Subgroups — C₂×C₆₇×#1543 #30696949

Order 17152.

1274 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×#532×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1543	C₂	1	8	yes	maximal, direct product factor
4	C₂×#333×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#331×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₆₇	C₄	1	2	yes	—
7	#532×C₆₇	—	2	4	no	—
8	#532×C₆₇	C₄	1	4	yes	—
9	#532×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#124×C₆₇	C₄	1	2	yes	—
13	C₂×#121×C₆₇	—	2	2	no	—
14	C₂×C₄×D₄×C₆₇	—	2	2	no	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#431×C₆₇	—	2	4	no	—
17	#431×C₆₇	C₂²	1	8	yes	complemented
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
20	C₂×C₄×C₈×C₆₇	—	2	2	no	—
21	#333×C₆₇	C₂²	1	8	yes	complemented
22	#331×C₆₇	C₂²	1	8	yes	complemented
23	C₂×#41×C₆₇	—	2	4	no	—
24	C₂×#41×C₆₇	—	4	8	no	—
25	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
26	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₆₇	—	2	2	no	—
28	C₂²×D₄×C₆₇	—	2	2	no	—
29	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₆₇	—	2	2	no	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	D₄	1	2	yes	—
33	C₂²×C₈×C₆₇	—	2	4	no	—
34	C₄⋊Q₈×C₆₇	—	2	2	no	—
35	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
37	#127×C₆₇	—	2	2	no	—
38	#127×C₆₇	C₂×C₄	1	2	yes	—
39	#127×C₆₇	C₂³	1	2	yes	—
40	#124×C₆₇	—	2	4	no	—
41	#124×C₆₇	C₂×C₄	1	8	yes	—
42	#121×C₆₇	—	2	8	no	—
43	#121×C₆₇	—	2	8	no	—
44	C₄×D₄×C₆₇	—	2	8	no	—
45	C₄×D₄×C₆₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	D₄	1	4	yes	—
51	C₂×C₄²×C₆₇	D₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
54	C₄⋊C₈×C₆₇	—	2	4	no	—
55	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
56	C₂²⋊C₈×C₆₇	—	2	8	no	—
57	C₄×C₈×C₆₇	—	2	8	no	—
58	C₂⁴×C₆₇	—	2	2	no	—
59	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	#41×C₆₇	—	4	16	no	—
61	#41×C₆₇	—	4	16	no	—
62	C₂×Q₈×C₆₇	—	2	2	no	—
63	C₂×Q₈×C₆₇	—	4	4	no	—
64	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
65	C₂×D₄×C₆₇	—	2	8	no	—
66	C₂×D₄×C₆₇	—	2	4	no	—
67	C₂×D₄×C₆₇	—	2	2	no	—
68	C₂×D₄×C₆₇	—	2	4	no	—
69	C₂×D₄×C₆₇	—	4	4	no	—
70	C₂×D₄×C₆₇	—	4	8	no	—
71	C₂×D₄×C₆₇	—	4	8	no	—
72	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₆₇	—	2	4	no	—
74	C₂²×C₄×C₆₇	—	2	8	no	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	8	no	—
77	C₂²×C₄×C₆₇	—	4	8	no	—
78	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₆₇	—	2	8	no	—
87	C₂×C₈×C₆₇	—	4	8	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₂²⋊C₄×C₆₇	—	2	16	no	—
91	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
92	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₆₇	C₂×D₄	1	2	yes	—
96	C₄²×C₆₇	C₂×D₄	1	4	yes	—
97	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	—	2	4	no	—
100	C₂³×C₆₇	—	2	4	no	—
101	C₂³×C₆₇	—	4	4	no	—
102	C₂³×C₆₇	—	4	8	no	—
103	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	Q₈×C₆₇	—	4	4	no	—
106	Q₈×C₆₇	—	4	4	no	—
107	D₄×C₆₇	—	4	4	no	—
108	D₄×C₆₇	—	4	4	no	—
109	D₄×C₆₇	—	4	16	no	—
110	D₄×C₆₇	—	4	8	no	—
111	D₄×C₆₇	—	4	8	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	2	8	no	—
115	C₂×C₄×C₆₇	—	2	4	no	—
116	C₂×C₄×C₆₇	—	2	4	no	—
117	C₂×C₄×C₆₇	—	2	8	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	4	4	no	—
120	C₂×C₄×C₆₇	—	4	16	no	—
121	C₂×C₄×C₆₇	#100	1	4	yes	—
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
123	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₆₇	#135	1	8	yes	—
127	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
129	C₈×C₆₇	—	4	8	no	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	—	2	4	no	—
132	C₂²×C₆₇	—	4	4	no	—
133	C₂²×C₆₇	—	4	4	no	—
134	C₂²×C₆₇	—	4	8	no	—
135	C₂²×C₆₇	—	8	8	no	—
136	C₂²×C₆₇	C₂×#100	1	2	yes	—
137	C₂²×C₆₇	#416	1	4	yes	—
138	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₆₇	—	2	1	no	—
140	C₄×C₆₇	—	2	1	no	—
141	C₄×C₆₇	—	2	4	no	—
142	C₄×C₆₇	—	4	4	no	—
143	C₄×C₆₇	C₂×#100	1	4	yes	—
144	C₄×C₆₇	#416	1	4	yes	—
145	C₄×C₆₇	C₂×#135	1	4	yes	—
146	C₂×#1543	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂×C₆₇	—	4	2	no	—
148	C₂×C₆₇	—	8	4	no	—
149	C₂×C₆₇	#1543	1	4	yes	direct product factor
150	C₂×C₆₇	C₂²×#100	1	1	yes	characteristic
151	C₂×C₆₇	C₂×#416	1	2	yes	—
152	C₂×#532	C₂×C₆₇	1	1	yes	characteristic
153	C₂×#431	C₂×C₆₇	1	2	yes	complemented
154	#1543	C₂×C₆₇	1	8	yes	direct product factor
155	C₂×#333	C₂×C₆₇	1	2	yes	complemented
156	C₂×#331	C₂×C₆₇	1	2	yes	complemented
157	C₆₇	C₂×#1543	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×#41	C₄×C₆₇	1	2	yes	—
159	#532	—	2	4	no	—
