Subgroups — #30748668

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22560189	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863148	C₂	1	2	yes	maximal, complemented
4	#22558043	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂³×C₆₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₂×C₄×C₆₇⋊C₈	—	2	4	no	—
11	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	#16863151	—	2	1	no	—
16	#16863148	—	2	4	no	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
20	#336×C₆₇	—	2	1	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	C₈	1	4	yes	complemented
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
34	C₂²⋊(C₆₇⋊C₈)	—	4	2	no	—
35	C₄×C₆₇⋊C₈	—	2	2	no	—
36	C₄×C₆₇⋊C₈	—	2	2	no	—
37	C₄×C₆₇⋊C₈	—	2	2	no	—
38	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	2	2	no	—
44	C₂×C₄²×C₆₇	—	2	2	no	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	—	2	2	no	—
50	C₄⋊C₈×C₆₇	—	2	2	no	—
51	C₂²⋊C₈×C₆₇	—	4	2	no	—
52	C₂×C₆₇⋊C₈	—	2	8	no	—
53	C₂×C₆₇⋊C₈	—	2	1	no	—
54	C₂×C₆₇⋊C₈	—	2	1	no	—
55	C₂×C₆₇⋊C₈	—	2	1	no	—
56	C₂×C₆₇⋊C₈	—	2	1	no	—
57	C₂×C₆₇⋊C₈	—	2	1	no	—
58	C₂×C₆₇⋊C₈	—	2	1	no	—
59	C₂×C₆₇⋊C₈	—	2	8	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
65	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
66	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
67	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
68	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
69	C₂⁴×C₆₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
84	C₂²×C₄×C₆₇	D₈	1	2	yes	—
85	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	—	4	2	no	—
91	C₂×C₈×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	—	2	1	no	—
93	C₄²×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	2	2	no	—
96	C₄²×C₆₇	C₂×D₄	1	2	yes	—
97	C₄²×C₆₇	#41	1	2	yes	—
98	C₂³×C₆₇	—	2	1	no	—
99	C₂³×C₆₇	—	2	1	no	—
100	C₂³×C₆₇	—	2	1	no	—
101	C₂³×C₆₇	—	2	1	no	—
102	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
103	C₂³×C₆₇	#107	1	2	yes	—
104	C₂³×C₆₇	#108	1	2	yes	—
105	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	—	2	8	no	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	2	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₆₇	#107	1	2	yes	—
123	C₂×C₄×C₆₇	#108	1	2	yes	—
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#121	1	2	yes	—
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₄×C₆₇	#135	1	2	yes	—
135	C₈×C₆₇	—	8	2	no	—
136	C₆₇⋊C₈	—	2	2	no	—
137	C₆₇⋊C₈	—	2	2	no	—
138	C₆₇⋊C₈	—	2	2	no	—
139	C₆₇⋊C₈	—	8	2	no	—
140	C₆₇⋊C₈	C₂²⋊C₈	1	4	yes	complemented
141	C₂²×C₆₇	—	2	2	no	—
142	C₂²×C₆₇	—	2	2	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	—	2	2	no	—
147	C₂²×C₆₇	#335	1	4	yes	complemented
148	C₂²×C₆₇	#380	1	1	yes	characteristic
149	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₂²×C₆₇	#423	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
152	C₂²×C₆₇	#427	1	1	yes	characteristic
153	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
154	C₂²×C₆₇	C₂×#108	1	1	yes	characteristic
155	C₄×C₆₇	—	2	2	no	—
156	C₄×C₆₇	—	2	2	no	—
157	C₄×C₆₇	—	2	2	no	—
158	C₄×C₆₇	—	2	2	no	—
159	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₄×C₆₇	#407	1	1	yes	characteristic
161	C₄×C₆₇	#431	1	1	yes	characteristic
162	C₄×C₆₇	#432	1	1	yes	characteristic
163	C₄×C₆₇	#466	1	2	yes	—
164	C₄×C₆₇	#467	1	2	yes	—
165	#8596	—	67	1	no	maximal, Sylow
166	C₂×C₆₇	—	2	4	no	—
167	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
168	C₂×C₆₇	#1564	1	1	yes	characteristic
169	C₂×C₆₇	#1567	1	1	yes	characteristic
170	C₂×C₆₇	#1568	1	1	yes	characteristic
171	C₂×C₆₇	#1797	1	1	yes	characteristic
172	C₂×C₆₇	#1808	1	1	yes	characteristic
173	C₂×C₆₇	#1810	1	1	yes	characteristic
174	C₂²×C₄×C₈	—	67	1	no	—
175	#1797	—	67	1	no	—
176	#1797	D₆₇	1	1	yes	characteristic
177	C₂×#335	—	67	2	no	—
178	#1277	—	67	2	no	—
179	C₆₇	#8596	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₈	—	67	1	no	—
181	C₂³×C₈	—	67	1	no	—
182	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
183	C₂×C₄⋊C₈	—	67	2	no	—
