Subgroups — C₂×#22560989 #30859371

Order 17152.

918 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863439	C₂	1	2	yes	maximal
3	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
4	#22560989	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16863164	C₂	1	2	yes	maximal
6	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂×#14523979	C₄	1	8	yes	complemented
10	C₂×#14523979	C₂²	1	4	yes	—
11	#16863439	—	2	4	no	—
12	#16863439	C₂²	1	8	yes	—
13	C₂×#14523976	—	2	2	no	—
14	C₂×#14523975	—	2	1	no	—
15	C₂×#14523975	—	2	1	no	—
16	C₂×#14523975	—	2	2	no	—
17	C₂×#14523975	C₂²	1	4	yes	—
18	C₂×#14523974	—	2	2	no	—
19	#16863164	C₂²	1	8	yes	—
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	#405×C₆₇	—	2	2	no	—
24	#405×C₆₇	C₂²	1	4	yes	—
25	C₂×#97×C₆₇	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	—	2	1	no	—
27	C₂³×Dic₆₇	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	4	no	—
30	C₂×#11761612	C₂×C₄	1	8	yes	—
31	C₂×#11761612	C₂³	1	4	yes	—
32	C₂²×C₆₇⋊C₈	—	2	2	no	—
33	C₂×C₄⋊Dic₆₇	—	2	2	no	—
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₄⋊Dic₆₇	—	2	2	no	—
36	C₂×C₄⋊Dic₆₇	—	2	2	no	—
37	C₂×C₄⋊Dic₆₇	—	2	2	no	—
38	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
39	#14523979	C₂×C₄	1	16	yes	complemented
40	#14523976	—	2	8	no	—
41	#14523975	—	2	4	no	—
42	#14523975	—	2	4	no	—
43	#14523974	—	2	8	no	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	1	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
48	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	—	2	1	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	D₄	1	2	yes	—
54	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	#117×C₆₇	—	2	1	no	—
57	#117×C₆₇	—	2	1	no	—
58	#117×C₆₇	D₄	1	2	yes	—
59	#117×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄²×C₆₇	—	2	1	no	—
63	#97×C₆₇	—	2	4	no	—
64	C₄×C₈×C₆₇	—	2	4	no	—
65	C₂²×Dic₆₇	—	2	4	no	—
66	C₂²×Dic₆₇	—	4	2	no	—
67	C₂²×Dic₆₇	—	4	2	no	—
68	C₂²×Dic₆₇	—	8	4	no	—
69	#11761612	—	2	8	no	—
70	#11761612	—	2	4	no	—
71	#11761612	C₂²×C₄	1	8	yes	—
72	C₂×C₆₇⋊C₈	—	2	8	no	—
73	C₂×C₆₇⋊C₈	—	2	4	no	—
74	C₄⋊Dic₆₇	—	2	4	no	—
75	C₄⋊Dic₆₇	—	2	4	no	—
76	C₄⋊Dic₆₇	—	4	2	no	—
77	C₄⋊Dic₆₇	—	4	2	no	—
78	C₂⁴×C₆₇	#41	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₆₇	—	2	4	no	—
91	M₁₆×C₆₇	—	2	2	no	—
92	M₁₆×C₆₇	C₂×Q₈	1	4	yes	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
96	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
97	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
98	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
104	C₂²⋊C₄×C₆₇	—	2	2	no	—
105	C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	—	2	4	no	—
109	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
110	C₂³×C₆₇	#121	1	1	yes	characteristic
111	C₂³×C₆₇	#121	1	1	yes	characteristic
112	C₂³×C₆₇	#125	1	2	yes	—
113	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	4	4	no	—
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₆₇	#125	1	2	yes	—
126	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₆₇	—	4	4	no	—
135	C₂×Dic₆₇	—	8	4	no	—
136	C₂×Dic₆₇	—	8	4	no	—
137	C₂×Dic₆₇	—	8	4	no	—
138	C₈×C₆₇	—	2	4	no	—
139	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
140	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
141	C₆₇⋊C₈	—	2	8	no	—
142	C₂²×C₆₇	—	2	4	no	—
143	C₂²×C₆₇	—	2	4	no	—
144	C₂²×C₆₇	—	2	2	no	—
145	C₂²×C₆₇	—	2	2	no	—
146	C₂²×C₆₇	#389	1	2	yes	—
147	C₂²×C₆₇	#389	1	2	yes	—
148	C₂²×C₆₇	#471	1	4	yes	—
149	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
151	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
152	C₄×C₆₇	—	4	4	no	—
153	C₄×C₆₇	#389	1	2	yes	—
154	C₄×C₆₇	#389	1	2	yes	—
155	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	Dic₆₇	—	8	4	no	—
160	C₂×#1951	—	67	1	no	maximal, Sylow
161	C₂×C₆₇	—	2	1	no	—
162	C₂×C₆₇	—	2	1	no	—
163	C₂×C₆₇	—	2	2	no	—
164	C₂×C₆₇	#1951	1	4	yes	complemented
165	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
166	C₂×C₆₇	C₂×#471	1	2	yes	—
167	C₂²×#108	—	67	1	no	—
168	C₂×#428	—	67	2	no	—
169	C₂²×#106	—	67	1	no	—
170	C₂×#405	D₆₇	1	1	yes	characteristic
171	#1951	—	67	8	no	—
172	C₂×#343	—	67	2	no	—
173	C₆₇	C₂×#1951	1	1	yes	characteristic, complemented, Sylow
174	C₂²×M₁₆	—	67	2	no	—
175	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
176	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
177	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
178	C₂²×C₄⋊C₄	—	67	2	no	—
179	C₂×#108	—	67	8	no	—
180	C₂×#108	—	67	4	no	—
181	#428	—	67	8	no	—
182	#428	—	134	4	no	—
183	C₂×#107	—	134	2	no	—
184	C₂×#106	—	67	4	no	—
185	C₂×#106	—	134	1	no	—
186	C₂×#106	—	134	1	no	—
187	C₂×#106	—	134	2	no	—
188	C₂×#106	—	134	2	no	—
189	#405	—	2	2	no	—
190	#405	C₂×D₆₇	1	4	yes	—
191	C₂×#97	—	2	1	no	—
192	C₂×C₄×C₈	—	2	1	no	—
193	#343	—	67	8	no	—
