Subgroups — C₂³×#104×C₆₇ #30739341

Order 17152.

5758 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#104×C₆₇	C₂	1	28	yes	maximal, direct product factor
5	C₂²×#41×C₆₇	—	2	7	no	—
6	C₂²×#41×C₆₇	—	2	7	no	—
7	C₂²×#41×C₆₇	C₄	1	14	yes	complemented
8	C₂²×#41×C₆₇	C₂²	1	14	yes	—
9	C₂³×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
10	C₂³×D₄×C₆₇	—	2	1	no	—
11	C₂³×D₄×C₆₇	C₄	1	1	yes	characteristic, complemented
12	C₂⁴×C₄×C₆₇	—	2	1	no	—
13	C₂⁴×C₄×C₆₇	—	2	1	no	—
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	—	2	7	no	—
16	C₂²×M₁₆×C₆₇	C₂²	1	14	yes	complemented
17	C₂³×C₈×C₆₇	—	2	1	no	—
18	C₂²×C₄²×C₆₇	—	2	7	no	—
19	C₂²×C₄²×C₆₇	C₂²	1	14	yes	complemented
20	C₂×#104×C₆₇	C₂²	1	112	yes	direct product factor
21	C₂⁵×C₆₇	—	2	1	no	—
22	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
23	C₂×#41×C₆₇	—	2	84	no	—
24	C₂×#41×C₆₇	—	2	42	no	—
25	C₂×#41×C₆₇	—	2	42	no	—
26	C₂×#41×C₆₇	C₂×C₄	1	84	yes	complemented
27	C₂×#41×C₆₇	C₂³	1	28	yes	—
28	C₂²×Q₈×C₆₇	—	2	7	no	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	14	yes	complemented
30	C₂²×D₄×C₆₇	—	2	14	no	—
31	C₂²×D₄×C₆₇	—	2	14	no	—
32	C₂²×D₄×C₆₇	—	2	7	no	—
33	C₂²×D₄×C₆₇	C₂×C₄	1	14	yes	complemented
34	C₂³×C₄×C₆₇	—	2	16	no	—
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	—	2	14	no	—
37	C₂³×C₄×C₆₇	—	2	14	no	—
38	C₂³×C₄×C₆₇	—	2	14	no	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂×C₄	1	7	yes	—
41	C₂³×C₄×C₆₇	D₄	1	7	yes	—
42	C₂³×C₄×C₆₇	D₄	1	7	yes	—
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
45	C₂×M₁₆×C₆₇	—	2	42	no	—
46	C₂×M₁₆×C₆₇	C₂³	1	28	yes	complemented
47	C₂²×C₈×C₆₇	—	2	14	no	—
48	C₂×C₄²×C₆₇	—	2	42	no	—
49	C₂×C₄²×C₆₇	C₂³	1	28	yes	complemented
50	#104×C₆₇	C₂³	1	64	yes	direct product factor
51	C₂⁴×C₆₇	—	2	8	no	—
52	C₂⁴×C₆₇	—	2	14	no	—
53	C₂⁴×C₆₇	—	4	8	no	—
54	C₂⁴×C₆₇	C₂²⋊C₄	1	7	yes	—
55	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂⁴×C₆₇	C₂×D₄	1	7	yes	—
57	#41×C₆₇	—	2	168	no	—
58	#41×C₆₇	—	2	28	no	—
59	#41×C₆₇	—	2	28	no	—
60	#41×C₆₇	C₂²×C₄	1	56	yes	complemented
61	#41×C₆₇	C₂⁴	1	8	yes	—
62	C₂×Q₈×C₆₇	—	2	42	no	—
63	C₂×Q₈×C₆₇	C₂²×C₄	1	28	yes	complemented
64	C₂×D₄×C₆₇	—	2	84	no	—
65	C₂×D₄×C₆₇	—	2	28	no	—
66	C₂×D₄×C₆₇	—	2	42	no	—
67	C₂×D₄×C₆₇	C₂²×C₄	1	28	yes	complemented
68	C₂²×C₄×C₆₇	—	2	112	no	—
69	C₂²×C₄×C₆₇	—	2	14	no	—
70	C₂²×C₄×C₆₇	—	2	28	no	—
71	C₂²×C₄×C₆₇	—	2	84	no	—
72	C₂²×C₄×C₆₇	—	2	28	no	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	7	yes	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	7	yes	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	42	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	7	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	21	yes	—
78	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	7	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	21	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	7	yes	—
