Subgroups — #30875534

Order 17152.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524065	C₂	1	1	yes	maximal, characteristic
2	#22568874	C₂	1	2	yes	maximal, complemented
3	#22561102	C₂	1	2	yes	maximal
4	#22560858	C₂	1	4	yes	maximal, complemented
5	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560230	C₂	1	2	yes	maximal, complemented
7	#22560093	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14524065	C₂²	1	1	yes	characteristic
10	C₄×#11761628	C₂²	1	2	yes	—
11	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic, complemented
12	C₄×Dic₂₆₈	C₄	1	1	yes	characteristic, complemented
13	C₄×#11761617	C₄	1	2	yes	complemented
14	C₄×#11761617	C₄	1	2	yes	complemented
15	C₄×D₅₃₆	C₄	1	1	yes	characteristic, complemented
16	C₄×D₅₃₆	C₄	1	1	yes	characteristic, complemented
17	#16863456	—	2	4	no	—
18	#16863452	—	2	2	no	—
19	#16863448	—	2	2	no	—
20	C₂×#14523979	C₂²	1	2	yes	—
21	C₂×#14523979	C₂²	1	2	yes	—
22	#16863434	C₂²	1	1	yes	characteristic
23	#16863420	C₂²	1	2	yes	—
24	C₂×#14523971	C₂²	1	4	yes	—
25	#16863393	C₂²	1	2	yes	—
26	#16863362	C₂²	1	4	yes	—
27	#16863351	C₂²	1	4	yes	—
28	#16863156	C₂²	1	4	yes	—
29	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	#336×C₆₇	C₂²	1	2	yes	—
34	C₂×#11761628	C₂³	1	2	yes	—
35	#14524065	—	2	2	no	—
36	#14524065	—	2	2	no	—
37	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
38	C₂×#11761617	C₂×C₄	1	2	yes	—
39	C₂×D₅₃₆	C₂×C₄	1	1	yes	characteristic
40	C₂×#11761612	C₂³	1	4	yes	—
41	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
43	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
44	C₄×D₂₆₈	C₂×C₄	1	2	yes	—
45	C₄×C₂²⋊D₆₇	—	2	2	no	—
46	C₄×C₂²⋊D₆₇	—	2	2	no	—
47	#14524002	C₂³	1	2	yes	—
48	#14524000	—	2	2	no	—
49	C₄²×D₆₇	—	2	2	no	—
50	#14523979	—	2	4	no	—
51	#14523979	—	4	2	no	—
52	#14523977	C₂×C₄	1	1	yes	characteristic
53	#14523977	C₂×C₄	1	1	yes	characteristic
54	#14523975	C₂×C₄	1	2	yes	—
55	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
57	#14523971	—	4	4	no	—
58	#14523968	C₂×C₄	1	2	yes	—
59	#14523968	C₂×C₄	1	2	yes	—
60	#14523965	C₂×C₄	1	2	yes	—
61	#14523965	C₂×C₄	1	2	yes	—
62	(C₂×D₆₇)⋊C₈	—	2	4	no	—
63	(C₂×D₆₇)⋊C₈	—	2	4	no	—
64	#14523953	—	2	4	no	—
65	C₈×Dic₆₇	—	2	4	no	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
73	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
74	#97×C₆₇	—	2	4	no	—
75	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	#95×C₆₇	—	2	1	no	—
80	#11761628	—	2	2	no	—
81	#11761628	—	2	2	no	—
82	#11761628	—	2	2	no	—
83	#11761628	—	2	2	no	—
84	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
85	C₂×D₂₆₈	C₂²×C₄	1	2	yes	—
86	C₂×C₂²⋊D₆₇	—	2	2	no	—
87	C₂×C₄×D₆₇	—	2	2	no	—
88	C₂×C₄×D₆₇	—	2	2	no	—
89	C₂×C₄×D₆₇	—	2	2	no	—
90	Dic₂₆₈	—	2	2	no	—
91	#11761617	—	2	4	no	—
92	D₅₃₆	—	2	2	no	—
93	#11761612	—	2	4	no	—
94	#11761612	—	2	4	no	—
95	C₂×C₆₇⋊C₈	—	2	4	no	—
96	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
97	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
98	Dic₆₇⋊C₄	—	2	2	no	—
99	Dic₆₇⋊C₄	—	2	2	no	—
100	C₂²⋊Dic₆₇	—	2	2	no	—
101	(C₂×D₆₇)⋊C₄	—	2	2	no	—
102	(C₂×D₆₇)⋊C₄	—	2	2	no	—
103	C₄×Dic₆₇	—	2	2	no	—
104	C₄×Dic₆₇	—	2	2	no	—
105	C₄×Dic₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
119	C₂×C₈×C₆₇	#41	1	2	yes	—
120	C₂×C₈×C₆₇	#41	1	2	yes	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
129	C₂²×D₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²⋊D₆₇	—	4	2	no	—
146	C₂²⋊D₆₇	—	4	2	no	—
147	C₂×Dic₆₇	—	2	2	no	—
148	C₂×Dic₆₇	—	2	2	no	—
149	C₂×Dic₆₇	—	2	2	no	—
150	D₂₆₈	—	2	2	no	—
