Subgroups — #30827330

Order 17152.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11761612	C₂	1	2	yes	maximal
2	#22559118	C₂	1	2	yes	maximal, complemented
3	#22559081	C₂	1	4	yes	maximal, complemented
4	#22559028	C₂	1	2	yes	maximal, complemented
5	#22558965	C₂	1	1	yes	maximal, characteristic
6	#22558638	C₂	1	2	yes	maximal
7	#22558498	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863527	—	2	4	no	—
10	#16863527	C₂²	1	4	yes	—
11	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	—
12	#16863449	C₂²	1	1	yes	characteristic
13	#16863420	—	2	2	no	—
14	#16863420	C₂²	1	2	yes	—
15	C₄×#11761612	—	2	2	no	—
16	C₄×#11761612	C₂²	1	1	yes	characteristic
17	#16863147	C₂²	1	2	yes	—
18	#16863147	C₂²	1	2	yes	—
19	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
20	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
21	#16863118	C₂²	1	2	yes	—
22	#16863117	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₆₇	C₄	1	2	yes	—
27	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
28	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
30	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
32	C₂×#103×C₆₇	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	—	2	2	no	—
35	C₂×#11761612	D₄	1	2	yes	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
40	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
41	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
42	#14523954	—	2	4	no	—
43	#14523954	D₄	1	2	yes	—
44	C₄×C₆₇⋊C₈	—	2	4	no	—
45	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
46	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₆₇	—	2	4	no	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×Q₈×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
59	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	C₂×C₄	1	2	yes	—
65	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
66	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#103×C₆₇	C₂×C₄	1	2	yes	—
68	#103×C₆₇	C₂×C₄	1	2	yes	—
69	#103×C₆₇	C₂×C₄	1	2	yes	—
70	#103×C₆₇	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	#11761612	—	2	4	no	—
76	#11761612	—	4	4	no	—
77	C₂×C₆₇⋊C₈	—	2	4	no	—
78	C₂×C₆₇⋊C₈	—	2	2	no	—
79	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	4	no	—
82	C₂×Q₈×C₆₇	—	4	2	no	—
83	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
90	Q₁₆×C₆₇	—	2	4	no	—
91	Q₁₆×C₆₇	—	2	2	no	—
92	Q₁₆×C₆₇	M₁₆	1	4	yes	complemented
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	4	1	no	—
95	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	#41	1	1	yes	characteristic
113	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
115	Q₈×C₆₇	—	2	2	no	—
116	Q₈×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	—	2	2	no	—
118	Q₈×C₆₇	—	4	4	no	—
119	Q₈×C₆₇	C₂×M₁₆	1	4	yes	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	8	2	no	—
129	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#137	1	4	yes	—
133	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	2	1	no	—
139	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
140	C₆₇⋊C₈	—	2	4	no	—
141	C₆₇⋊C₈	—	4	2	no	—
142	C₆₇⋊C₈	—	4	4	no	—
143	C₂²×C₆₇	—	2	1	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	#441	1	2	yes	—
146	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₆₇	—	2	1	no	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	4	2	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	#441	1	2	yes	—
155	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
156	C₄×C₆₇	#568	1	1	yes	characteristic
157	C₄×C₆₇	C₂×#137	1	2	yes	—
158	#18042	—	67	1	no	maximal, Sylow
159	C₂×C₆₇	—	2	2	no	—
160	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
161	C₂×C₆₇	C₂×#441	1	1	yes	characteristic
162	C₂×C₆₇	#2957	1	1	yes	characteristic
163	Q₈×M₁₆	—	67	2	no	—
164	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
165	#1593	—	67	2	no	—
166	#1587	—	67	4	no	—
167	#1579	—	67	2	no	—
168	#1566	—	67	1	no	—
169	#1487	—	67	2	no	—
170	#1451	—	67	1	no	—
171	C₆₇	#18042	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₁₆	Dic₆₇	1	1	yes	characteristic
173	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