82	M₁₆×C₆₇	—	2	28	no	—
83	M₁₆×C₆₇	C₂⁴	1	8	yes	complemented
84	C₂×C₈×C₆₇	—	2	28	no	—
85	C₄²×C₆₇	—	2	28	no	—
86	C₄²×C₆₇	C₂⁴	1	8	yes	complemented
87	C₂³×C₆₇	—	2	28	no	—
88	C₂³×C₆₇	—	2	28	no	—
89	C₂³×C₆₇	—	2	28	no	—
90	C₂³×C₆₇	—	4	56	no	—
91	C₂³×C₆₇	#104	1	8	yes	direct product factor
92	C₂³×C₆₇	C₂×C₂²⋊C₄	1	21	yes	—
93	C₂³×C₆₇	C₂×C₂²⋊C₄	1	7	yes	—
94	C₂³×C₆₇	C₂²×D₄	1	7	yes	—
95	Q₈×C₆₇	—	2	28	no	—
96	Q₈×C₆₇	C₂³×C₄	1	8	yes	complemented
97	D₄×C₆₇	—	2	56	no	—
98	D₄×C₆₇	—	2	8	no	—
99	D₄×C₆₇	—	2	28	no	—
100	D₄×C₆₇	C₂³×C₄	1	8	yes	complemented
101	C₂×C₄×C₆₇	—	2	112	no	—
102	C₂×C₄×C₆₇	—	2	28	no	—
103	C₂×C₄×C₆₇	—	2	8	no	—
104	C₂×C₄×C₆₇	—	2	56	no	—
105	C₂×C₄×C₆₇	—	2	8	no	—
106	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	21	yes	—
107	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	21	yes	—
108	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	42	yes	—
109	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
110	C₂×C₄×C₆₇	C₂³×C₄	1	7	yes	—
111	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	7	yes	—
114	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
115	C₈×C₆₇	—	2	8	no	—
116	C₂²×C₆₇	—	2	14	no	—
117	C₂²×C₆₇	—	2	42	no	—
118	C₂²×C₆₇	—	2	8	no	—
119	C₂²×C₆₇	—	4	56	no	—
120	C₂²×C₆₇	C₂×#104	1	28	yes	direct product factor
121	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	7	yes	—
122	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	7	yes	—
123	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
124	C₄×C₆₇	—	2	16	no	—
125	C₄×C₆₇	—	2	8	no	—
126	C₄×C₆₇	C₂²×C₂²⋊C₄	1	7	yes	—
127	C₄×C₆₇	C₂²×C₂²⋊C₄	1	7	yes	—
128	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
129	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
130	C₂³×#104	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₆₇	—	2	1	no	—
132	C₂×C₆₇	—	2	7	no	—
133	C₂×C₆₇	—	4	8	no	—
134	C₂×C₆₇	C₂²×#104	1	14	yes	direct product factor
135	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
136	C₂³×#41	C₂×C₆₇	1	1	yes	characteristic
137	C₂³×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
138	C₂³×C₄²	C₂×C₆₇	1	1	yes	characteristic, complemented
139	C₂²×#104	C₂×C₆₇	1	28	yes	direct product factor
140	C₆₇	C₂³×#104	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²×#41	—	2	7	no	—
142	C₂²×#41	—	2	7	no	—
143	C₂²×#41	C₄×C₆₇	1	14	yes	complemented
144	C₂²×#41	C₂²×C₆₇	1	14	yes	—
145	C₂³×Q₈	C₄×C₆₇	1	1	yes	characteristic, complemented
146	C₂³×D₄	—	2	1	no	—
147	C₂³×D₄	C₄×C₆₇	1	1	yes	characteristic, complemented
148	C₂⁴×C₄	—	2	1	no	—
149	C₂⁴×C₄	—	2	1	no	—
150	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
151	C₂²×M₁₆	—	2	7	no	—
152	C₂²×M₁₆	C₂²×C₆₇	1	14	yes	complemented
153	C₂³×C₈	—	2	1	no	—
154	C₂²×C₄²	—	2	7	no	—
155	C₂²×C₄²	C₂²×C₆₇	1	14	yes	complemented
156	C₂×#104	C₂²×C₆₇	1	112	yes	direct product factor
157	C₂⁵	—	2	1	no	—