194	C₂³×C₄	—	134	2	no	—
195	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	67	8	no	—
199	C₂×M₁₆	—	67	4	no	—
200	C₂×M₁₆	—	134	2	no	—
201	C₂×M₁₆	—	134	4	no	—
202	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
203	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	134	2	no	—
206	C₂²×C₈	Dic₁₃₄	1	4	yes	—
207	C₂²×C₈	C₄×D₆₇	1	4	yes	—
208	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
209	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
210	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
211	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
212	#117	—	2	1	no	—
213	#117	—	2	1	no	—
214	#117	D₂₆₈	1	2	yes	—
215	#117	C₂²×D₆₇	1	2	yes	—
216	C₂×C₄⋊C₄	—	67	4	no	—
217	C₂×C₄⋊C₄	—	134	2	no	—
218	C₂×C₄⋊C₄	—	134	2	no	—
219	C₂×C₄⋊C₄	—	134	2	no	—
220	C₂×C₄⋊C₄	—	134	2	no	—
221	C₂×C₄⋊C₄	—	134	2	no	—
222	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₄²	—	2	1	no	—
225	#108	—	67	16	no	—
226	#107	—	134	8	no	—
227	#106	—	134	4	no	—
228	#106	—	134	4	no	—
229	#106	—	134	8	no	—
230	#97	—	2	4	no	—
231	C₄×C₈	—	2	4	no	—
232	C₂⁴	#11761628	1	1	yes	characteristic
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	268	2	no	—
235	C₂²×C₄	—	268	2	no	—
236	C₂²×C₄	—	536	4	no	—
237	C₂²×C₄	—	134	4	no	—
238	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
239	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
244	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
245	C₂²×C₄	#11761628	1	2	yes	—
246	C₂²×C₄	#11761628	1	1	yes	characteristic
247	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	67	8	no	—
251	M₁₆	—	134	8	no	—
252	M₁₆	—	134	4	no	—
253	M₁₆	C₂×Dic₁₃₄	1	4	yes	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	134	8	no	—
257	C₂×C₈	—	134	4	no	—
258	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
259	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
260	C₂×C₈	Dic₆₇⋊C₄	1	8	yes	—
261	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
262	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
263	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
264	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	268	2	no	—
267	C₄⋊C₄	—	268	2	no	—
268	C₄⋊C₄	—	134	4	no	—
269	C₄⋊C₄	—	134	4	no	—
270	C₄⋊C₄	C₂×D₂₆₈	1	4	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	C₄×Dic₁₃₄	1	2	yes	—
277	C₂³	#14524017	1	1	yes	characteristic
278	C₂³	#14524017	1	1	yes	characteristic
279	C₂³	#14524035	1	2	yes	—
280	C₂³	C₂×#11761628	1	1	yes	characteristic
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	268	4	no	—
284	C₂×C₄	—	536	4	no	—
285	C₂×C₄	—	536	4	no	—
286	C₂×C₄	—	536	4	no	—
287	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
288	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
289	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
290	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
291	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
292	C₂×C₄	#14524017	1	1	yes	characteristic
293	C₂×C₄	#14524017	1	1	yes	characteristic
294	C₂×C₄	#14524022	1	2	yes	—
295	C₂×C₄	#14524022	1	2	yes	—
296	C₂×C₄	#14524035	1	2	yes	—
297	C₂×C₄	#14524043	1	2	yes	—
298	C₂×C₄	#14524043	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
302	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
303	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
304	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
305	C₈	—	2	4	no	—
306	C₈	—	134	8	no	—
307	C₈	C₂×Dic₆₇⋊C₄	1	4	yes	—
308	C₈	C₂×Dic₆₇⋊C₄	1	4	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	#16863294	1	2	yes	—
314	C₂²	#16863294	1	2	yes	—
315	C₂²	#16863659	1	4	yes	—
316	C₂²	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
317	C₂²	C₂×#14524017	1	1	yes	characteristic
318	C₂²	C₂×#14524035	1	1	yes	characteristic
319	C₄	—	4	4	no	—
320	C₄	—	536	4	no	—
321	C₄	#16863294	1	2	yes	—
322	C₄	#16863294	1	2	yes	—
323	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
324	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
325	C₄	C₂×#14524022	1	1	yes	characteristic
326	C₄	C₂×#14524043	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	#22560989	1	4	yes	direct product factor
331	C₂	C₂×#16863294	1	1	yes	characteristic
332	C₂	C₂×#16863659	1	2	yes	—

	subgroup	G/H
center	C₂³	#14524017
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	C₂×#1951
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #22560989	C₂	direct (N × H)
9. C₂×#14523979	C₄	semidirect
39. #14523979	C₂×C₄	semidirect
164. C₂×C₆₇	#1951	semidirect
173. C₆₇	C₂×#1951	semidirect
330. C₂	#22560989	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×#14523979	2
C₂×#16863439	2
C₂²×#14523975	2
#22560989	2
C₂×#16863164	2
C₂×#405×C₆₇	2
C₂×#1951	67

Subgroups — C2×#22560989 #30859371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22560989 #30859371