151	D₂₆₈	—	2	2	no	—
152	C₄×D₆₇	—	4	2	no	—
153	C₄×D₆₇	—	4	2	no	—
154	C₄×D₆₇	—	4	4	no	—
155	C₄×D₆₇	—	8	2	no	—
156	Dic₁₃₄	—	2	2	no	—
157	Dic₁₃₄	—	2	2	no	—
158	C₈×C₆₇	—	2	1	no	—
159	C₈×C₆₇	—	2	1	no	—
160	C₈×C₆₇	—	2	4	no	—
161	C₈×C₆₇	#117	1	2	yes	—
162	C₈×C₆₇	#117	1	2	yes	—
163	C₆₇⋊C₈	—	4	4	no	—
164	C₂²×C₆₇	—	2	1	no	—
165	C₂²×C₆₇	—	2	1	no	—
166	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₆₇	#555	1	1	yes	characteristic
168	C₂²×C₆₇	#556	1	1	yes	characteristic
169	C₂×D₆₇	—	4	2	no	—
170	C₂×D₆₇	—	4	2	no	—
171	C₂×D₆₇	—	4	2	no	—
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	8	2	no	—
176	C₄×C₆₇	#444	1	2	yes	—
177	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
178	C₄×C₆₇	#520	1	1	yes	characteristic
179	C₄×C₆₇	#547	1	1	yes	characteristic
180	C₄×C₆₇	#550	1	1	yes	characteristic
181	Dic₆₇	—	4	2	no	—
182	Dic₆₇	—	4	2	no	—
183	Dic₆₇	—	4	2	no	—
184	#22803	—	67	1	no	maximal, Sylow
185	C₂×C₆₇	—	2	1	no	—
186	C₂×C₆₇	—	2	1	no	—
187	C₂×C₆₇	#2354	1	1	yes	characteristic
188	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
189	C₂×C₆₇	#2957	1	1	yes	characteristic
190	D₆₇	—	8	2	no	—
191	C₄×#135	—	67	1	no	—
192	#2925	—	67	2	no	—
193	#1975	—	67	2	no	—
194	#1929	—	67	4	no	—
195	#1848	—	67	1	no	—
196	#1806	—	67	2	no	—
197	#1773	—	67	2	no	—
198	#1770	D₆₇	1	1	yes	characteristic, complemented
199	C₆₇	#22803	1	1	yes	characteristic, complemented, Sylow
200	C₂×#135	—	67	1	no	—
201	C₄×#41	—	67	2	no	—
202	C₄×Q₁₆	—	67	1	no	—
203	C₄×Q₁₆	—	67	1	no	—
204	C₄×SD₁₆	—	67	2	no	—
205	C₄×SD₁₆	—	67	2	no	—
206	C₄×D₈	—	67	1	no	—
207	C₄×D₈	—	67	1	no	—
208	#433	—	134	4	no	—
209	#432	—	134	2	no	—
210	#431	—	134	2	no	—
211	C₂×#108	—	67	2	no	—
212	C₂×#108	—	67	2	no	—
213	#427	—	67	1	no	—
214	#423	—	67	2	no	—
215	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
216	C₂×#104	—	67	4	no	—
217	#416	—	67	2	no	—
218	#408	—	67	4	no	—
219	#405	—	67	4	no	—
220	C₂×#97	C₂×D₆₇	1	2	yes	—
221	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
222	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
223	#339	—	67	4	no	—
224	#336	C₂×D₆₇	1	2	yes	—
225	C₂×#41	—	67	2	no	—
226	#135	—	134	2	no	—
227	#135	—	134	2	no	—
228	C₂×Q₁₆	—	67	1	no	—
229	C₂×SD₁₆	—	67	2	no	—
230	C₂×D₈	—	67	1	no	—
231	C₂×M₁₆	—	67	4	no	—
232	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
234	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
235	C₄×Q₈	—	67	2	no	—
236	C₄×Q₈	—	67	2	no	—
237	C₄×D₄	—	67	2	no	—
238	C₄×D₄	—	67	2	no	—
239	C₄×D₄	—	134	2	no	—
240	C₄×D₄	—	134	2	no	—
241	#117	—	67	2	no	—
242	#117	—	134	2	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄²	—	134	2	no	—
245	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
246	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
247	#108	—	268	2	no	—
248	#108	—	134	4	no	—
249	#107	—	67	1	no	—
250	#107	—	67	1	no	—
251	#106	—	67	2	no	—
252	C₄⋊C₈	—	134	2	no	—
253	C₄⋊C₈	—	134	2	no	—
254	C₄⋊C₈	D₂₆₈	1	2	yes	—
255	C₄⋊C₈	D₂₆₈	1	2	yes	—
256	#104	—	268	4	no	—
257	#103	—	67	2	no	—
258	#103	—	67	2	no	—
259	#102	—	67	2	no	—
260	#102	—	67	2	no	—
261	C₂²⋊C₈	—	134	4	no	—
262	C₂²⋊C₈	—	134	4	no	—
263	#97	—	2	4	no	—
264	#97	—	134	4	no	—
265	C₄×C₈	—	134	4	no	—
266	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
267	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
268	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
269	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
270	#95	—	2	1	no	—
271	#41	—	134	2	no	—
272	#41	—	134	2	no	—
273	#41	—	134	2	no	—
274	#41	—	134	2	no	—
