Subgroups — C₂×#22562298 #31010253

Order 17152.

702 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523989	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863637	C₂	1	1	yes	maximal, characteristic
4	#22562298	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863221	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863200	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16863199	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
10	C₂×#14524022	—	2	1	no	—
11	C₂×#14523989	—	2	4	no	—
12	C₂×#14523989	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	4	yes	—
16	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#16863637	—	2	2	no	—
18	#16863637	C₂²	1	4	yes	—
19	C₂×#14523977	C₂²	1	1	yes	characteristic
20	C₂×#14523968	—	2	1	no	—
21	#16863221	C₂²	1	4	yes	complemented
22	#16863200	C₂²	1	4	yes	complemented
23	#16863199	C₂²	1	4	yes	complemented
24	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
27	#467×C₆₇	—	2	2	no	—
28	#467×C₆₇	C₂²	1	4	yes	—
29	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#103×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	#14523989	—	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	—
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×Q₈×C₆₇	—	2	1	no	—
51	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂×Q₁₆×C₆₇	—	2	2	no	—
53	C₂×Q₁₆×C₆₇	—	2	1	no	—
54	C₂×Q₁₆×C₆₇	—	2	2	no	—
55	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	—	2	2	no	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	D₄	1	2	yes	—
59	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²⋊Q₈×C₆₇	—	2	4	no	—
63	C₂²⋊Q₈×C₆₇	—	4	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
67	#107×C₆₇	—	2	1	no	—
68	#107×C₆₇	C₂³	1	2	yes	—
69	#103×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	—
85	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	4	1	no	—
88	C₂×Q₈×C₆₇	—	4	2	no	—
89	C₂×Q₈×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	4	1	no	—
93	C₂²×C₄×C₆₇	—	4	1	no	—
94	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	D₈	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₆₇	—	2	2	no	—
99	Q₁₆×C₆₇	—	2	2	no	—
100	Q₁₆×C₆₇	—	4	2	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₆₇	#41	1	2	yes	—
109	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₄⋊C₄×C₆₇	—	4	4	no	—
113	C₂²⋊C₄×C₆₇	—	4	4	no	—
114	C₂³×C₆₇	—	2	2	no	—
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	Q₃₂	1	4	yes	complemented
117	C₂³×C₆₇	C₂×D₈	1	2	yes	—
118	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₆₇	—	4	2	no	—
120	Q₈×C₆₇	—	4	2	no	—
121	Q₈×C₆₇	—	8	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	8	2	no	—
129	C₂×C₄×C₆₇	—	8	2	no	—
130	C₂×C₄×C₆₇	#121	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	#135	1	2	yes	—
134	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₆₇	—	4	2	no	—
136	C₂×Dic₆₇	—	4	2	no	—
137	C₂×Dic₆₇	—	8	2	no	—
138	C₂×Dic₆₇	—	8	2	no	—
139	Dic₁₃₄	—	4	2	no	—
140	Dic₁₃₄	—	4	2	no	—
141	Dic₁₃₄	—	8	2	no	—
142	C₈×C₆₇	—	2	2	no	—
143	C₈×C₆₇	#121	1	2	yes	—
144	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂²×C₆₇	—	2	2	no	—
147	C₂²×C₆₇	—	2	8	no	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	#466	1	2	yes	—
150	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
151	C₂²×C₆₇	C₂×Q₃₂	1	4	yes	complemented
152	C₂²×C₆₇	#508	1	2	yes	—
153	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₆₇	—	2	2	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	#466	1	2	yes	—
158	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
159	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
160	Dic₆₇	—	8	2	no	—
161	Dic₆₇	—	8	2	no	—
162	C₂×#2216	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	—	2	4	no	—
164	C₂×C₆₇	#2216	1	4	yes	complemented
165	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
166	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
167	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
168	C₂²×Q₃₂	—	67	1	no	—
169	C₂³×C₁₆	—	67	1	no	—
170	C₂×#467	—	67	1	no	—
171	C₂×#467	D₆₇	1	1	yes	characteristic
172	#2216	—	67	8	no	—
173	C₂×#366	—	67	1	no	—
174	C₂×#358	—	67	1	no	—
175	C₂×#358	—	67	1	no	—
176	C₆₇	C₂×#2216	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₁₆	—	67	1	no	—
178	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
179	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
180	C₂×C₂²⋊Q₈	—	2	1	no	—
181	C₂×C₂²⋊Q₈	—	134	1	no	—
182	C₂×Q₃₂	—	67	4	no	—
183	C₂×Q₃₂	—	134	4	no	—
184	C₂²×C₁₆	—	67	1	no	—
