Subgroups — C₆₇×#21158 #30700409

Order 17152.

982 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2316	C₂	1	2	yes	maximal
5	C₆₇×#1943	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1942	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1736	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
13	#427×C₆₇	—	2	4	no	—
14	C₂×#107×C₆₇	C₄	1	8	yes	complemented
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	#405×C₆₇	—	2	4	no	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#389×C₆₇	—	2	4	no	—
23	#389×C₆₇	C₂²	1	8	yes	—
24	#380×C₆₇	—	2	4	no	—
25	#340×C₆₇	C₂²	1	8	yes	—
26	C₂³×C₄×C₆₇	—	2	2	no	—
27	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₆₇	—	2	2	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₆₇	D₄	1	4	yes	—
33	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	#117×C₆₇	—	2	2	no	—
38	#117×C₆₇	—	2	2	no	—
39	#117×C₆₇	—	2	4	no	—
40	#117×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
50	C₂×C₄²×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
52	#107×C₆₇	—	2	4	no	—
53	#107×C₆₇	C₂×C₄	1	8	yes	—
54	#107×C₆₇	C₂×C₄	1	8	yes	—
55	#106×C₆₇	—	2	4	no	—
56	#97×C₆₇	—	2	4	no	—
57	C₄×C₈×C₆₇	—	2	4	no	—
58	#95×C₆₇	—	2	8	no	—
59	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	—	2	8	no	—
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	8	4	no	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₆₇	#41	1	2	yes	—
71	C₂²×C₄×C₆₇	#41	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₆₇	—	2	8	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₂×C₈×C₆₇	—	2	2	no	—
76	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	complemented
77	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
84	C₄⋊C₄×C₆₇	—	2	8	no	—
85	C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	4	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	8	8	no	—
90	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
91	C₂²⋊C₄×C₆₇	—	2	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	4	no	—
93	C₄²×C₆₇	—	2	2	no	—
94	C₄²×C₆₇	—	2	2	no	—
95	C₄²×C₆₇	—	4	4	no	—
96	C₂³×C₆₇	—	2	2	no	—
97	C₂³×C₆₇	—	2	2	no	—
98	C₂³×C₆₇	C₄×D₄	1	2	yes	—
99	C₂³×C₆₇	#121	1	2	yes	—
100	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	—	2	8	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	4	8	no	—
107	C₂×C₄×C₆₇	—	4	4	no	—
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	8	8	no	—
111	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₆₇	#121	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
116	C₂×C₄×C₆₇	#125	1	4	yes	—
117	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₈×C₆₇	—	2	4	no	—
125	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
126	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
127	C₂²×C₆₇	—	2	4	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	2	4	no	—
131	C₂²×C₆₇	#389	1	4	yes	—
132	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
134	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₂²×C₆₇	#576	1	2	yes	—
136	C₂²×C₆₇	#578	1	2	yes	—
137	C₄×C₆₇	—	4	8	no	—
138	C₄×C₆₇	—	4	4	no	—
139	C₄×C₆₇	—	8	4	no	—
140	C₄×C₆₇	#389	1	4	yes	—
141	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
145	#21158	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂×C₆₇	—	2	4	no	—
147	C₂×C₆₇	C₂×#389	1	1	yes	characteristic
148	C₂×C₆₇	#2948	1	1	yes	characteristic
149	C₂×C₆₇	#2950	1	1	yes	characteristic
150	C₂×C₆₇	#3061	1	2	yes	—
151	C₂×C₆₇	#3164	1	2	yes	—
152	C₂×#427	C₂×C₆₇	1	1	yes	characteristic
153	C₂²×#107	C₂×C₆₇	1	1	yes	characteristic
154	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
155	#2316	C₂×C₆₇	1	2	yes	—
156	#1943	C₂×C₆₇	1	4	yes	complemented
157	#1942	C₂×C₆₇	1	4	yes	complemented
158	#1736	C₂×C₆₇	1	2	yes	—
159	C₆₇	#21158	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
161	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
162	C₂×#117	C₂²×C₆₇	1	2	yes	—
163	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
165	#427	—	2	4	no	—
166	C₂×#107	C₄×C₆₇	1	8	yes	complemented
167	C₂×#107	C₂²×C₆₇	1	2	yes	—
168	C₂×#107	C₂²×C₆₇	1	2	yes	—
169	C₂×#107	C₂²×C₆₇	1	2	yes	—
170	C₂×#106	C₂²×C₆₇	1	2	yes	—
171	#405	—	2	4	no	—
172	C₂×#97	C₂²×C₆₇	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
174	#389	—	2	4	no	—
175	#389	C₂²×C₆₇	1	8	yes	—
176	#380	—	2	4	no	—
177	#340	C₂²×C₆₇	1	8	yes	—
178	C₂³×C₄	—	2	2	no	—
179	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	Q₈×C₆₇	1	4	yes	—
183	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
184	C₂²×C₈	D₄×C₆₇	1	4	yes	—
185	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
186	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
187	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	4	no	—
192	#117	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	8	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
197	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
199	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
204	#107	—	2	4	no	—
205	#107	C₂×C₄×C₆₇	1	8	yes	—
206	#107	C₂×C₄×C₆₇	1	8	yes	—
207	#106	—	2	4	no	—
208	#97	—	2	4	no	—
209	C₄×C₈	—	2	4	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	8	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	8	4	no	—
216	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
218	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
221	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
222	C₂²×C₄	#41×C₆₇	1	2	yes	—
223	C₂²×C₄	#41×C₆₇	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
225	M₁₆	—	2	8	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₄⋊C₄×C₆₇	1	16	yes	complemented
229	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
230	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
231	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
233	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
234	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
235	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	8	8	no	—
242	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	4	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₄×D₄×C₆₇	1	2	yes	—
251	C₂³	#121×C₆₇	1	2	yes	—
252	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
253	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	8	no	—
263	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
264	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
265	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
266	C₂×C₄	#121×C₆₇	1	2	yes	—
267	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
268	C₂×C₄	#125×C₆₇	1	4	yes	—
269	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
270	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
275	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
278	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	#389×C₆₇	1	4	yes	—
284	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
285	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
286	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
287	C₂²	#576×C₆₇	1	2	yes	—
288	C₂²	#578×C₆₇	1	2	yes	—
289	C₄	—	4	4	no	—
290	C₄	—	4	8	no	—
291	C₄	—	8	4	no	—
292	C₄	#389×C₆₇	1	4	yes	—
293	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
294	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
295	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
296	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
297	C₂	—	2	4	no	—
298	C₂	C₂×#389×C₆₇	1	1	yes	characteristic
299	C₂	C₆₇×#2948	1	1	yes	characteristic
300	C₂	C₆₇×#2950	1	1	yes	characteristic
301	C₂	C₆₇×#3061	1	2	yes	—
302	C₂	C₆₇×#3164	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₇×#1943	C₂	semidirect
6. C₆₇×#1942	C₂	semidirect
14. C₂×#107×C₆₇	C₄	semidirect
76. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
145. #21158	C₆₇	direct (N × H)
156. #1943	C₂×C₆₇	semidirect
157. #1942	C₂×C₆₇	semidirect
159. C₆₇	#21158	direct (N × H)
166. C₂×#107	C₄×C₆₇	semidirect
228. 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
C₂×#427×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2316	2
C₆₇×#1943	2
C₆₇×#1942	2
C₆₇×#1736	2
#21158	67

Subgroups — C67×#21158 #30700409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21158 #30700409