Subgroups — C₂×C₄×#14523984 #31009090

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523984	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14523984	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863221	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863197	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863196	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₆₈	—	2	1	no	—
11	C₂×#14523984	C₄	1	8	yes	direct product factor
12	C₂×#14523984	C₂²	1	4	yes	—
13	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×D₅₃₆	—	2	2	no	—
16	C₄×D₅₃₆	C₂²	1	4	yes	complemented
17	C₂×#14523977	C₂²	1	1	yes	characteristic
18	C₂×#14523965	—	2	1	no	—
19	#16863221	C₂²	1	4	yes	complemented
20	#16863197	C₂²	1	4	yes	complemented
21	#16863196	C₂²	1	4	yes	complemented
22	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	—	2	1	no	—
25	C₄×D₈×C₆₇	—	2	2	no	—
26	C₄×D₈×C₆₇	C₂²	1	4	yes	complemented
27	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	—	2	1	no	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×D₂₆₈	—	2	1	no	—
31	C₂²×C₄×D₆₇	—	4	1	no	—
32	C₂×D₅₃₆	—	2	1	no	—
33	C₂×D₅₃₆	—	2	2	no	—
34	C₂×D₅₃₆	C₂×C₄	1	4	yes	complemented
35	C₂×D₅₃₆	C₂³	1	2	yes	—
36	C₄×D₂₆₈	—	2	4	no	—
37	C₄×D₂₆₈	—	4	2	no	—
38	C₂×C₄⋊Dic₆₇	—	2	1	no	—
39	C₂×(C₂×D₆₇)⋊C₄	—	4	1	no	—
40	#14523984	C₂×C₄	1	16	yes	direct product factor
41	C₂×C₆₇⋊C₁₆	—	2	2	no	—
42	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
45	#14523977	—	2	1	no	—
46	#14523977	C₂³	1	2	yes	—
47	#14523965	—	2	4	no	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂³×C₄×C₆₇	—	4	1	no	—
50	C₂×D₈×C₆₇	—	2	1	no	—
51	C₂×D₈×C₆₇	—	2	2	no	—
52	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
53	C₂×D₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₆₇	—	2	4	no	—
57	C₄×D₄×C₆₇	—	4	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
60	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
61	#107×C₆₇	—	2	1	no	—
62	#107×C₆₇	C₂³	1	2	yes	—
63	#102×C₆₇	—	2	4	no	—
64	C₄×C₈×C₆₇	D₄	1	2	yes	—
65	C₄×C₈×C₆₇	C₂³	1	2	yes	—
66	C₂³×D₆₇	—	4	1	no	—
67	C₂×D₂₆₈	—	2	4	no	—
68	C₂×D₂₆₈	—	2	2	no	—
69	C₂×D₂₆₈	—	4	1	no	—
70	C₂×D₂₆₈	—	4	2	no	—
71	C₂²×Dic₆₇	—	4	1	no	—
72	C₂×C₄×D₆₇	—	4	4	no	—
73	C₂×C₄×D₆₇	—	8	4	no	—
74	D₅₃₆	—	2	4	no	—
75	D₅₃₆	—	2	2	no	—
76	D₅₃₆	C₂²×C₄	1	4	yes	complemented
77	C₄⋊Dic₆₇	—	2	2	no	—
78	C₄⋊Dic₆₇	—	4	1	no	—
79	(C₂×D₆₇)⋊C₄	—	4	4	no	—
80	C₆₇⋊C₁₆	—	2	2	no	—
81	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	complemented
82	C₂⁴×C₆₇	—	4	1	no	—
83	C₂×D₄×C₆₇	—	2	4	no	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	4	1	no	—
86	C₂×D₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	—	4	4	no	—
89	C₂²×C₄×C₆₇	—	8	4	no	—
90	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	D₈×C₆₇	—	2	4	no	—
94	D₈×C₆₇	—	2	2	no	—
95	D₈×C₆₇	C₂²×C₄	1	4	yes	complemented
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	#41	1	2	yes	—
101	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	4	1	no	—
104	C₂²⋊C₄×C₆₇	—	4	4	no	—
105	C₄²×C₆₇	D₈	1	2	yes	—
106	C₄²×C₆₇	C₂×D₄	1	2	yes	—
107	C₂³×C₆₇	—	4	2	no	—
108	C₂³×C₆₇	—	4	4	no	—
109	C₂³×C₆₇	—	8	2	no	—
110	C₂³×C₆₇	—	8	2	no	—
111	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₆₇	—	4	2	no	—
113	C₂²×D₆₇	—	4	4	no	—
114	C₂²×D₆₇	—	8	2	no	—
115	C₂²×D₆₇	—	8	2	no	—
116	D₄×C₆₇	—	2	4	no	—
117	D₄×C₆₇	—	4	2	no	—
118	D₄×C₆₇	—	4	4	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	8	2	no	—
122	C₂×C₄×C₆₇	—	8	8	no	—
123	C₂×C₄×C₆₇	D₁₆	1	4	yes	complemented
124	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₆₇	#135	1	2	yes	—
