Subgroups — C₂×#22562303 #31010259

Order 17152.

754 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523987	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16863637	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562303	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863223	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863199	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863197	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
10	C₂×#14524022	—	2	1	no	—
11	C₂×#14523987	—	2	4	no	—
12	C₂×#14523987	C₂²	1	4	yes	complemented
13	C₂²×C₆₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	complemented
17	#16863637	—	2	2	no	—
18	#16863637	C₂²	1	4	yes	complemented
19	C₂×#14523977	C₂²	1	1	yes	characteristic
20	C₂×#14523968	—	2	1	no	—
21	#16863223	C₂²	1	4	yes	complemented
22	#16863199	C₂²	1	4	yes	complemented
23	#16863197	C₂²	1	4	yes	complemented
24	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#121×C₆₇	—	2	1	no	—
27	#466×C₆₇	—	2	2	no	—
28	#466×C₆₇	C₂²	1	4	yes	—
29	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₆₇	—	2	1	no	—
31	C₂²×Dic₁₃₄	—	2	1	no	—
32	C₂×Dic₂₆₈	—	2	2	no	—
33	C₂×Dic₂₆₈	—	2	1	no	—
34	C₂×Dic₂₆₈	—	2	2	no	—
35	C₂×Dic₂₆₈	C₂³	1	2	yes	—
36	#14524022	—	2	4	no	—
37	#14524022	—	4	2	no	—
38	C₂×C₄⋊Dic₆₇	—	2	1	no	—
39	C₂×Dic₆₇⋊C₄	—	4	1	no	—
40	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
41	#14523987	—	2	8	no	—
42	C₂×C₆₇⋊C₁₆	—	2	8	no	—
43	C₂×C₆₇⋊C₁₆	—	2	2	no	—
44	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	complemented
45	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
46	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
47	#14523977	—	2	1	no	—
48	#14523977	C₂³	1	2	yes	—
49	#14523968	—	2	4	no	—
50	C₂²×D₄×C₆₇	—	2	1	no	—
51	C₂²×D₄×C₆₇	—	4	1	no	—
52	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂×D₈×C₆₇	—	2	2	no	—
54	C₂×D₈×C₆₇	—	2	1	no	—
55	C₂×D₈×C₆₇	—	2	2	no	—
56	C₂×D₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	D₄	1	2	yes	—
60	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	#121×C₆₇	—	2	4	no	—
64	#121×C₆₇	—	4	2	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
67	#107×C₆₇	—	2	1	no	—
68	#107×C₆₇	C₂³	1	2	yes	—
69	#102×C₆₇	—	2	4	no	—
70	C₂×Dic₁₃₄	—	2	2	no	—
71	C₂×Dic₁₃₄	—	4	2	no	—
72	C₂×Dic₁₃₄	—	4	2	no	—
73	C₂×Dic₁₃₄	—	4	1	no	—
74	C₂²×Dic₆₇	—	4	1	no	—
75	C₂²×Dic₆₇	—	4	1	no	—
76	Dic₂₆₈	—	2	2	no	—
77	Dic₂₆₈	—	2	2	no	—
78	Dic₂₆₈	—	4	2	no	—
79	C₄⋊Dic₆₇	—	2	2	no	—
80	C₄⋊Dic₆₇	—	4	1	no	—
81	Dic₆₇⋊C₄	—	4	4	no	—
82	C₂²⋊Dic₆₇	—	4	4	no	—
83	C₆₇⋊C₁₆	—	2	2	no	—
84	C₆₇⋊C₁₆	C₂×D₄	1	4	yes	complemented
85	C₂⁴×C₆₇	—	4	1	no	—
86	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	4	4	no	—
89	C₂×D₄×C₆₇	—	4	2	no	—
90	C₂×D₄×C₆₇	—	4	1	no	—
91	C₂×D₄×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	8	4	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	4	1	no	—
96	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	D₈	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	D₈×C₆₇	—	2	2	no	—
101	D₈×C₆₇	—	2	2	no	—
102	D₈×C₆₇	—	4	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	2	no	—
107	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	#41	1	2	yes	—
111	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	4	1	no	—
114	C₂²⋊C₄×C₆₇	—	4	4	no	—
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	—	4	2	no	—
118	C₂³×C₆₇	—	8	2	no	—
119	C₂³×C₆₇	—	8	2	no	—
120	C₂³×C₆₇	—	8	2	no	—
121	C₂³×C₆₇	SD₃₂	1	4	yes	complemented
122	C₂³×C₆₇	C₂×D₈	1	2	yes	—
123	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