174	#446	—	67	4	no	—
175	#446	—	134	4	no	—
176	C₈×Q₈	—	67	4	no	—
177	C₄×Q₁₆	Dic₆₇	1	2	yes	—
178	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
179	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
180	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
181	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
182	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
183	#431	—	67	1	no	—
184	C₂×#107	Dic₆₇	1	1	yes	characteristic
185	#423	—	67	2	no	—
186	#423	—	134	2	no	—
187	C₂×#103	Dic₆₇	1	2	yes	—
188	C₄×M₁₆	—	67	1	no	—
189	C₄×M₁₆	—	134	2	no	—
190	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
191	#334	—	67	2	no	—
192	#334	—	67	2	no	—
193	Q₈⋊C₈	—	67	4	no	—
194	Q₈⋊C₈	—	67	4	no	—
195	#322	—	67	2	no	—
196	#322	—	67	2	no	—
197	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
198	C₂×Q₁₆	—	2	4	no	—
199	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
200	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
201	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
202	C₂×M₁₆	—	67	2	no	—
203	C₂×M₁₆	—	134	2	no	—
204	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
205	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
209	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
210	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
211	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
216	#107	C₂×Dic₆₇	1	2	yes	—
217	#107	C₂×Dic₆₇	1	1	yes	characteristic
218	#107	C₂×Dic₆₇	1	1	yes	characteristic
219	C₄⋊C₈	—	67	4	no	—
220	C₄⋊C₈	—	134	4	no	—
221	C₄⋊C₈	—	134	4	no	—
222	C₄⋊C₈	—	134	2	no	—
223	C₄⋊C₈	—	134	1	no	—
224	#103	C₂×Dic₆₇	1	2	yes	—
225	#103	C₂×Dic₆₇	1	2	yes	—
226	#103	C₂×Dic₆₇	1	2	yes	—
227	#103	C₂×Dic₆₇	1	2	yes	—
228	C₂²⋊C₈	—	134	1	no	—
229	#97	—	67	2	no	—
230	#97	—	134	4	no	—
231	C₄×C₈	—	67	2	no	—
232	C₄×C₈	—	67	2	no	—
233	C₄×C₈	—	134	4	no	—
234	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
235	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
242	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
247	C₂²×C₄	#11761627	1	1	yes	characteristic
248	Q₁₆	—	2	4	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	#11761612	1	4	yes	complemented
251	M₁₆	—	268	4	no	—
252	M₁₆	—	134	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	67	4	no	—
256	C₂×C₈	—	134	4	no	—
257	C₂×C₈	—	134	2	no	—
258	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
268	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
274	C₄²	D₄×D₆₇	1	1	yes	characteristic
275	C₄²	#11761627	1	1	yes	characteristic
276	C₄²	C₂³×D₆₇	1	1	yes	characteristic
277	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
278	Q₈	—	2	2	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	4	4	no	—
282	Q₈	C₂×#11761612	1	4	yes	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
293	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
294	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
295	C₂×C₄	#14524075	1	4	yes	—
296	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
298	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	268	2	no	—
303	C₈	—	268	4	no	—
304	C₈	—	134	4	no	—
305	C₈	C₂×#11761612	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#16863503	1	2	yes	—
309	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	4	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	#16863503	1	2	yes	—
318	C₄	C₂²×#11761612	1	1	yes	characteristic
319	C₄	#16864163	1	1	yes	characteristic
320	C₄	C₂×#14524075	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	D₄×#11761612	1	1	yes	characteristic
323	C₂	C₂×#16863503	1	1	yes	characteristic
324	C₂	#22569126	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#18042
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22559118	C₂	semidirect
3. #22559081	C₂	semidirect
4. #22559028	C₂	semidirect
92. Q₁₆×C₆₇	M₁₆	semidirect
171. C₆₇	#18042	semidirect
250. Q₁₆	#11761612	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11761612	2
#22559118	2
#22559081	2
#22559028	2
#22558965	2
#22558638	2
#22558498	2
C₂×C₄×Q₁₆×C₆₇	2
#18042	67