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	8	2	no	—
132	D₂₆₈	—	2	4	no	—
133	D₂₆₈	—	4	2	no	—
134	D₂₆₈	—	4	4	no	—
135	C₄×D₆₇	—	8	8	no	—
136	C₈×C₆₇	—	2	2	no	—
137	C₈×C₆₇	C₄×D₄	1	2	yes	—
138	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₂²×C₆₇	—	4	4	no	—
141	C₂²×C₆₇	—	8	2	no	—
142	C₂²×C₆₇	—	8	2	no	—
143	C₂²×C₆₇	—	8	8	no	—
144	C₂²×C₆₇	C₄×D₈	1	2	yes	—
145	C₂²×C₆₇	C₂×D₁₆	1	2	yes	—
146	C₂²×C₆₇	#508	1	2	yes	—
147	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₆₇	—	4	4	no	—
149	C₂×D₆₇	—	8	2	no	—
150	C₂×D₆₇	—	8	2	no	—
151	C₂×D₆₇	—	8	8	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	C₄×D₈	1	2	yes	—
155	C₄×C₆₇	C₂×D₁₆	1	4	yes	complemented
156	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
158	Dic₆₇	—	8	2	no	—
159	C₂×C₄×D₁₆	—	67	1	no	maximal, Sylow
160	C₂×C₆₇	—	8	4	no	—
161	C₂×C₆₇	C₄×D₁₆	1	4	yes	complemented
162	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
164	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
165	D₆₇	—	8	4	no	—
166	C₂²×D₁₆	—	67	1	no	—
167	C₂×C₄×D₈	—	67	1	no	—
168	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic, complemented
169	C₄×D₁₆	—	67	8	no	—
170	C₂×#366	—	67	1	no	—
171	C₂×#357	—	67	1	no	—
172	C₂×#357	—	67	1	no	—
173	C₂×C₄×C₁₆	—	67	1	no	—
174	C₆₇	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
175	C₂²×D₈	—	67	1	no	—
176	C₂²×D₈	C₂×D₆₇	1	1	yes	characteristic
177	C₂×C₄×D₄	—	2	1	no	—
178	C₂×C₄×D₄	—	134	1	no	—
179	C₂×D₁₆	—	67	8	no	—
180	C₂×D₁₆	—	67	4	no	—
181	C₂²×C₁₆	—	67	1	no	—
182	C₂²×C₁₆	—	67	1	no	—
183	C₄×D₈	—	2	2	no	—
184	C₄×D₈	—	67	4	no	—
185	C₄×D₈	—	134	2	no	—
186	C₄×D₈	C₂×D₆₇	1	4	yes	complemented
187	C₂×#107	—	67	1	no	—
188	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
189	C₂×#102	—	2	1	no	—
190	C₂×#102	—	134	1	no	—
191	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
192	#366	—	67	4	no	—
193	#357	—	67	4	no	—
194	#357	—	67	4	no	—
195	C₄×C₁₆	—	67	4	no	—
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	134	1	no	—
198	C₂³×C₄	—	4	1	no	—
199	C₂³×C₄	—	268	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	2	no	—
202	C₂×D₈	—	67	4	no	—
203	C₂×D₈	—	67	2	no	—
204	C₂×D₈	—	134	1	no	—
205	C₂×D₈	—	134	2	no	—
206	C₂×D₈	C₄×D₆₇	1	4	yes	complemented
207	C₂×D₈	C₂²×D₆₇	1	2	yes	—
208	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	C₄×D₄	—	2	4	no	—
211	C₄×D₄	—	4	2	no	—
212	C₄×D₄	—	268	2	no	—
213	C₄×D₄	—	134	4	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	134	1	no	—
216	C₂×C₂²⋊C₄	—	4	1	no	—
217	C₂×C₂²⋊C₄	—	268	1	no	—
218	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
219	D₁₆	—	67	16	no	—
220	C₂×C₁₆	—	67	4	no	—
221	C₂×C₁₆	—	67	2	no	—
222	C₂×C₁₆	—	67	2	no	—
223	C₂×C₁₆	—	134	2	no	—
224	#107	—	2	1	no	—
225	#107	—	67	2	no	—
226	#107	—	134	1	no	—
227	#107	C₂²×D₆₇	1	2	yes	—
228	#102	—	2	4	no	—
229	#102	—	134	4	no	—
230	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
231	C₄×C₈	C₂²×D₆₇	1	2	yes	—
232	C₂⁴	—	4	1	no	—
233	C₂⁴	—	268	1	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	268	1	no	—
239	C₂×D₄	—	268	2	no	—
240	C₂×D₄	—	134	4	no	—
241	C₂×D₄	—	134	2	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	8	4	no	—
245	C₂²×C₄	—	268	1	no	—
246	C₂²×C₄	—	268	4	no	—
247	C₂²×C₄	—	536	4	no	—
248	C₂²×C₄	#11761613	1	1	yes	characteristic
249	C₂²×C₄	#11761613	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
251	D₈	—	2	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	67	4	no	—
254	D₈	—	134	4	no	—
255	D₈	—	134	2	no	—
256	D₈	C₂×C₄×D₆₇	1	4	yes	complemented
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
