Subgroups — C₂²×#332×C₆₇ #30696945

Order 17152.

1334 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#332×C₆₇	C₂	1	12	yes	maximal, direct product factor
5	C₂³×Q₈×C₆₇	—	2	1	no	—
6	C₂³×C₈×C₆₇	—	2	1	no	—
7	C₂³×C₈×C₆₇	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₆₇	—	2	3	no	—
9	C₂×C₄⋊Q₈×C₆₇	—	2	3	no	—
10	C₂×C₄⋊Q₈×C₆₇	C₄	1	6	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	6	yes	—
12	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	—	2	3	no	—
16	C₂×C₄⋊C₈×C₆₇	—	2	3	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	6	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	6	yes	—
19	#332×C₆₇	C₂²	1	16	yes	direct product factor
20	C₂²×Q₈×C₆₇	—	2	6	no	—
21	C₂²×Q₈×C₆₇	—	2	6	no	—
22	C₂²×Q₈×C₆₇	—	4	4	no	—
23	C₂²×Q₈×C₆₇	—	4	4	no	—
24	C₂³×C₄×C₆₇	—	2	2	no	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	—	2	6	no	—
29	C₂²×C₈×C₆₇	—	2	6	no	—
30	C₂²×C₈×C₆₇	—	4	4	no	—
31	C₂²×C₈×C₆₇	—	4	4	no	—
32	C₄⋊Q₈×C₆₇	—	2	12	no	—
33	C₄⋊Q₈×C₆₇	—	2	6	no	—
34	C₄⋊Q₈×C₆₇	—	2	6	no	—
35	C₄⋊Q₈×C₆₇	C₂×C₄	1	12	yes	—
36	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	12	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₆₇	D₄	1	3	yes	—
43	C₂×C₄²×C₆₇	D₄	1	3	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
45	C₄⋊C₈×C₆₇	—	2	6	no	—
46	C₄⋊C₈×C₆₇	—	2	6	no	—
47	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
48	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
49	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Q₈×C₆₇	—	2	12	no	—
51	C₂×Q₈×C₆₇	—	2	4	no	—
52	C₂×Q₈×C₆₇	—	4	24	no	—
53	C₂×Q₈×C₆₇	—	4	12	no	—
54	C₂×Q₈×C₆₇	—	4	12	no	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	2	12	no	—
57	C₂²×C₄×C₆₇	—	4	8	no	—
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
61	C₂²×C₄×C₆₇	D₈	1	4	yes	—
62	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
63	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
64	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
68	C₂×C₈×C₆₇	—	2	4	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	4	12	no	—
71	C₂×C₈×C₆₇	—	4	12	no	—
72	C₄⋊C₄×C₆₇	—	2	12	no	—
73	C₄⋊C₄×C₆₇	—	2	12	no	—
74	C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
76	C₄²×C₆₇	C₂²⋊C₄	1	6	yes	—
77	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
78	C₄²×C₆₇	C₂×D₄	1	3	yes	—
79	C₄²×C₆₇	C₂×D₄	1	3	yes	—
80	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
81	C₂³×C₆₇	#101	1	4	yes	—
82	C₂³×C₆₇	#102	1	4	yes	—
83	C₂³×C₆₇	#103	1	4	yes	—
84	C₂³×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
85	Q₈×C₆₇	—	4	4	no	—
86	Q₈×C₆₇	—	4	4	no	—
87	Q₈×C₆₇	—	4	24	no	—
88	C₂×C₄×C₆₇	—	2	6	no	—
89	C₂×C₄×C₆₇	—	2	6	no	—
90	C₂×C₄×C₆₇	—	2	8	no	—
91	C₂×C₄×C₆₇	—	4	24	no	—
92	C₂×C₄×C₆₇	#102	1	12	yes	—
93	C₂×C₄×C₆₇	#103	1	12	yes	—
94	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
97	C₂×C₄×C₆₇	C₂×D₈	1	6	yes	—
98	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
99	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
100	C₂×C₄×C₆₇	C₂×Q₁₆	1	6	yes	—
101	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
104	C₈×C₆₇	—	4	4	no	—
105	C₈×C₆₇	—	4	4	no	—
106	C₂²×C₆₇	#332	1	16	yes	direct product factor
107	C₂²×C₆₇	C₂×#101	1	6	yes	—
108	C₂²×C₆₇	C₂×#102	1	6	yes	—
109	C₂²×C₆₇	C₂×#103	1	6	yes	—
110	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₆₇	—	2	3	no	—
112	C₄×C₆₇	—	2	1	no	—
113	C₄×C₆₇	—	4	8	no	—
114	C₄×C₆₇	C₂×#102	1	6	yes	—
115	C₄×C₆₇	C₂×#103	1	6	yes	—
116	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
117	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
118	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
119	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
120	C₂²×#332	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₆₇	C₂×#332	1	12	yes	direct product factor
122	C₂×C₆₇	C₂²×#101	1	1	yes	characteristic
123	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
124	C₂×C₆₇	C₂²×#103	1	1	yes	characteristic
125	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
126	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
127	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
128	C₂×#332	C₂×C₆₇	1	12	yes	direct product factor
129	C₆₇	C₂²×#332	1	1	yes	characteristic, direct product factor, Sylow
130	C₂³×Q₈	—	2	1	no	—
131	C₂³×C₈	—	2	1	no	—
132	C₂³×C₈	—	2	1	no	—
133	C₂×C₄⋊Q₈	—	2	3	no	—
134	C₂×C₄⋊Q₈	—	2	3	no	—
135	C₂×C₄⋊Q₈	C₄×C₆₇	1	6	yes	—
136	C₂×C₄⋊Q₈	C₂²×C₆₇	1	6	yes	—
137	C₂²×C₄⋊C₄	—	2	1	no	—
138	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
