Subgroups — C₆₇×#10843 #30690094

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#340×C₆₇	C₂	1	4	yes	maximal
5	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
7	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#97×C₆₇	—	2	2	no	—
11	C₂×#97×C₆₇	—	2	2	no	—
12	C₂×#97×C₆₇	C₄	1	4	yes	complemented
13	#384×C₆₇	—	2	2	no	—
14	#384×C₆₇	C₄	1	4	yes	complemented
15	#382×C₆₇	—	2	2	no	—
16	#382×C₆₇	C₄	1	4	yes	complemented
17	C₂×#95×C₆₇	—	2	2	no	—
18	#340×C₆₇	—	2	8	no	—
19	C₂³×C₄×C₆₇	—	2	2	no	—
20	C₂³×C₄×C₆₇	—	2	2	no	—
21	C₂³×C₄×C₆₇	D₄	1	2	yes	—
22	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	—	2	2	no	—
24	C₂²×C₈×C₆₇	—	2	2	no	—
25	C₂²×C₈×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	—	2	2	no	—
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₆₇	D₄	1	4	yes	—
29	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
30	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₆₇	D₄	1	2	yes	—
44	C₂×C₄²×C₆₇	D₄	1	2	yes	—
45	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
46	#97×C₆₇	—	2	8	no	—
47	#95×C₆₇	—	2	8	no	—
48	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	—	2	4	no	—
51	C₂²×C₄×C₆₇	—	2	4	no	—
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	—	4	2	no	—
54	C₂²×C₄×C₆₇	—	4	2	no	—
55	C₂²×C₄×C₆₇	—	4	2	no	—
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	—	4	8	no	—
58	C₂²×C₄×C₆₇	—	4	4	no	—
59	C₂²×C₄×C₆₇	—	4	4	no	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
62	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₆₇	#41	1	2	yes	—
72	C₂²×C₄×C₆₇	#41	1	2	yes	—
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	4	no	—
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	4	no	—
79	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	complemented
80	C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₄⋊C₄×C₆₇	—	2	8	no	—
82	C₄⋊C₄×C₆₇	—	4	16	no	—
83	C₄⋊C₄×C₆₇	—	4	4	no	—
84	C₄⋊C₄×C₆₇	—	4	4	no	—
85	C₄²×C₆₇	C₄²	1	4	yes	—
86	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
87	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
88	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
89	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₆₇	#124	1	1	yes	characteristic
93	C₂³×C₆₇	#124	1	1	yes	characteristic
94	C₂³×C₆₇	#127	1	1	yes	characteristic
95	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
96	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
97	C₂³×C₆₇	#137	1	4	yes	—
98	C₂×C₄×C₆₇	—	2	8	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	2	8	no	—
102	C₂×C₄×C₆₇	—	4	4	no	—
103	C₂×C₄×C₆₇	—	4	4	no	—
104	C₂×C₄×C₆₇	—	4	8	no	—
105	C₂×C₄×C₆₇	—	4	8	no	—
106	C₂×C₄×C₆₇	—	8	4	no	—
107	C₂×C₄×C₆₇	—	8	4	no	—
108	C₂×C₄×C₆₇	—	8	2	no	—
109	C₂×C₄×C₆₇	—	8	4	no	—
110	C₂×C₄×C₆₇	—	8	2	no	—
111	C₂×C₄×C₆₇	#95	1	8	yes	—
112	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#117	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	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₆₇	#121	1	2	yes	—
123	C₂×C₄×C₆₇	#121	1	2	yes	—
124	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	#125	1	2	yes	—
127	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
128	C₈×C₆₇	—	2	8	no	—
129	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
130	C₂²×C₆₇	#382	1	1	yes	characteristic
131	C₂²×C₆₇	#384	1	1	yes	characteristic
132	C₂²×C₆₇	#390	1	1	yes	characteristic
133	C₂²×C₆₇	#390	1	1	yes	characteristic
134	C₂²×C₆₇	#391	1	1	yes	characteristic
135	C₂²×C₆₇	#391	1	1	yes	characteristic
136	C₂²×C₆₇	#417	1	4	yes	—
137	C₂²×C₆₇	#418	1	2	yes	—
138	C₂²×C₆₇	#419	1	2	yes	—
139	C₂²×C₆₇	#428	1	4	yes	—
140	C₂²×C₆₇	#489	1	4	yes	—
141	C₂²×C₆₇	#490	1	2	yes	—
142	C₂²×C₆₇	#491	1	2	yes	—
143	C₂²×C₆₇	#496	1	2	yes	—
144	C₂²×C₆₇	#497	1	2	yes	—
145	C₂²×C₆₇	#501	1	4	yes	—
146	C₄×C₆₇	—	2	8	no	—
147	C₄×C₆₇	—	8	4	no	—
148	C₄×C₆₇	—	8	4	no	—
149	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
150	C₄×C₆₇	#380	1	2	yes	—
151	C₄×C₆₇	#389	1	2	yes	—
152	C₄×C₆₇	#389	1	2	yes	—
153	#10843	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
155	C₂×C₆₇	#1736	1	2	yes	—
156	C₂×C₆₇	#1836	1	2	yes	—
157	C₂×C₆₇	#1847	1	2	yes	—
158	C₂×C₆₇	#1959	1	2	yes	—
159	C₂×C₆₇	#1960	1	1	yes	characteristic
160	C₂×C₆₇	#1961	1	1	yes	characteristic
161	C₂×C₆₇	#1992	1	2	yes	—
162	C₂×C₆₇	#1993	1	2	yes	—
163	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic
164	C₂×#384	C₂×C₆₇	1	1	yes	characteristic
165	C₂×#382	C₂×C₆₇	1	1	yes	characteristic
166	C₂×#340	C₂×C₆₇	1	4	yes	—
167	C₆₇	#10843	1	1	yes	characteristic, direct product factor, Sylow
168	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
169	C₂²×C₄⋊C₄	—	2	2	no	—
170	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
171	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