124	D₄×C₆₇	—	4	2	no	—
125	D₄×C₆₇	—	4	2	no	—
126	D₄×C₆₇	—	8	8	no	—
127	D₄×C₆₇	—	8	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	8	2	no	—
134	C₂×C₄×C₆₇	#121	1	2	yes	—
135	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	#135	1	2	yes	—
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₆₇	—	4	2	no	—
140	C₂×Dic₆₇	—	4	2	no	—
141	C₂×Dic₆₇	—	8	2	no	—
142	C₂×Dic₆₇	—	8	2	no	—
143	Dic₁₃₄	—	4	2	no	—
144	Dic₁₃₄	—	4	2	no	—
145	Dic₁₃₄	—	8	2	no	—
146	C₈×C₆₇	—	2	2	no	—
147	C₈×C₆₇	#121	1	2	yes	—
148	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	8	no	—
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	2	no	—
155	C₂²×C₆₇	—	8	2	no	—
156	C₂²×C₆₇	—	16	4	no	—
157	C₂²×C₆₇	#466	1	2	yes	—
158	C₂²×C₆₇	C₂×SD₃₂	1	2	yes	—
159	C₂²×C₆₇	C₂×SD₃₂	1	4	yes	complemented
160	C₂²×C₆₇	#508	1	2	yes	—
161	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₆₇	—	2	2	no	—
163	C₄×C₆₇	—	8	2	no	—
164	C₄×C₆₇	#466	1	2	yes	—
165	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
166	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
167	Dic₆₇	—	8	2	no	—
168	Dic₆₇	—	8	2	no	—
169	C₂×#2217	—	67	1	no	maximal, Sylow
170	C₂×C₆₇	—	2	4	no	—
171	C₂×C₆₇	—	16	2	no	—
172	C₂×C₆₇	#2217	1	4	yes	complemented
173	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
174	C₂×C₆₇	C₂²×SD₃₂	1	1	yes	characteristic
175	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
176	C₂²×SD₃₂	—	67	1	no	—
177	C₂³×C₁₆	—	67	1	no	—
178	C₂×#467	—	67	1	no	—
179	C₂×#466	D₆₇	1	1	yes	characteristic
180	#2217	—	67	8	no	—
181	C₂×#367	—	67	1	no	—
182	C₂×#358	—	67	1	no	—
183	C₂×#357	—	67	1	no	—
184	C₆₇	C₂×#2217	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₁₆	—	67	1	no	—
186	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
187	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
188	C₂×C₂²⋊Q₈	—	134	1	no	—
189	C₂×#121	—	2	1	no	—
190	C₂×SD₃₂	—	67	4	no	—
191	C₂×SD₃₂	—	134	4	no	—
192	C₂²×C₁₆	—	67	1	no	—
193	C₂²×C₁₆	—	67	1	no	—
194	C₂²×C₁₆	—	67	4	no	—
195	C₂²×C₁₆	—	134	4	no	—
196	#467	—	67	4	no	—
197	#467	—	134	2	no	—
198	#466	—	2	2	no	—
199	#466	C₂×D₆₇	1	4	yes	—
200	C₂×#107	—	67	1	no	—
201	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
202	C₂×#103	—	134	1	no	—
203	C₂×#102	—	2	1	no	—
204	#367	—	67	4	no	—
205	#358	—	67	4	no	—
206	#357	—	67	4	no	—
207	C₂²×Q₈	—	134	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	4	1	no	—
210	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
211	C₂×Q₁₆	—	67	2	no	—
212	C₂×Q₁₆	—	134	2	no	—
213	C₂×Q₁₆	—	134	1	no	—
214	C₂×Q₁₆	—	134	2	no	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	—	2	2	no	—
218	C₂×D₈	C₂²×D₆₇	1	2	yes	—
219	C₂²×C₈	—	2	2	no	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
222	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
223	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
224	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
225	C₂²⋊Q₈	—	268	2	no	—
226	C₂²⋊Q₈	—	134	4	no	—
227	#121	—	2	4	no	—
228	#121	—	4	2	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	268	1	no	—
231	C₂×C₄⋊C₄	—	134	1	no	—
232	C₂×C₂²⋊C₄	—	4	1	no	—
233	C₂×C₂²⋊C₄	—	268	1	no	—
234	SD₃₂	—	134	8	no	—
235	C₂×C₁₆	—	67	4	no	—
236	C₂×C₁₆	—	67	2	no	—
237	C₂×C₁₆	—	67	2	no	—
238	C₂×C₁₆	—	134	8	no	—
239	C₂×C₁₆	—	134	2	no	—
240	#107	—	2	1	no	—
241	#107	—	67	2	no	—
242	#107	—	134	1	no	—
243	#107	C₂²×D₆₇	1	2	yes	—
244	#103	—	134	4	no	—
245	#102	—	2	4	no	—
246	C₂⁴	—	4	1	no	—
247	C₂⁴	#11761613	1	1	yes	characteristic
248	C₂×Q₈	—	268	1	no	—
249	C₂×Q₈	—	268	2	no	—
