Subgroups — #30874103

Order 17152.

862 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11761612	C₂	1	2	yes	maximal
2	C₂×#16863445	C₂	1	2	yes	maximal, complemented
3	#22560985	C₂	1	2	yes	maximal, complemented
4	#22560416	C₂	1	4	yes	maximal, complemented
5	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
6	#22560251	C₂	1	2	yes	maximal, complemented
7	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11761612	C₂²	1	4	yes	—
10	#16863474	—	2	2	no	—
11	#16863468	—	2	4	no	—
12	D₄×C₆₇⋊C₈	—	2	2	no	—
13	#16863445	—	2	8	no	—
14	C₂×#14523979	C₂²	1	1	yes	characteristic
15	C₂×#14523979	C₂²	1	2	yes	—
16	C₂×#14523979	C₂²	1	4	yes	—
17	C₂×#14523979	C₂²	1	2	yes	—
18	C₂×#14523979	C₂²	1	1	yes	characteristic
19	#16863420	C₂²	1	2	yes	—
20	#16863360	C₂²	1	4	yes	—
21	C₄×#11761612	C₂²	1	2	yes	—
22	#16863161	C₂²	1	2	yes	—
23	#16863161	C₂²	1	4	yes	—
24	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	#436×C₆₇	C₄	1	4	yes	—
27	#436×C₆₇	C₄	1	4	yes	—
28	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#11761612	—	2	8	no	—
32	C₂×#11761612	—	2	4	no	—
33	C₂×#11761612	—	4	4	no	—
34	C₂×#11761612	C₂³	1	2	yes	—
35	C₂×#11761612	C₂³	1	4	yes	—
36	C₂×#11761612	C₂³	1	2	yes	—
37	C₂²×C₆₇⋊C₈	—	2	4	no	—
38	#14523979	—	2	8	no	—
39	#14523979	—	2	4	no	—
40	#14523979	—	2	2	no	—
41	#14523979	—	4	2	no	—
42	#14523979	D₄	1	4	yes	complemented
43	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
45	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
46	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
47	#14523954	—	2	2	no	—
48	C₄×C₆₇⋊C₈	—	2	2	no	—
49	C₂²×D₄×C₆₇	Q₈	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
51	C₂×M₁₆×C₆₇	—	2	4	no	—
52	C₂×M₁₆×C₆₇	—	2	4	no	—
53	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₄×D₄×C₆₇	D₄	1	2	yes	—
60	C₄×D₄×C₆₇	D₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	Q₈	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
65	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
67	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
69	#11761612	—	2	2	no	—
70	#11761612	—	2	4	no	—
71	#11761612	—	2	2	no	—
72	#11761612	—	2	4	no	—
73	#11761612	—	2	2	no	—
74	#11761612	—	4	8	no	—
75	#11761612	C₂×D₄	1	4	yes	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	—	2	4	no	—
79	C₂×C₆₇⋊C₈	—	4	8	no	—
80	C₂⁴×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂×D₄×C₆₇	—	4	2	no	—
82	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
83	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
90	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
94	M₁₆×C₆₇	—	2	16	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	1	no	—
98	C₂×C₈×C₆₇	—	2	1	no	—
99	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₆₇	#41	1	2	yes	—
104	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
105	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
106	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
107	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
108	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₆₇	—	2	2	no	—
113	C₂³×C₆₇	—	2	2	no	—
114	C₂³×C₆₇	—	4	2	no	—
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
118	D₄×C₆₇	—	2	4	no	—
119	D₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	4	no	—
123	C₂×C₄×C₆₇	—	2	4	no	—
124	C₂×C₄×C₆₇	—	2	4	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	4	1	no	—
128	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
139	C₈×C₆₇	—	2	4	no	—
140	C₈×C₆₇	—	2	2	no	—
141	C₈×C₆₇	C₄×D₄	1	4	yes	—
142	C₆₇⋊C₈	—	2	4	no	—
143	C₆₇⋊C₈	—	4	4	no	—
144	C₆₇⋊C₈	—	4	2	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	2	1	no	—
147	C₂²×C₆₇	—	2	4	no	—
148	C₂²×C₆₇	—	2	4	no	—
149	C₂²×C₆₇	—	2	4	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	—	8	2	no	—
152	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
154	C₂²×C₆₇	#554	1	1	yes	characteristic
155	C₄×C₆₇	—	2	4	no	—
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	2	1	no	—
158	C₄×C₆₇	—	2	1	no	—
159	C₄×C₆₇	#430	1	2	yes	complemented
160	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₆₇	#520	1	1	yes	characteristic
162	C₄×C₆₇	D₄²	1	1	yes	characteristic
163	C₄×C₆₇	#547	1	1	yes	characteristic
164	#22669	—	67	1	no	maximal, Sylow
165	C₂×C₆₇	—	2	1	no	—
166	C₂×C₆₇	—	2	1	no	—
167	C₂×C₆₇	—	4	4	no	—
168	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
170	C₂×C₆₇	#2916	1	1	yes	characteristic
171	D₄×M₁₆	—	67	2	no	—
172	C₂×#436	D₆₇	1	1	yes	characteristic
173	C₂×#430	—	67	2	no	—
174	#1950	—	67	2	no	—
175	#1855	—	67	4	no	—
176	#1848	—	67	1	no	—
177	#1811	—	67	2	no	—
178	C₄×#108	—	67	1	no	—
179	C₆₇	#22669	1	1	yes	characteristic, complemented, Sylow
180	C₂²×M₁₆	—	67	4	no	—
181	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
182	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
183	#436	—	134	2	no	—
184	#436	Dic₆₇	1	4	yes	—
185	#436	Dic₆₇	1	4	yes	—
186	#435	—	134	4	no	—