172	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
173	C₂×#97	—	2	2	no	—
174	C₂×#97	—	2	2	no	—
175	C₂×#97	C₄×C₆₇	1	4	yes	complemented
176	#384	—	2	2	no	—
177	#384	C₄×C₆₇	1	4	yes	complemented
178	#382	—	2	2	no	—
179	#382	C₄×C₆₇	1	4	yes	complemented
180	C₂×#95	—	2	2	no	—
181	#340	—	2	8	no	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	—	2	2	no	—
184	C₂³×C₄	D₄×C₆₇	1	2	yes	—
185	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
191	C₂²×C₈	D₄×C₆₇	1	4	yes	—
192	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	8	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	4	8	no	—
201	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
203	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
204	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
205	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
206	C₂×C₄²	D₄×C₆₇	1	2	yes	—
207	C₂×C₄²	D₄×C₆₇	1	2	yes	—
208	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
209	#97	—	2	8	no	—
210	#95	—	2	8	no	—
211	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	8	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
224	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
225	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
232	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
234	C₂²×C₄	#41×C₆₇	1	2	yes	—
235	C₂²×C₄	#41×C₆₇	1	2	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₄⋊C₄×C₆₇	1	8	yes	complemented
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	4	16	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄²	C₄²×C₆₇	1	4	yes	—
249	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
250	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
251	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
252	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
253	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
254	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
255	C₂³	#124×C₆₇	1	1	yes	characteristic
256	C₂³	#124×C₆₇	1	1	yes	characteristic
257	C₂³	#127×C₆₇	1	1	yes	characteristic
258	C₂³	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
259	C₂³	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
260	C₂³	#137×C₆₇	1	4	yes	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	4	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	#95×C₆₇	1	8	yes	—
275	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
278	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
280	C₂×C₄	#117×C₆₇	1	2	yes	—
281	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
282	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
283	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
284	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
285	C₂×C₄	#121×C₆₇	1	2	yes	—
286	C₂×C₄	#121×C₆₇	1	2	yes	—
287	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
288	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
289	C₂×C₄	#125×C₆₇	1	2	yes	—
290	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
291	C₈	—	2	8	no	—
292	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
293	C₂²	#382×C₆₇	1	1	yes	characteristic
294	C₂²	#384×C₆₇	1	1	yes	characteristic
295	C₂²	#390×C₆₇	1	1	yes	characteristic
296	C₂²	#390×C₆₇	1	1	yes	characteristic
297	C₂²	#391×C₆₇	1	1	yes	characteristic
298	C₂²	#391×C₆₇	1	1	yes	characteristic
299	C₂²	#417×C₆₇	1	4	yes	—
300	C₂²	#418×C₆₇	1	2	yes	—
301	C₂²	#419×C₆₇	1	2	yes	—
302	C₂²	#428×C₆₇	1	4	yes	—
303	C₂²	#489×C₆₇	1	4	yes	—
304	C₂²	#490×C₆₇	1	2	yes	—
305	C₂²	#491×C₆₇	1	2	yes	—
306	C₂²	#496×C₆₇	1	2	yes	—
307	C₂²	#497×C₆₇	1	2	yes	—
308	C₂²	#501×C₆₇	1	4	yes	—
309	C₄	—	2	8	no	—
310	C₄	—	8	4	no	—
311	C₄	—	8	4	no	—
312	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
313	C₄	#380×C₆₇	1	2	yes	—
314	C₄	#389×C₆₇	1	2	yes	—
315	C₄	#389×C₆₇	1	2	yes	—
316	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
317	C₂	C₆₇×#1736	1	2	yes	—
318	C₂	C₆₇×#1836	1	2	yes	—
319	C₂	C₆₇×#1847	1	2	yes	—
320	C₂	C₆₇×#1959	1	2	yes	—
321	C₂	C₆₇×#1960	1	1	yes	characteristic
322	C₂	C₆₇×#1961	1	1	yes	characteristic
323	C₂	C₆₇×#1992	1	2	yes	—
324	C₂	C₆₇×#1993	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×#97×C₆₇	C₄	semidirect
14. #384×C₆₇	C₄	semidirect
16. #382×C₆₇	C₄	semidirect
79. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
153. #10843	C₆₇	direct (N × H)
167. C₆₇	#10843	direct (N × H)
175. C₂×#97	C₄×C₆₇	semidirect
177. #384	C₄×C₆₇	semidirect
179. #382	C₄×C₆₇	semidirect
242. C₂×C₈	C₄⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#97×C₆₇	2
C₂×#384×C₆₇	2
C₂×#382×C₆₇	2
C₂×#340×C₆₇	2
#10843	67

Subgroups — C67×#10843 #30690094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10843 #30690094