185	C₂²×C₁₆	—	67	1	no	—
186	C₂²×C₁₆	—	67	4	no	—
187	C₂²×C₁₆	—	134	4	no	—
188	#467	—	2	2	no	—
189	#467	—	67	4	no	—
190	#467	—	134	2	no	—
191	#467	C₂×D₆₇	1	4	yes	—
192	C₂×#107	—	67	1	no	—
193	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
194	C₂×#103	—	2	1	no	—
195	C₂×#103	—	134	1	no	—
196	#366	—	67	4	no	—
197	#358	—	67	4	no	—
198	#358	—	67	4	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	134	1	no	—
201	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	67	2	no	—
206	C₂×Q₁₆	—	134	2	no	—
207	C₂×Q₁₆	—	134	1	no	—
208	C₂×Q₁₆	—	134	2	no	—
209	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
214	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
215	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
216	C₂²⋊Q₈	—	2	4	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	C₂²⋊Q₈	—	268	2	no	—
219	C₂²⋊Q₈	—	134	4	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	4	1	no	—
222	C₂×C₄⋊C₄	—	268	1	no	—
223	C₂×C₄⋊C₄	—	134	1	no	—
224	C₂×C₂²⋊C₄	—	4	1	no	—
225	C₂×C₂²⋊C₄	—	268	1	no	—
226	Q₃₂	—	134	8	no	—
227	C₂×C₁₆	—	67	4	no	—
228	C₂×C₁₆	—	67	2	no	—
229	C₂×C₁₆	—	67	2	no	—
230	C₂×C₁₆	—	134	8	no	—
231	C₂×C₁₆	—	134	2	no	—
232	#107	—	2	1	no	—
233	#107	—	67	2	no	—
234	#107	—	134	1	no	—
235	#107	C₂²×D₆₇	1	2	yes	—
236	#103	—	2	4	no	—
237	#103	—	134	4	no	—
238	C₂⁴	#11761613	1	1	yes	characteristic
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	268	1	no	—
244	C₂×Q₈	—	268	2	no	—
245	C₂×Q₈	—	268	2	no	—
246	C₂×Q₈	—	134	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	268	1	no	—
252	C₂²×C₄	—	268	1	no	—
253	C₂²×C₄	#11761613	1	2	yes	—
254	C₂²×C₄	#11761613	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
256	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	2	2	no	—
259	Q₁₆	—	4	2	no	—
260	Q₁₆	—	268	2	no	—
261	Q₁₆	—	134	2	no	—
262	Q₁₆	—	134	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	D₄×D₆₇	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
270	C₂×C₈	#11761627	1	2	yes	—
271	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	268	4	no	—
276	C₄⋊C₄	—	268	1	no	—
277	C₄⋊C₄	—	134	2	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₂²⋊C₄	—	268	4	no	—
280	C₁₆	—	67	4	no	—
281	C₁₆	—	134	2	no	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	#14523989	1	4	yes	complemented
285	C₂³	C₂×#11761613	1	2	yes	—
286	C₂³	C₂×#11761613	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	268	2	no	—
291	Q₈	—	268	2	no	—
292	Q₈	—	536	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	268	2	no	—
302	C₂×C₄	—	268	2	no	—
303	C₂×C₄	—	536	2	no	—
304	C₂×C₄	—	536	2	no	—
305	C₂×C₄	#14524015	1	2	yes	—
306	C₂×C₄	C₂×#11761613	1	2	yes	—
307	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
308	C₂×C₄	#14524063	1	2	yes	—
309	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	#14524015	1	2	yes	—
312	C₈	C₂×D₄×D₆₇	1	1	yes	characteristic
313	C₈	C₂×#11761627	1	1	yes	characteristic
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	8	no	—
317	C₂²	#16863630	1	2	yes	—
318	C₂²	C₂×#14523989	1	2	yes	—
319	C₂²	C₂×#14523989	1	4	yes	complemented
320	C₂²	#16863826	1	2	yes	—
321	C₂²	C₂²×#11761613	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	536	2	no	—
326	C₄	—	536	2	no	—
327	C₄	#16863630	1	2	yes	—
328	C₄	C₂×#14524015	1	1	yes	characteristic
329	C₄	C₂×#14524063	1	1	yes	characteristic
330	C₂	—	2	4	no	—
331	C₂	#22562298	1	4	yes	direct product factor
332	C₂	C₂×#16863630	1	1	yes	characteristic
333	C₂	C₂²×#14523989	1	1	yes	characteristic
334	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	C₂×#467×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
hypercenter	C₂×#467	D₆₇
hypocenter	C₆₇	C₂×#2216
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523989	C₂	semidirect
4. #22562298	C₂	direct (N × H)
5. C₂×#16863221	C₂	semidirect
6. C₂×#16863200	C₂	semidirect
7. C₂×#16863199	C₂	semidirect
12. C₂×#14523989	C₂²	semidirect
21. #16863221	C₂²	semidirect
22. #16863200	C₂²	semidirect
23. #16863199	C₂²	semidirect
116. C₂³×C₆₇	Q₃₂	semidirect
151. C₂²×C₆₇	C₂×Q₃₂	semidirect
164. C₂×C₆₇	#2216	semidirect
176. C₆₇	C₂×#2216	semidirect
284. C₂³	#14523989	semidirect
319. C₂²	C₂×#14523989	semidirect
331. C₂	#22562298	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×#14523989	2
C₂³×C₆₇⋊C₁₆	2
C₂×#16863637	2
#22562298	2
C₂×#16863221	2
C₂×#16863200	2
C₂×#16863199	2
C₂×#467×C₆₇	2
C₂×#2216	67

Subgroups — C2×#22562298 #31010253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562298 #31010253