#41	—	67	1	no	—
251	#131	—	67	4	no	—
252	#131	—	134	2	no	—
253	C₂×M₁₆	—	67	1	no	—
254	C₂×M₁₆	—	67	1	no	—
255	C₂×M₁₆	—	67	2	no	—
256	C₂×M₁₆	—	134	1	no	—
257	C₂²×C₈	—	134	1	no	—
258	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
259	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
260	C₄⋊Q₈	—	134	2	no	—
261	C₄⋊Q₈	—	134	2	no	—
262	#125	—	134	2	no	—
263	#125	—	134	2	no	—
264	C₂²⋊Q₈	—	268	2	no	—
265	C₄×Q₈	—	134	2	no	—
266	C₄×D₄	—	268	2	no	—
267	C₄×D₄	—	134	2	no	—
268	#117	—	67	2	no	—
269	#117	—	67	1	no	—
270	#117	—	134	2	no	—
271	C₂×C₄⋊C₄	—	134	2	no	—
272	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
274	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
275	#108	—	134	2	no	—
276	#107	—	2	2	no	—
277	#107	—	2	2	no	—
278	#107	—	67	2	no	—
279	#107	—	134	2	no	—
280	#107	—	134	2	no	—
281	#107	—	134	2	no	—
282	#106	—	2	1	no	—
283	#106	—	2	1	no	—
284	#106	—	2	1	no	—
285	#106	—	2	1	no	—
286	#106	—	67	1	no	—
287	#106	—	67	1	no	—
288	#106	—	134	2	no	—
289	#106	—	134	2	no	—
290	#106	—	134	2	no	—
291	C₄⋊C₈	—	134	2	no	—
292	C₄⋊C₈	—	134	2	no	—
293	#104	—	67	4	no	—
294	#103	—	67	2	no	—
295	#103	—	134	2	no	—
296	#102	—	67	2	no	—
297	#102	—	134	2	no	—
298	#97	—	134	1	no	—
299	C₄×C₈	—	2	2	no	—
300	C₄×C₈	—	134	1	no	—
301	#95	—	4	2	no	—
302	#41	—	67	2	no	—
303	#41	—	67	2	no	—
304	#41	—	134	1	no	—
305	#41	—	134	1	no	—
306	C₂×Q₈	—	268	2	no	—
307	C₂×Q₈	—	67	1	no	—
308	C₂×D₄	—	67	1	no	—
309	C₂×D₄	—	134	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	268	2	no	—
314	C₂²×C₄	—	134	1	no	—
315	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
316	M₁₆	—	67	2	no	—
317	M₁₆	—	67	2	no	—
318	M₁₆	—	134	2	no	—
319	M₁₆	—	134	2	no	—
320	M₁₆	—	134	1	no	—
321	M₁₆	—	134	1	no	—
322	M₁₆	—	134	4	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	134	2	no	—
326	C₂×C₈	—	134	1	no	—
327	C₂×C₈	—	134	1	no	—
328	C₂×C₈	—	134	4	no	—
329	C₂×C₈	D₄×D₆₇	1	2	yes	—
330	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
331	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
332	C₂×C₈	Q₈×D₆₇	1	2	yes	—
333	C₂×C₈	#11761628	1	2	yes	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	268	2	no	—
339	C₄⋊C₄	—	268	2	no	—
340	C₄⋊C₄	—	268	2	no	—
341	C₄⋊C₄	—	268	2	no	—
342	C₄⋊C₄	—	67	2	no	—
343	C₄⋊C₄	—	67	2	no	—
344	C₄⋊C₄	—	67	2	no	—
345	C₂²⋊C₄	—	268	2	no	—
346	C₂²⋊C₄	—	134	2	no	—
347	C₂²⋊C₄	—	134	1	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	268	2	no	—
350	C₄²	—	134	1	no	—
351	C₄²	—	134	2	no	—
352	C₄²	Q₈×D₆₇	1	2	yes	—
353	C₂³	—	134	1	no	—
354	C₂³	#14524088	1	1	yes	characteristic
355	Q₈	—	1072	2	no	—
356	Q₈	—	67	2	no	—
357	Q₈	—	134	1	no	—
358	D₄	—	67	2	no	—
359	D₄	—	134	2	no	—
360	D₄	—	134	2	no	—
361	D₄	—	134	1	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	8	2	no	—
368	C₂×C₄	—	1072	2	no	—
369	C₂×C₄	—	134	2	no	—
370	C₂×C₄	—	134	2	no	—
371	C₂×C₄	—	134	2	no	—
372	C₂×C₄	—	134	1	no	—
373	C₂×C₄	—	134	2	no	—
374	C₂×C₄	—	134	2	no	—
375	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
377	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
378	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
380	C₂×C₄	#14524091	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	268	2	no	—
383	C₈	—	134	2	no	—
384	C₈	—	134	2	no	—
385	C₈	#14524021	1	2	yes	—
386	C₈	#14524021	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	268	2	no	—
389	C₂²	—	134	2	no	—
390	C₂²	#16864023	1	1	yes	characteristic
391	C₂²	#16864031	1	1	yes	characteristic
392	C₂²	#16864101	1	1	yes	characteristic
393	C₄	—	4	2	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	—	268	2	no	—
397	C₄	—	268	2	no	—
398	C₄	—	268	2	no	—
399	C₄	—	134	2	no	—
400	C₄	C₂×#14524021	1	1	yes	characteristic
401	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
402	C₄	#16864047	1	1	yes	characteristic
403	C₄	#16864061	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	268	2	no	—
407	C₂	#22563914	1	1	yes	characteristic
408	C₂	#22571994	1	2	yes	—

	subgroup	G/H
center	C₂²	#16864031
commutator, cosocle	C₂²×C₄×C₆₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1846	C₂
socle	C₂²×C₆₇	D₄²
prime radical	C₂×#11761628	C₂³
hypercenter	#1846	D₆₇
hypocenter	C₆₇	#26080
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1846

Complements

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

N	H ≅ G/N	type
1. #22570737	C₂	semidirect
4. #22561715	C₂	semidirect
5. #22561559	C₂	semidirect
6. #22560987	C₂	semidirect
7. #22560957	C₂	semidirect
11. C₆₇×#1846	C₂	semidirect
47. #14524049	Q₈	semidirect
77. #14523971	Q₈	semidirect
213. #1846	D₆₇	semidirect
216. C₆₇	#26080	semidirect

Maximal subgroups

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

subgroup	index
#22570737	2
#22570065	2
#22568800	2
#22561715	2
#22561559	2
#22560987	2
#22560957	2
#22560441	2
#22560269	2
#22560046	2
C₆₇×#1846	2
#26080	67