Subgroups — C₂×#22560995 #30859378

Order 17152.

918 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863438	C₂	1	2	yes	maximal
2	C₂²×#14523977	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
4	#22560995	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863164	C₂	1	2	yes	maximal
6	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	1	yes	characteristic
8	C₂×#14524002	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
10	#16863438	—	2	4	no	—
11	#16863438	C₂²	1	8	yes	—
12	C₂×#14523977	—	2	2	no	—
13	C₂×#14523977	—	2	1	no	—
14	C₂×#14523977	—	2	1	no	—
15	C₂×#14523977	—	2	2	no	—
16	C₂×#14523977	C₂²	1	4	yes	—
17	C₂×#14523975	—	2	2	no	—
18	#16863351	—	2	2	no	—
19	#16863351	C₂²	1	4	yes	—
20	C₂×#14523953	—	2	1	no	—
21	C₂×C₈×Dic₆₇	—	2	1	no	—
22	#16863164	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
24	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#108×C₆₇	C₄	1	8	yes	complemented
26	C₂×#108×C₆₇	C₂²	1	4	yes	—
27	C₂³×Dic₆₇	—	2	2	no	—
28	C₂×#11761612	—	2	1	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	Q₈	1	4	yes	—
31	C₂×#11761612	C₂³	1	2	yes	—
32	C₂²×C₆₇⋊C₈	—	2	1	no	—
33	#14524002	—	2	1	no	—
34	#14524002	—	2	1	no	—
35	#14524002	D₄	1	2	yes	—
36	#14524002	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₆₇	—	2	2	no	—
38	C₂×C₄⋊Dic₆₇	—	2	2	no	—
39	C₂×C₄⋊Dic₆₇	—	2	2	no	—
40	C₂×C₄⋊Dic₆₇	—	2	2	no	—
41	C₂×C₄⋊Dic₆₇	—	2	2	no	—
42	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
45	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
46	C₂×C₄×Dic₆₇	—	2	1	no	—
47	#14523977	—	2	8	no	—
48	#14523977	—	2	4	no	—
49	#14523977	—	2	4	no	—
50	#14523975	—	2	8	no	—
51	#14523953	—	2	4	no	—
52	C₈×Dic₆₇	—	2	4	no	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	4	no	—
56	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
57	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	D₄	1	2	yes	—
63	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	#108×C₆₇	C₂×C₄	1	16	yes	complemented
66	C₂²×Dic₆₇	—	2	1	no	—
67	C₂²×Dic₆₇	—	2	1	no	—
68	C₂²×Dic₆₇	—	2	4	no	—
69	C₂²×Dic₆₇	—	4	2	no	—
70	C₂²×Dic₆₇	—	4	2	no	—
71	C₂²×Dic₆₇	—	8	4	no	—
72	#11761612	—	2	4	no	—
73	#11761612	—	2	2	no	—
74	#11761612	C₂×Q₈	1	4	yes	—
75	C₂×C₆₇⋊C₈	—	2	4	no	—
76	C₂×C₆₇⋊C₈	—	2	2	no	—
77	C₄⋊Dic₆₇	—	2	1	no	—
78	C₄⋊Dic₆₇	—	2	1	no	—
79	C₄⋊Dic₆₇	—	2	4	no	—
80	C₄⋊Dic₆₇	—	2	4	no	—
81	C₄⋊Dic₆₇	—	4	2	no	—
82	C₄⋊Dic₆₇	—	4	2	no	—
83	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
84	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
85	C₂²⋊Dic₆₇	—	2	2	no	—
86	C₂²⋊Dic₆₇	—	2	2	no	—
87	C₄×Dic₆₇	—	2	2	no	—
88	C₄×Dic₆₇	—	2	2	no	—
89	C₂⁴×C₆₇	#41	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₆₇	#41	1	2	yes	—
98	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	M₁₆×C₆₇	—	2	8	no	—
101	M₁₆×C₆₇	—	2	4	no	—
102	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
103	C₂×C₈×C₆₇	—	2	8	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
106	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
107	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
108	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
112	C₂³×C₆₇	—	2	2	no	—
113	C₂³×C₆₇	—	2	2	no	—
114	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
115	C₂³×C₆₇	#121	1	1	yes	characteristic
116	C₂³×C₆₇	#121	1	1	yes	characteristic
117	C₂³×C₆₇	#125	1	2	yes	—
118	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	#125	1	2	yes	—
129	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
137	C₂×Dic₆₇	—	2	2	no	—
138	C₂×Dic₆₇	—	2	2	no	—
139	C₂×Dic₆₇	—	4	2	no	—
140	C₂×Dic₆₇	—	4	2	no	—
141	C₂×Dic₆₇	—	4	4	no	—
142	C₂×Dic₆₇	—	8	4	no	—
143	C₂×Dic₆₇	—	8	4	no	—
144	C₂×Dic₆₇	—	8	4	no	—
145	C₈×C₆₇	—	2	8	no	—