259	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
260	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
261	C₂×C₈	#11761628	1	2	yes	—
262	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	268	1	no	—
266	C₄⋊C₄	—	134	2	no	—
267	C₂²⋊C₄	—	4	4	no	—
268	C₂²⋊C₄	—	268	4	no	—
269	C₄²	#11761613	1	2	yes	—
270	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
271	C₁₆	—	67	4	no	—
272	C₁₆	—	134	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	4	no	—
275	C₂³	—	8	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	—	268	2	no	—
278	C₂³	—	268	4	no	—
279	C₂³	—	536	2	no	—
280	C₂³	—	536	2	no	—
281	C₂³	C₂×#11761613	1	1	yes	characteristic
282	D₄	—	2	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	4	no	—
285	D₄	—	268	2	no	—
286	D₄	—	268	4	no	—
287	D₄	—	134	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	8	no	—
292	C₂×C₄	—	268	2	no	—
293	C₂×C₄	—	536	2	no	—
294	C₂×C₄	—	536	8	no	—
295	C₂×C₄	#14523984	1	4	yes	direct product factor
296	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
297	C₂×C₄	C₂×#11761613	1	2	yes	—
298	C₂×C₄	C₂×#11761613	1	2	yes	—
299	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
300	C₂×C₄	#14524063	1	2	yes	—
301	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
304	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
305	C₈	C₂×#11761628	1	1	yes	characteristic
306	C₂²	—	4	4	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	8	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	268	4	no	—
311	C₂²	—	536	2	no	—
312	C₂²	—	536	8	no	—
313	C₂²	—	536	2	no	—
314	C₂²	C₄×#11761613	1	2	yes	—
315	C₂²	C₂×#14523984	1	2	yes	—
316	C₂²	#16863826	1	2	yes	—
317	C₂²	C₂²×#11761613	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	536	2	no	—
321	C₄	C₄×#11761613	1	2	yes	—
322	C₄	C₂×#14523984	1	4	yes	direct product factor
323	C₄	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
324	C₄	C₂×#14524063	1	1	yes	characteristic
325	C₂	—	8	4	no	—
326	C₂	—	536	4	no	—
327	C₂	C₄×#14523984	1	4	yes	direct product factor
328	C₂	C₂×C₄×#11761613	1	1	yes	characteristic
329	C₂	C₂²×#14523984	1	1	yes	characteristic
330	C₂	C₂×#16863826	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂²×#14523984	C₂
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅₃₆	C₂	semidirect
3. C₄×#14523984	C₂	direct (N × H)
4. C₂×#16863221	C₂	semidirect
5. C₂×#16863197	C₂	semidirect
6. C₂×#16863196	C₂	semidirect
7. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
8. C₂×C₄×D₈×C₆₇	C₂	semidirect
11. C₂×#14523984	C₄	direct (N × H)
16. C₄×D₅₃₆	C₂²	semidirect
19. #16863221	C₂²	semidirect
20. #16863197	C₂²	semidirect
21. #16863196	C₂²	semidirect
22. C₄×C₆₇⋊C₁₆	C₂²	semidirect
26. C₄×D₈×C₆₇	C₂²	semidirect
34. C₂×D₅₃₆	C₂×C₄	semidirect
40. #14523984	C₂×C₄	direct (N × H)
42. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
52. C₂×D₈×C₆₇	C₂×C₄	semidirect
76. D₅₃₆	C₂²×C₄	semidirect
81. C₆₇⋊C₁₆	C₂²×C₄	semidirect
95. D₈×C₆₇	C₂²×C₄	semidirect
123. C₂×C₄×C₆₇	D₁₆	semidirect
155. C₄×C₆₇	C₂×D₁₆	semidirect
161. C₂×C₆₇	C₄×D₁₆	semidirect
168. C₂×C₄×D₈	D₆₇	semidirect
174. C₆₇	C₂×C₄×D₁₆	semidirect
186. C₄×D₈	C₂×D₆₇	semidirect
206. C₂×D₈	C₄×D₆₇	semidirect
256. D₈	C₂×C₄×D₆₇	semidirect
295. C₂×C₄	#14523984	direct (N × H)
322. C₄	C₂×#14523984	direct (N × H)
327. C₂	C₄×#14523984	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×#14523984	2
C₂×C₄×D₅₃₆	2
C₄×#14523984	2
C₂×#16863221	2
C₂×#16863197	2
C₂×#16863196	2
C₂×C₄×C₆₇⋊C₁₆	2
C₂×C₄×D₈×C₆₇	2
C₂×C₄×D₁₆	67

Subgroups — C2×C4×#14523984 #31009090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14523984 #31009090