158	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
159	C₂×#41	—	2	84	no	—
160	C₂×#41	—	2	42	no	—
161	C₂×#41	—	2	42	no	—
162	C₂×#41	C₂×C₄×C₆₇	1	84	yes	complemented
163	C₂×#41	C₂³×C₆₇	1	28	yes	—
164	C₂²×Q₈	—	2	7	no	—
165	C₂²×Q₈	C₂×C₄×C₆₇	1	14	yes	complemented
166	C₂²×D₄	—	2	14	no	—
167	C₂²×D₄	—	2	14	no	—
168	C₂²×D₄	—	2	7	no	—
169	C₂²×D₄	C₂×C₄×C₆₇	1	14	yes	complemented
170	C₂³×C₄	—	2	16	no	—
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	—	2	14	no	—
173	C₂³×C₄	—	2	14	no	—
174	C₂³×C₄	—	2	14	no	—
175	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
176	C₂³×C₄	C₂×C₄×C₆₇	1	7	yes	—
177	C₂³×C₄	D₄×C₆₇	1	7	yes	—
178	C₂³×C₄	D₄×C₆₇	1	7	yes	—
179	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
180	C₂³×C₄	C₂³×C₆₇	1	7	yes	—
181	C₂×M₁₆	—	2	42	no	—
182	C₂×M₁₆	C₂³×C₆₇	1	28	yes	complemented
183	C₂²×C₈	—	2	14	no	—
184	C₂×C₄²	—	2	42	no	—
185	C₂×C₄²	C₂³×C₆₇	1	28	yes	complemented
186	#104	C₂³×C₆₇	1	64	yes	direct product factor
187	C₂⁴	—	2	8	no	—
188	C₂⁴	—	2	14	no	—
189	C₂⁴	—	4	8	no	—
190	C₂⁴	C₂²⋊C₄×C₆₇	1	7	yes	—
191	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
192	C₂⁴	C₂×D₄×C₆₇	1	7	yes	—
193	#41	—	2	168	no	—
194	#41	—	2	28	no	—
195	#41	—	2	28	no	—
196	#41	C₂²×C₄×C₆₇	1	56	yes	complemented
197	#41	C₂⁴×C₆₇	1	8	yes	—
198	C₂×Q₈	—	2	42	no	—
199	C₂×Q₈	C₂²×C₄×C₆₇	1	28	yes	complemented
200	C₂×D₄	—	2	84	no	—
201	C₂×D₄	—	2	28	no	—
202	C₂×D₄	—	2	42	no	—
203	C₂×D₄	C₂²×C₄×C₆₇	1	28	yes	complemented
204	C₂²×C₄	—	2	112	no	—
205	C₂²×C₄	—	2	14	no	—
206	C₂²×C₄	—	2	28	no	—
207	C₂²×C₄	—	2	84	no	—
208	C₂²×C₄	—	2	28	no	—
209	C₂²×C₄	C₂²⋊C₄×C₆₇	1	7	yes	—
210	C₂²×C₄	C₂²⋊C₄×C₆₇	1	7	yes	—
211	C₂²×C₄	C₂²⋊C₄×C₆₇	1	42	yes	—
212	C₂²×C₄	C₂²×C₄×C₆₇	1	7	yes	—
213	C₂²×C₄	C₂²×C₄×C₆₇	1	21	yes	—
214	C₂²×C₄	C₂×D₄×C₆₇	1	21	yes	—
215	C₂²×C₄	C₂×D₄×C₆₇	1	7	yes	—
216	C₂²×C₄	C₂×D₄×C₆₇	1	21	yes	—
217	C₂²×C₄	C₂⁴×C₆₇	1	7	yes	—
218	M₁₆	—	2	28	no	—
219	M₁₆	C₂⁴×C₆₇	1	8	yes	complemented
220	C₂×C₈	—	2	28	no	—
221	C₄²	—	2	28	no	—
222	C₄²	C₂⁴×C₆₇	1	8	yes	complemented
223	C₂³	—	2	28	no	—
224	C₂³	—	2	28	no	—
225	C₂³	—	2	28	no	—
226	C₂³	—	4	56	no	—
227	C₂³	#104×C₆₇	1	8	yes	direct product factor
228	C₂³	C₂×C₂²⋊C₄×C₆₇	1	21	yes	—
229	C₂³	C₂×C₂²⋊C₄×C₆₇	1	7	yes	—
230	C₂³	C₂²×D₄×C₆₇	1	7	yes	—
231	Q₈	—	2	28	no	—
232	Q₈	C₂³×C₄×C₆₇	1	8	yes	complemented
233	D₄	—	2	56	no	—
234	D₄	—	2	8	no	—
235	D₄	—	2	28	no	—
236	D₄	C₂³×C₄×C₆₇	1	8	yes	complemented
237	C₂×C₄	—	2	112	no	—
238	C₂×C₄	—	2	28	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	56	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	21	yes	—
243	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	21	yes	—
244	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	42	yes	—
245	C₂×C₄	C₂³×C₄×C₆₇	1	7	yes	—
246	C₂×C₄	C₂³×C₄×C₆₇	1	7	yes	—
247	C₂×C₄	C₂²×D₄×C₆₇	1	7	yes	—
248	C₂×C₄	C₂²×D₄×C₆₇	1	7	yes	—
249	C₂×C₄	C₂²×D₄×C₆₇	1	7	yes	—