146	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
147	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
148	C₆₇⋊C₈	—	2	4	no	—
149	C₂²×C₆₇	—	2	4	no	—
150	C₂²×C₆₇	—	2	4	no	—
151	C₂²×C₆₇	—	2	1	no	—
152	C₂²×C₆₇	—	2	1	no	—
153	C₂²×C₆₇	—	2	1	no	—
154	C₂²×C₆₇	—	2	1	no	—
155	C₂²×C₆₇	#389	1	2	yes	—
156	C₂²×C₆₇	#389	1	2	yes	—
157	C₂²×C₆₇	#472	1	2	yes	—
158	C₂²×C₆₇	#473	1	2	yes	—
159	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
161	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
162	C₄×C₆₇	#389	1	2	yes	—
163	C₄×C₆₇	#389	1	2	yes	—
164	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
167	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
168	Dic₆₇	—	4	2	no	—
169	Dic₆₇	—	4	2	no	—
170	Dic₆₇	—	8	4	no	—
171	C₂×#1952	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	—	2	1	no	—
173	C₂×C₆₇	—	2	1	no	—
174	C₂×C₆₇	—	2	2	no	—
175	C₂×C₆₇	#1952	1	4	yes	complemented
176	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
177	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
178	C₂×C₆₇	C₂×#473	1	1	yes	characteristic
179	C₂²×#108	D₆₇	1	1	yes	characteristic
180	C₂×#428	—	67	2	no	—
181	C₂²×#107	—	67	1	no	—
182	C₂×#405	—	67	1	no	—
183	#1952	—	67	8	no	—
184	C₂×#343	—	67	2	no	—
185	C₆₇	C₂×#1952	1	1	yes	characteristic, complemented, Sylow
186	C₂²×M₁₆	—	67	1	no	—
187	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
188	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
189	C₂×#117	—	67	1	no	—
190	C₂²×C₄⋊C₄	—	67	2	no	—
191	C₂×#108	Dic₆₇	1	8	yes	complemented
192	C₂×#108	C₂×D₆₇	1	4	yes	—
193	#428	—	67	8	no	—
194	#428	—	134	4	no	—
195	C₂×#107	—	67	4	no	—
196	C₂×#107	—	134	2	no	—
197	C₂×#107	—	134	1	no	—
198	C₂×#107	—	134	1	no	—
199	C₂×#107	—	134	2	no	—
200	C₂×#106	—	134	2	no	—
201	#405	—	67	4	no	—
202	#405	—	134	2	no	—
203	C₂×#97	—	134	1	no	—
204	C₂×C₄×C₈	—	134	1	no	—
205	#343	—	67	8	no	—
206	C₂³×C₄	—	134	2	no	—
207	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	2	4	no	—
210	C₂×M₁₆	—	67	4	no	—
211	C₂×M₁₆	—	67	2	no	—
212	C₂×M₁₆	—	134	1	no	—
213	C₂×M₁₆	—	134	2	no	—
214	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
215	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
216	C₂²×C₈	—	2	2	no	—
217	C₂²×C₈	—	134	1	no	—
218	C₂²×C₈	Dic₁₃₄	1	4	yes	—
219	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
220	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
221	C₂²×C₈	D₂₆₈	1	2	yes	—
222	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
223	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
224	#117	—	67	2	no	—
225	#117	—	67	2	no	—
226	#117	—	134	1	no	—
227	#117	—	134	1	no	—
228	C₂×C₄⋊C₄	—	67	1	no	—
229	C₂×C₄⋊C₄	—	67	1	no	—
230	C₂×C₄⋊C₄	—	67	4	no	—
231	C₂×C₄⋊C₄	—	134	2	no	—
232	C₂×C₄⋊C₄	—	134	2	no	—
233	C₂×C₄⋊C₄	—	134	2	no	—
234	C₂×C₄⋊C₄	—	134	2	no	—
235	C₂×C₄⋊C₄	—	134	2	no	—
236	C₂×C₂²⋊C₄	—	134	1	no	—
237	C₂×C₄²	—	134	1	no	—
238	#108	C₂×Dic₆₇	1	16	yes	complemented
239	#107	—	134	8	no	—
240	#107	—	134	4	no	—
241	#107	—	134	4	no	—
242	#106	—	134	8	no	—
243	#97	—	134	4	no	—
244	C₄×C₈	—	134	4	no	—
245	C₂⁴	#11761629	1	1	yes	characteristic
246	C₂²×C₄	—	268	2	no	—
247	C₂²×C₄	—	268	2	no	—
248	C₂²×C₄	—	536	4	no	—
249	C₂²×C₄	—	134	1	no	—
250	C₂²×C₄	—	134	1	no	—
251	C₂²×C₄	—	134	4	no	—
252	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
253	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
257	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
258	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
259	C₂²×C₄	#11761627	1	2	yes	—
260	C₂²×C₄	#11761629	1	1	yes	characteristic
261	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
262	M₁₆	—	2	8	no	—
263	M₁₆	—	2	4	no	—
264	M₁₆	—	67	4	no	—