160	#532	C₄×C₆₇	1	4	yes	—
161	#532	C₂²×C₆₇	1	4	yes	—
162	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
163	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
164	C₂×#124	C₄×C₆₇	1	2	yes	—
165	C₂×#121	—	2	2	no	—
166	C₂×C₄×D₄	—	2	2	no	—
167	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
168	#431	—	2	4	no	—
169	#431	C₂²×C₆₇	1	8	yes	complemented
170	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
171	C₂×C₂²⋊C₈	—	2	2	no	—
172	C₂×C₄×C₈	—	2	2	no	—
173	#333	C₂²×C₆₇	1	8	yes	complemented
174	#331	C₂²×C₆₇	1	8	yes	complemented
175	C₂×#41	—	2	4	no	—
176	C₂×#41	—	4	8	no	—
177	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
178	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	2	2	no	—
181	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
184	C₂³×C₄	D₄×C₆₇	1	2	yes	—
185	C₂²×C₈	—	2	4	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
188	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
189	#127	—	2	2	no	—
190	#127	C₂×C₄×C₆₇	1	2	yes	—
191	#127	C₂³×C₆₇	1	2	yes	—
192	#124	—	2	4	no	—
193	#124	C₂×C₄×C₆₇	1	8	yes	—
194	#121	—	2	8	no	—
195	#121	—	2	8	no	—
196	C₄×D₄	—	2	8	no	—
197	C₄×D₄	—	2	8	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	2	4	no	—
200	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
201	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
202	C₂×C₄²	D₄×C₆₇	1	2	yes	—
203	C₂×C₄²	D₄×C₆₇	1	4	yes	—
204	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
208	C₂²⋊C₈	—	2	8	no	—
209	C₄×C₈	—	2	8	no	—
210	C₂⁴	—	2	2	no	—
211	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
212	#41	—	4	16	no	—
213	#41	—	4	16	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	4	4	no	—
216	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
217	C₂×D₄	—	2	8	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	4	4	no	—
222	C₂×D₄	—	4	8	no	—
223	C₂×D₄	—	4	8	no	—
224	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	4	8	no	—
230	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
231	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
232	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
233	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
237	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	4	8	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	2	16	no	—
243	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
244	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
245	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
246	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₆₇	1	4	yes	—
248	C₄²	C₂×D₄×C₆₇	1	2	yes	—
249	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	4	8	no	—
255	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
256	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	16	no	—
262	D₄	—	4	8	no	—
263	D₄	—	4	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	#100×C₆₇	1	4	yes	—
274	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
277	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
278	C₂×C₄	#135×C₆₇	1	8	yes	—
279	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
280	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
281	C₈	—	4	8	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	8	no	—
287	C₂²	—	8	8	no	—
288	C₂²	C₂×#100×C₆₇	1	2	yes	—
289	C₂²	#416×C₆₇	1	4	yes	—
290	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	4	no	—
294	C₄	—	4	4	no	—
295	C₄	C₂×#100×C₆₇	1	4	yes	—
296	C₄	#416×C₆₇	1	4	yes	—
297	C₄	C₂×#135×C₆₇	1	4	yes	—
298	C₂	—	4	2	no	—
299	C₂	—	8	4	no	—
300	C₂	C₆₇×#1543	1	4	yes	direct product factor
301	C₂	C₂²×#100×C₆₇	1	1	yes	characteristic
302	C₂	C₂×#416×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂×#532×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#431×C₆₇	C₂	semidirect
3. C₆₇×#1543	C₂	direct (N × H)
4. C₂×#333×C₆₇	C₂	semidirect
5. C₂×#331×C₆₇	C₂	semidirect
17. #431×C₆₇	C₂²	semidirect
21. #333×C₆₇	C₂²	semidirect
22. #331×C₆₇	C₂²	semidirect
146. C₂×#1543	C₆₇	direct (N × H)
149. C₂×C₆₇	#1543	direct (N × H)
153. C₂×#431	C₂×C₆₇	semidirect
154. #1543	C₂×C₆₇	direct (N × H)
155. C₂×#333	C₂×C₆₇	semidirect
156. C₂×#331	C₂×C₆₇	semidirect
157. C₆₇	C₂×#1543	direct (N × H)
169. #431	C₂²×C₆₇	semidirect
173. #333	C₂²×C₆₇	semidirect
174. #331	C₂²×C₆₇	semidirect
300. C₂	C₆₇×#1543	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×#532×C₆₇	2
C₂×#431×C₆₇	2
C₆₇×#1543	2
C₂×#333×C₆₇	2
C₂×#331×C₆₇	2
C₂×#1543	67

Subgroups — C2×C67×#1543 #30696949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1543 #30696949