250	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₂²	—	2	14	no	—
253	C₂²	—	2	42	no	—
254	C₂²	—	2	8	no	—
255	C₂²	—	4	56	no	—
256	C₂²	C₂×#104×C₆₇	1	28	yes	direct product factor
257	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	7	yes	—
258	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	7	yes	—
259	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
260	C₄	—	2	16	no	—
261	C₄	—	2	8	no	—
262	C₄	C₂²×C₂²⋊C₄×C₆₇	1	7	yes	—
263	C₄	C₂²×C₂²⋊C₄×C₆₇	1	7	yes	—
264	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
265	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	7	no	—
268	C₂	—	4	8	no	—
269	C₂	C₂²×#104×C₆₇	1	14	yes	direct product factor
270	C₂	C₂³×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₆₇	D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂⁴×C₆₇	C₂²⋊C₄
prime radical	C₂³×#41×C₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆×C₆₇	C₂	semidirect
3. C₂³×C₄²×C₆₇	C₂	semidirect
4. C₂²×#104×C₆₇	C₂	direct (N × H)
7. C₂²×#41×C₆₇	C₄	semidirect
9. C₂³×Q₈×C₆₇	C₄	semidirect
11. C₂³×D₄×C₆₇	C₄	semidirect
16. C₂²×M₁₆×C₆₇	C₂²	semidirect
19. C₂²×C₄²×C₆₇	C₂²	semidirect
20. C₂×#104×C₆₇	C₂²	direct (N × H)
26. C₂×#41×C₆₇	C₂×C₄	semidirect
29. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
33. C₂²×D₄×C₆₇	C₂×C₄	semidirect
46. C₂×M₁₆×C₆₇	C₂³	semidirect
49. C₂×C₄²×C₆₇	C₂³	semidirect
50. #104×C₆₇	C₂³	direct (N × H)
60. #41×C₆₇	C₂²×C₄	semidirect
63. C₂×Q₈×C₆₇	C₂²×C₄	semidirect
67. C₂×D₄×C₆₇	C₂²×C₄	semidirect
83. M₁₆×C₆₇	C₂⁴	semidirect
86. C₄²×C₆₇	C₂⁴	semidirect
91. C₂³×C₆₇	#104	direct (N × H)
96. Q₈×C₆₇	C₂³×C₄	semidirect
100. D₄×C₆₇	C₂³×C₄	semidirect
120. C₂²×C₆₇	C₂×#104	direct (N × H)
130. C₂³×#104	C₆₇	direct (N × H)
134. C₂×C₆₇	C₂²×#104	direct (N × H)
137. C₂³×M₁₆	C₂×C₆₇	semidirect
138. C₂³×C₄²	C₂×C₆₇	semidirect
139. C₂²×#104	C₂×C₆₇	direct (N × H)
140. C₆₇	C₂³×#104	direct (N × H)
143. C₂²×#41	C₄×C₆₇	semidirect
145. C₂³×Q₈	C₄×C₆₇	semidirect
147. C₂³×D₄	C₄×C₆₇	semidirect
152. C₂²×M₁₆	C₂²×C₆₇	semidirect
155. C₂²×C₄²	C₂²×C₆₇	semidirect
156. C₂×#104	C₂²×C₆₇	direct (N × H)
162. C₂×#41	C₂×C₄×C₆₇	semidirect
165. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
169. C₂²×D₄	C₂×C₄×C₆₇	semidirect
182. C₂×M₁₆	C₂³×C₆₇	semidirect
185. C₂×C₄²	C₂³×C₆₇	semidirect
186. #104	C₂³×C₆₇	direct (N × H)
196. #41	C₂²×C₄×C₆₇	semidirect
199. C₂×Q₈	C₂²×C₄×C₆₇	semidirect
203. C₂×D₄	C₂²×C₄×C₆₇	semidirect
219. M₁₆	C₂⁴×C₆₇	semidirect
222. C₄²	C₂⁴×C₆₇	semidirect
227. C₂³	#104×C₆₇	direct (N × H)
232. Q₈	C₂³×C₄×C₆₇	semidirect
236. D₄	C₂³×C₄×C₆₇	semidirect
256. C₂²	C₂×#104×C₆₇	direct (N × H)
269. C₂	C₂²×#104×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₆₇	2
C₂³×M₁₆×C₆₇	2
C₂³×C₄²×C₆₇	2
C₂²×#104×C₆₇	2
C₂³×#104	67

Subgroups — C23×#104×C67 #30739341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#104×C₆₇ #30739341