187	C₈×D₄	—	134	2	no	—
188	#430	—	134	8	no	—
189	C₂×#108	—	67	2	no	—
190	C₂×#108	—	67	1	no	—
191	C₂×#108	—	67	1	no	—
192	C₂×#108	—	67	2	no	—
193	C₂×#108	—	67	4	no	—
194	#423	—	67	2	no	—
195	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
196	#407	—	67	4	no	—
197	C₂×C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
198	C₄×M₁₆	—	67	2	no	—
199	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
200	#341	—	67	4	no	—
201	#341	—	67	2	no	—
202	C₂²×D₄	Dic₁₃₄	1	1	yes	characteristic
203	C₂³×C₄	C₂²×D₆₇	1	2	yes	—
204	C₂×M₁₆	—	2	4	no	—
205	C₂×M₁₆	—	2	4	no	—
206	C₂×M₁₆	—	268	4	no	—
207	C₂×M₁₆	—	67	2	no	—
208	C₂×M₁₆	—	67	4	no	—
209	C₂×M₁₆	—	67	2	no	—
210	C₂×M₁₆	—	134	8	no	—
211	C₂×M₁₆	—	134	4	no	—
212	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
213	C₂²×C₈	—	134	4	no	—
214	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
215	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
217	C₄×D₄	D₂₆₈	1	2	yes	—
218	C₄×D₄	D₂₆₈	1	2	yes	—
219	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
220	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
221	C₂×C₄⋊C₄	Dic₁₃₄	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	Dic₁₃₄	1	2	yes	—
223	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
224	#108	—	268	2	no	—
225	#108	—	67	4	no	—
226	#108	—	134	8	no	—
227	#108	—	134	4	no	—
228	#108	—	134	2	no	—
229	C₄⋊C₈	—	134	2	no	—
230	C₄⋊C₈	—	134	2	no	—
231	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
232	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
233	C₂²⋊C₈	—	134	4	no	—
234	C₂²⋊C₈	—	134	4	no	—
235	C₂²⋊C₈	C₂×Dic₆₇	1	8	yes	—
236	#97	—	134	2	no	—
237	C₄×C₈	—	134	2	no	—
238	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
239	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
240	C₂⁴	C₂×Dic₁₃₄	1	2	yes	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
243	C₂×D₄	C₄⋊Dic₆₇	1	2	yes	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
249	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
250	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
252	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
254	M₁₆	—	2	16	no	—
255	M₁₆	—	268	8	no	—
256	M₁₆	—	67	4	no	—
257	M₁₆	—	134	4	no	—
258	M₁₆	—	134	2	no	—
259	M₁₆	—	134	4	no	—
260	M₁₆	—	134	2	no	—
261	M₁₆	—	134	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	268	8	no	—
267	C₂×C₈	—	134	2	no	—
268	C₂×C₈	—	134	4	no	—
269	C₂×C₈	—	134	2	no	—
270	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
271	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
272	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
273	C₂×C₈	D₄×D₆₇	1	2	yes	—
274	C₂×C₈	#11761627	1	2	yes	—
275	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
276	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
277	C₂²⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
278	C₂²⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
279	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
280	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
281	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
282	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	C₂×C₄⋊Dic₆₇	1	4	yes	—
288	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
289	D₄	—	2	4	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
303	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
307	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
308	C₂×C₄	#14524089	1	1	yes	characteristic
309	C₂×C₄	#14524090	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	2	2	no	—
312	C₈	—	268	4	no	—
313	C₈	—	268	2	no	—
314	C₈	—	134	4	no	—
315	C₈	D₄×Dic₆₇	1	4	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
324	C₂²	D₄×Dic₁₃₄	1	1	yes	characteristic
325	C₂²	#16864103	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	#16863445	1	2	yes	complemented
331	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
332	C₄	#16863879	1	1	yes	characteristic
333	C₄	D₄×D₂₆₈	1	1	yes	characteristic
334	C₄	#16864050	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	4	4	no	—
338	C₂	D₄×C₄⋊Dic₆₇	1	1	yes	characteristic
339	C₂	C₂×#16863445	1	1	yes	characteristic
340	C₂	#22568820	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#436×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂²×D₄×C₆₇	Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#436	D₆₇
hypocenter	C₆₇	#22669
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
2. C₂×#16863445	C₂	semidirect
3. #22560985	C₂	semidirect
4. #22560416	C₂	semidirect
5. #22560381	C₂	semidirect
6. #22560251	C₂	semidirect
7. C₄×#14523979	C₂	semidirect
42. #14523979	D₄	semidirect
159. C₄×C₆₇	#430	semidirect
179. C₆₇	#22669	semidirect
330. C₄	#16863445	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11761612	2
C₂×#16863445	2
#22560985	2
#22560416	2
#22560381	2
#22560251	2
C₄×#14523979	2
C₂×#436×C₆₇	2
#22669	67