275	C₂×Q₈	—	67	2	no	—
276	C₂×D₄	—	67	2	no	—
277	C₂×D₄	—	134	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	134	2	no	—
280	C₂²×C₄	—	134	2	no	—
281	C₂²×C₄	—	134	2	no	—
282	C₂²×C₄	#11761628	1	1	yes	characteristic
283	C₂²×C₄	#11761628	1	1	yes	characteristic
284	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
285	Q₁₆	—	134	2	no	—
286	SD₁₆	—	134	4	no	—
287	D₈	—	134	2	no	—
288	M₁₆	—	134	4	no	—
289	M₁₆	—	134	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	134	4	no	—
295	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
299	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
300	C₂×C₈	#11761627	1	2	yes	—
301	C₂×C₈	#11761629	1	2	yes	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	67	2	no	—
304	C₄⋊C₄	—	67	2	no	—
305	C₄⋊C₄	—	134	2	no	—
306	C₄⋊C₄	—	134	2	no	—
307	C₂²⋊C₄	—	134	2	no	—
308	C₂²⋊C₄	—	134	2	no	—
309	C₂²⋊C₄	—	134	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	134	2	no	—
313	C₄²	—	134	2	no	—
314	C₄²	—	134	2	no	—
315	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
317	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
318	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
319	C₂³	—	134	2	no	—
320	C₂³	C₂×#11761628	1	1	yes	characteristic
321	Q₈	—	134	2	no	—
322	Q₈	—	134	2	no	—
323	D₄	—	268	2	no	—
324	D₄	—	268	2	no	—
325	D₄	—	134	2	no	—
326	D₄	—	134	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	268	4	no	—
333	C₂×C₄	—	268	2	no	—
334	C₂×C₄	—	268	2	no	—
335	C₂×C₄	—	536	2	no	—
336	C₂×C₄	—	134	2	no	—
337	C₂×C₄	—	134	2	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
340	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
341	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
342	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
345	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
346	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
347	C₂×C₄	#14524089	1	1	yes	characteristic
348	C₂×C₄	#14524090	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	268	4	no	—
353	C₈	#14524001	1	2	yes	—
354	C₈	#14524001	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	268	2	no	—
358	C₂²	—	268	2	no	—
359	C₂²	—	268	2	no	—
360	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
361	C₂²	#16864111	1	1	yes	characteristic
362	C₂²	#16864119	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	268	2	no	—
368	C₄	—	268	2	no	—
369	C₄	—	268	2	no	—
370	C₄	#16863519	1	2	yes	—
371	C₄	C₂×#14524001	1	1	yes	characteristic
372	C₄	#16863876	1	1	yes	characteristic
373	C₄	#16864050	1	1	yes	characteristic
374	C₄	#16864068	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	536	2	no	—
378	C₂	#22563156	1	1	yes	characteristic
379	C₂	C₂×#16863519	1	1	yes	characteristic
380	C₂	#22569127	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1770	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524065	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1770	D₆₇
hypocenter	C₆₇	#22803
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #22568874	C₂	semidirect
4. #22560858	C₂	semidirect
5. #22560381	C₂	semidirect
6. #22560230	C₂	semidirect
7. #22560093	C₂	semidirect
8. C₆₇×#1770	C₂	semidirect
11. C₄×Dic₂₆₈	C₄	semidirect
12. C₄×Dic₂₆₈	C₄	semidirect
13. C₄×#11761617	C₄	semidirect
14. C₄×#11761617	C₄	semidirect
15. C₄×D₅₃₆	C₄	semidirect
16. C₄×D₅₃₆	C₄	semidirect
198. #1770	D₆₇	semidirect
199. C₆₇	#22803	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524065	2
#22568874	2
#22561102	2
#22560858	2
#22560381	2
#22560230	2
#22560093	2
C₆₇×#1770	2
#22803	67