184	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
185	C₂×C₂²⋊C₈	—	2	1	no	—
186	C₂×C₂²⋊C₈	—	134	1	no	—
187	C₂×C₄×C₈	—	67	1	no	—
188	C₂×C₄×C₈	—	67	1	no	—
189	C₂×C₄×C₈	—	67	1	no	—
190	C₂×C₄×C₈	—	67	1	no	—
191	C₂×C₄×C₈	—	134	4	no	—
192	#336	—	2	1	no	—
193	#336	—	134	1	no	—
194	#335	—	134	4	no	—
195	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
196	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
197	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	67	1	no	—
200	C₂²×C₈	—	67	4	no	—
201	C₂²×C₈	—	67	1	no	—
202	C₂²×C₈	—	67	1	no	—
203	C₂²×C₈	—	67	1	no	—
204	C₂²×C₈	—	134	2	no	—
205	C₂²×C₈	—	134	2	no	—
206	C₂²×C₈	—	134	2	no	—
207	C₂²×C₈	—	134	2	no	—
208	C₂²×C₈	—	134	2	no	—
209	C₂²×C₈	—	134	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	134	2	no	—
219	C₄⋊C₈	—	134	2	no	—
220	C₂²⋊C₈	—	4	2	no	—
221	C₂²⋊C₈	—	268	2	no	—
222	C₄×C₈	—	67	4	no	—
223	C₄×C₈	—	134	2	no	—
224	C₄×C₈	—	134	2	no	—
225	C₄×C₈	—	134	2	no	—
226	C₂⁴	Dic₆₇⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
239	C₂²×C₄	C₈×D₆₇	1	2	yes	—
240	C₂²×C₄	#11761611	1	2	yes	—
241	C₂²×C₄	#11761613	1	2	yes	—
242	C₂²×C₄	#11761618	1	2	yes	—
243	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
245	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	268	2	no	—
250	C₂×C₈	—	268	2	no	—
251	C₂×C₈	—	268	2	no	—
252	C₂×C₈	—	67	2	no	—
253	C₂×C₈	—	67	2	no	—
254	C₂×C₈	—	67	2	no	—
255	C₂×C₈	—	67	2	no	—
256	C₂×C₈	—	67	2	no	—
257	C₂×C₈	—	67	2	no	—
258	C₂×C₈	—	134	8	no	—
259	C₂×C₈	—	134	8	no	—
260	C₂×C₈	—	134	1	no	—
261	C₂×C₈	—	134	1	no	—
262	C₂×C₈	—	134	1	no	—
263	C₂×C₈	—	134	1	no	—
264	C₂×C₈	—	134	1	no	—
265	C₂×C₈	—	134	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	D₄×D₆₇	1	2	yes	—
271	C₄²	#11761627	1	2	yes	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	Dic₆₇⋊C₈	1	2	yes	—
277	C₂³	#14523976	1	2	yes	—
278	C₂³	#14523978	1	2	yes	—
279	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
296	C₂×C₄	#14523976	1	2	yes	—
297	C₂×C₄	#14523978	1	2	yes	—
298	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
299	C₂×C₄	#14523999	1	1	yes	characteristic
300	C₂×C₄	#14524015	1	2	yes	—
301	C₂×C₄	#14524020	1	2	yes	—
302	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
303	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
304	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
306	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
307	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
308	C₂×C₄	#14524063	1	2	yes	—
309	C₈	—	8	2	no	—
310	C₈	—	536	2	no	—
311	C₈	—	67	4	no	—
312	C₈	—	134	2	no	—
313	C₈	—	134	2	no	—
314	C₈	—	134	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	#16863148	1	4	yes	complemented
322	C₂²	#16863254	1	1	yes	characteristic
323	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
324	C₂²	#16863419	1	1	yes	characteristic
325	C₂²	C₂×#14523976	1	1	yes	characteristic
326	C₂²	#16863432	1	1	yes	characteristic
327	C₂²	C₂×#14523978	1	1	yes	characteristic
328	C₂²	C₂×#14523978	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
334	C₄	#16863358	1	1	yes	characteristic
335	C₄	#16863446	1	1	yes	characteristic
336	C₄	#16863450	1	1	yes	characteristic
337	C₄	#16863630	1	2	yes	—
338	C₄	#16863635	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	C₂×#16863148	1	1	yes	characteristic
341	C₂	#22558955	1	1	yes	characteristic
342	C₂	#22558967	1	1	yes	characteristic
343	C₂	#22558970	1	1	yes	characteristic
344	C₂	#22560187	1	1	yes	characteristic
345	C₂	#22560236	1	1	yes	characteristic
346	C₂	#22560245	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1797	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1797	D₆₇
hypocenter	C₆₇	#8596
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#16863148	C₂	semidirect
4. #22558043	C₂	semidirect
27. C₂²×C₆₇⋊C₈	C₈	semidirect
140. C₆₇⋊C₈	C₂²⋊C₈	semidirect
147. C₂²×C₆₇	#335	semidirect
179. C₆₇	#8596	semidirect
321. C₂²	#16863148	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₆₇⋊C₈	2
#22560189	2
C₂×#16863148	2
#22558043	2
C₆₇×#1797	2
#8596	67