139	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
140	C₂×C₄⋊C₈	—	2	3	no	—
141	C₂×C₄⋊C₈	—	2	3	no	—
142	C₂×C₄⋊C₈	C₂²×C₆₇	1	6	yes	—
143	C₂×C₄⋊C₈	C₂²×C₆₇	1	6	yes	—
144	#332	C₂²×C₆₇	1	16	yes	direct product factor
145	C₂²×Q₈	—	2	6	no	—
146	C₂²×Q₈	—	2	6	no	—
147	C₂²×Q₈	—	4	4	no	—
148	C₂²×Q₈	—	4	4	no	—
149	C₂³×C₄	—	2	2	no	—
150	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
151	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
152	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
153	C₂²×C₈	—	2	6	no	—
154	C₂²×C₈	—	2	6	no	—
155	C₂²×C₈	—	4	4	no	—
156	C₂²×C₈	—	4	4	no	—
157	C₄⋊Q₈	—	2	12	no	—
158	C₄⋊Q₈	—	2	6	no	—
159	C₄⋊Q₈	—	2	6	no	—
160	C₄⋊Q₈	C₂×C₄×C₆₇	1	12	yes	—
161	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	6	no	—
163	C₂×C₄⋊C₄	—	2	6	no	—
164	C₂×C₄⋊C₄	—	2	6	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	12	yes	—
166	C₂×C₄²	C₂×C₄×C₆₇	1	3	yes	—
167	C₂×C₄²	D₄×C₆₇	1	3	yes	—
168	C₂×C₄²	D₄×C₆₇	1	3	yes	—
169	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
170	C₄⋊C₈	—	2	6	no	—
171	C₄⋊C₈	—	2	6	no	—
172	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
173	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
174	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
175	C₂×Q₈	—	2	12	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	4	24	no	—
178	C₂×Q₈	—	4	12	no	—
179	C₂×Q₈	—	4	12	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	4	8	no	—
183	C₂²×C₄	C₂²⋊C₄×C₆₇	1	3	yes	—
184	C₂²×C₄	C₂²⋊C₄×C₆₇	1	3	yes	—
185	C₂²×C₄	C₂²⋊C₄×C₆₇	1	3	yes	—
186	C₂²×C₄	D₈×C₆₇	1	4	yes	—
187	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
188	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
189	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₆₇	1	3	yes	—
191	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
192	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	12	no	—
196	C₂×C₈	—	4	12	no	—
197	C₄⋊C₄	—	2	12	no	—
198	C₄⋊C₄	—	2	12	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
201	C₄²	C₂²⋊C₄×C₆₇	1	6	yes	—
202	C₄²	C₂²×C₄×C₆₇	1	3	yes	—
203	C₄²	C₂×D₄×C₆₇	1	3	yes	—
204	C₄²	C₂×D₄×C₆₇	1	3	yes	—
205	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
206	C₂³	#101×C₆₇	1	4	yes	—
207	C₂³	#102×C₆₇	1	4	yes	—
208	C₂³	#103×C₆₇	1	4	yes	—
209	C₂³	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
210	Q₈	—	4	4	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	4	24	no	—
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	2	6	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	24	no	—
217	C₂×C₄	#102×C₆₇	1	12	yes	—
218	C₂×C₄	#103×C₆₇	1	12	yes	—
219	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
220	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
221	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
222	C₂×C₄	C₂×D₈×C₆₇	1	6	yes	—
223	C₂×C₄	C₂×SD₁₆×C₆₇	1	6	yes	—
224	C₂×C₄	C₂×SD₁₆×C₆₇	1	6	yes	—
225	C₂×C₄	C₂×Q₁₆×C₆₇	1	6	yes	—
226	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
228	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
229	C₈	—	4	4	no	—
230	C₈	—	4	4	no	—
231	C₂²	#332×C₆₇	1	16	yes	direct product factor
232	C₂²	C₂×#101×C₆₇	1	6	yes	—
233	C₂²	C₂×#102×C₆₇	1	6	yes	—
234	C₂²	C₂×#103×C₆₇	1	6	yes	—
235	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
236	C₄	—	2	3	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	8	no	—
239	C₄	C₂×#102×C₆₇	1	6	yes	—
240	C₄	C₂×#103×C₆₇	1	6	yes	—
241	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
242	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
243	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
244	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
245	C₂	C₂×#332×C₆₇	1	12	yes	direct product factor
246	C₂	C₂²×#101×C₆₇	1	1	yes	characteristic
247	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic
248	C₂	C₂²×#103×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
4. C₂×#332×C₆₇	C₂	direct (N × H)
19. #332×C₆₇	C₂²	direct (N × H)
106. C₂²×C₆₇	#332	direct (N × H)
120. C₂²×#332	C₆₇	direct (N × H)
121. C₂×C₆₇	C₂×#332	direct (N × H)
128. C₂×#332	C₂×C₆₇	direct (N × H)
129. C₆₇	C₂²×#332	direct (N × H)
144. #332	C₂²×C₆₇	direct (N × H)
231. C₂²	#332×C₆₇	direct (N × H)
245. C₂	C₂×#332×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
C₂²×C₄⋊Q₈×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂×#332×C₆₇	2
C₂²×#332	67

Subgroups — C22×#332×C67 #30696945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#332×C₆₇ #30696945