265	M₁₆	—	134	4	no	—
266	M₁₆	—	134	2	no	—
267	M₁₆	C₂²×Dic₆₇	1	8	yes	—
268	C₂×C₈	—	2	8	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	134	4	no	—
271	C₂×C₈	—	134	2	no	—
272	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
273	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
274	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
275	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
276	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
277	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
278	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
279	C₄⋊C₄	—	268	2	no	—
280	C₄⋊C₄	—	268	2	no	—
281	C₄⋊C₄	—	67	2	no	—
282	C₄⋊C₄	—	67	2	no	—
283	C₄⋊C₄	—	134	4	no	—
284	C₄⋊C₄	—	134	4	no	—
285	C₄⋊C₄	—	134	1	no	—
286	C₄⋊C₄	—	134	1	no	—
287	C₂²⋊C₄	—	134	2	no	—
288	C₂²⋊C₄	—	134	2	no	—
289	C₄²	—	134	2	no	—
290	C₄²	—	134	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	Q₈×Dic₆₇	1	2	yes	—
294	C₂³	#14524018	1	1	yes	characteristic
295	C₂³	#14524018	1	1	yes	characteristic
296	C₂³	#14524034	1	2	yes	—
297	C₂³	C₂×#11761629	1	1	yes	characteristic
298	C₂×C₄	—	268	2	no	—
299	C₂×C₄	—	268	2	no	—
300	C₂×C₄	—	268	4	no	—
301	C₂×C₄	—	536	4	no	—
302	C₂×C₄	—	536	4	no	—
303	C₂×C₄	—	536	4	no	—
304	C₂×C₄	—	134	2	no	—
305	C₂×C₄	—	134	2	no	—
306	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
308	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
309	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
310	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
311	C₂×C₄	#14524018	1	1	yes	characteristic
312	C₂×C₄	#14524018	1	1	yes	characteristic
313	C₂×C₄	#14524023	1	2	yes	—
314	C₂×C₄	#14524023	1	2	yes	—
315	C₂×C₄	#14524034	1	2	yes	—
316	C₂×C₄	#14524042	1	2	yes	—
317	C₂×C₄	#14524042	1	2	yes	—
318	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
321	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
322	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
323	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
324	C₈	—	2	8	no	—
325	C₈	—	134	4	no	—
326	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
327	C₈	C₂×C₄⋊Dic₆₇	1	4	yes	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	4	no	—
333	C₂²	—	2	4	no	—
334	C₂²	#16863295	1	2	yes	—
335	C₂²	#16863295	1	2	yes	—
336	C₂²	#16863667	1	2	yes	—
337	C₂²	#16863672	1	2	yes	—
338	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
339	C₂²	C₂×#14524018	1	1	yes	characteristic
340	C₂²	C₂×#14524034	1	1	yes	characteristic
341	C₄	—	268	2	no	—
342	C₄	—	268	2	no	—
343	C₄	—	536	4	no	—
344	C₄	#16863295	1	2	yes	—
345	C₄	#16863295	1	2	yes	—
346	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
347	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
348	C₄	C₂×#14524023	1	1	yes	characteristic
349	C₄	C₂×#14524042	1	1	yes	characteristic
350	C₂	—	2	2	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	#22560995	1	4	yes	direct product factor
354	C₂	C₂×#16863295	1	1	yes	characteristic
355	C₂	C₂×#16863667	1	1	yes	characteristic
356	C₂	C₂×#16863672	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14524018
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	C₂×#1952
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #22560995	C₂	direct (N × H)
25. C₂×#108×C₆₇	C₄	semidirect
65. #108×C₆₇	C₂×C₄	semidirect
175. C₂×C₆₇	#1952	semidirect
185. C₆₇	C₂×#1952	semidirect
191. C₂×#108	Dic₆₇	semidirect
238. #108	C₂×Dic₆₇	semidirect
353. C₂	#22560995	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16863438	2
C₂²×#14523977	2
C₂×#16863351	2
#22560995	2
C₂×#16863164	2
C₂²×#108×C₆₇	2
C₂×#1952	67

Subgroups — C2×#22560995 #30859378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560995 #30859378