250	C₂×Q₈	—	268	2	no	—
251	C₂×Q₈	—	134	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	8	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	268	1	no	—
262	C₂²×C₄	—	268	1	no	—
263	C₂²×C₄	#11761613	1	2	yes	—
264	C₂²×C₄	#11761613	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
266	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
267	Q₁₆	—	268	2	no	—
268	Q₁₆	—	134	2	no	—
269	Q₁₆	—	134	2	no	—
270	D₈	—	2	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	D₄×D₆₇	1	2	yes	—
278	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
279	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
280	C₂×C₈	#11761627	1	2	yes	—
281	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	268	4	no	—
285	C₄⋊C₄	—	268	1	no	—
286	C₄⋊C₄	—	134	2	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	268	4	no	—
289	C₁₆	—	67	4	no	—
290	C₁₆	—	134	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	—	8	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	#14523987	1	4	yes	complemented
298	C₂³	C₂×#11761613	1	2	yes	—
299	C₂³	C₂×#11761613	1	1	yes	characteristic
300	Q₈	—	268	2	no	—
301	Q₈	—	268	2	no	—
302	Q₈	—	536	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	8	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	268	2	no	—
314	C₂×C₄	—	268	2	no	—
315	C₂×C₄	—	536	2	no	—
316	C₂×C₄	—	536	2	no	—
317	C₂×C₄	#14524015	1	2	yes	—
318	C₂×C₄	C₂×#11761613	1	2	yes	—
319	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
320	C₂×C₄	#14524063	1	2	yes	—
321	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	#14524015	1	2	yes	—
324	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
325	C₈	C₂×#11761627	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	8	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	16	4	no	—
333	C₂²	#16863630	1	2	yes	—
334	C₂²	C₂×#14523987	1	2	yes	—
335	C₂²	C₂×#14523987	1	4	yes	complemented
336	C₂²	#16863826	1	2	yes	—
337	C₂²	C₂²×#11761613	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	536	2	no	—
341	C₄	—	536	2	no	—
342	C₄	#16863630	1	2	yes	—
343	C₄	C₂×#14524015	1	1	yes	characteristic
344	C₄	C₂×#14524063	1	1	yes	characteristic
345	C₂	—	2	4	no	—
346	C₂	—	16	2	no	—
347	C₂	#22562303	1	4	yes	direct product factor
348	C₂	C₂×#16863630	1	1	yes	characteristic
349	C₂	C₂²×#14523987	1	1	yes	characteristic
350	C₂	C₂×#16863826	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting, prime radical	C₂×#466×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×#466	D₆₇
hypocenter	C₆₇	C₂×#2217
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523987	C₂	semidirect
2. C₂³×C₆₇⋊C₁₆	C₂	semidirect
3. C₂×#16863637	C₂	semidirect
4. #22562303	C₂	direct (N × H)
5. C₂×#16863223	C₂	semidirect
6. C₂×#16863199	C₂	semidirect
7. C₂×#16863197	C₂	semidirect
12. C₂×#14523987	C₂²	semidirect
16. C₂²×C₆₇⋊C₁₆	C₂²	semidirect
18. #16863637	C₂²	semidirect
21. #16863223	C₂²	semidirect
22. #16863199	C₂²	semidirect
23. #16863197	C₂²	semidirect
44. C₂×C₆₇⋊C₁₆	D₄	semidirect
84. C₆₇⋊C₁₆	C₂×D₄	semidirect
121. C₂³×C₆₇	SD₃₂	semidirect
159. C₂²×C₆₇	C₂×SD₃₂	semidirect
172. C₂×C₆₇	#2217	semidirect
184. C₆₇	C₂×#2217	semidirect
297. C₂³	#14523987	semidirect
335. C₂²	C₂×#14523987	semidirect
347. C₂	#22562303	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523987	2
C₂³×C₆₇⋊C₁₆	2
C₂×#16863637	2
#22562303	2
C₂×#16863223	2
C₂×#16863199	2
C₂×#16863197	2
C₂×#466×C₆₇	2
C₂×#2217	67

Subgroups — C2×#22562303 #31010259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562303 #31010259