Subgroups — C₂²×#453×C₆₇ #30739629

Order 17152.

5670 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#453×C₆₇	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#104×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×#100×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×H(5,𝔽₂)×C₆₇	—	2	12	no	—
8	C₂×H(5,𝔽₂)×C₆₇	—	2	6	no	—
9	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	12	yes	complemented
10	C₂²×#41×C₆₇	—	2	2	no	—
11	C₂²×#41×C₆₇	C₂²	1	2	yes	—
12	C₂³×D₄×C₆₇	—	2	2	no	—
13	C₂³×D₄×C₆₇	—	2	2	no	—
14	C₂³×D₄×C₆₇	—	2	2	no	—
15	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	C₂×AGL(1,ℤ/8)×C₆₇	—	2	12	no	—
18	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	24	yes	complemented
19	C₂²×SD₁₆×C₆₇	—	2	2	no	—
20	C₂²×D₈×C₆₇	—	2	2	no	—
21	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
22	C₂×#127×C₆₇	—	2	6	no	—
23	C₂×#127×C₆₇	C₂²	1	12	yes	complemented
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic, complemented
25	#453×C₆₇	C₂²	1	64	yes	direct product factor
26	C₂×#104×C₆₇	C₂²	1	24	yes	complemented
27	C₂×#100×C₆₇	C₂²	1	12	yes	complemented
28	C₂⁵×C₆₇	—	2	2	no	—
29	C₂⁵×C₆₇	—	2	1	no	—
30	C₂⁵×C₆₇	D₄	1	2	yes	—
31	H(5,𝔽₂)×C₆₇	—	2	48	no	—
32	H(5,𝔽₂)×C₆₇	—	2	48	no	—
33	H(5,𝔽₂)×C₆₇	—	2	24	no	—
34	H(5,𝔽₂)×C₆₇	C₂³	1	16	yes	complemented
35	C₂×#41×C₆₇	—	2	24	no	—
36	C₂×#41×C₆₇	—	2	24	no	—
37	C₂×#41×C₆₇	—	2	6	no	—
38	C₂×#41×C₆₇	—	2	6	no	—
39	C₂×#41×C₆₇	D₄	1	12	yes	complemented
40	C₂×#41×C₆₇	C₂³	1	12	yes	—
41	C₂²×Q₈×C₆₇	D₄	1	2	yes	complemented
42	C₂²×D₄×C₆₇	—	2	4	no	—
43	C₂²×D₄×C₆₇	—	2	24	no	—
44	C₂²×D₄×C₆₇	—	2	24	no	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	—	2	24	no	—
47	C₂²×D₄×C₆₇	—	2	24	no	—
48	C₂²×D₄×C₆₇	—	2	24	no	—
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	—	2	6	no	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	—	2	6	no	—
53	C₂²×D₄×C₆₇	—	2	3	no	—
54	C₂²×D₄×C₆₇	—	4	16	no	—
55	C₂²×D₄×C₆₇	D₄	1	6	yes	—
56	C₂²×D₄×C₆₇	D₄	1	2	yes	complemented
57	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
58	C₂²×D₄×C₆₇	C₂³	1	6	yes	complemented
59	C₂³×C₄×C₆₇	—	2	2	no	—
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂³×C₄×C₆₇	—	2	2	no	—
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	AGL(1,ℤ/8)×C₆₇	—	2	48	no	—
64	AGL(1,ℤ/8)×C₆₇	C₂³	1	32	yes	complemented
65	C₂×SD₁₆×C₆₇	—	2	24	no	—
66	C₂×D₈×C₆₇	—	2	24	no	—
67	C₂×M₁₆×C₆₇	—	2	6	no	—
68	C₂×M₁₆×C₆₇	C₂³	1	12	yes	complemented
69	C₂²×C₈×C₆₇	—	2	2	no	—
70	#127×C₆₇	—	2	24	no	—
71	#127×C₆₇	C₂³	1	16	yes	complemented
72	C₂×C₄²×C₆₇	—	2	3	no	—
73	C₂×C₄²×C₆₇	C₂³	1	6	yes	complemented
74	#104×C₆₇	C₂³	1	32	yes	complemented
75	#100×C₆₇	C₂³	1	16	yes	complemented
76	C₂⁴×C₆₇	—	2	24	no	—
77	C₂⁴×C₆₇	—	2	2	no	—
78	C₂⁴×C₆₇	—	2	2	no	—
79	C₂⁴×C₆₇	—	2	6	no	—
80	C₂⁴×C₆₇	—	2	6	no	—
81	C₂⁴×C₆₇	—	4	8	no	—
82	C₂⁴×C₆₇	—	4	8	no	—
83	C₂⁴×C₆₇	—	4	3	no	—
84	C₂⁴×C₆₇	—	4	1	no	—
85	C₂⁴×C₆₇	—	8	8	no	—
86	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂⁴×C₆₇	C₂×D₄	1	12	yes	—
88	#41×C₆₇	—	2	96	no	—
89	#41×C₆₇	—	2	32	no	—
90	#41×C₆₇	—	2	24	no	—
91	#41×C₆₇	—	2	12	no	—
92	#41×C₆₇	—	2	12	no	—
93	#41×C₆₇	C₂×D₄	1	24	yes	complemented
94	#41×C₆₇	C₂⁴	1	8	yes	—
95	C₂×Q₈×C₆₇	—	2	6	no	—
96	C₂×Q₈×C₆₇	C₂×D₄	1	12	yes	complemented
97	C₂×D₄×C₆₇	—	2	48	no	—
98	C₂×D₄×C₆₇	—	2	96	no	—
99	C₂×D₄×C₆₇	—	2	96	no	—
100	C₂×D₄×C₆₇	—	2	12	no	—
101	C₂×D₄×C₆₇	—	2	12	no	—
102	C₂×D₄×C₆₇	—	2	32	no	—
103	C₂×D₄×C₆₇	—	2	32	no	—
104	C₂×D₄×C₆₇	—	2	32	no	—
105	C₂×D₄×C₆₇	—	2	12	no	—
106	C₂×D₄×C₆₇	—	2	6	no	—
107	C₂×D₄×C₆₇	—	2	12	no	—
108	C₂×D₄×C₆₇	—	2	12	no	—
109	C₂×D₄×C₆₇	—	2	12	no	—
110	C₂×D₄×C₆₇	—	2	12	no	—
111	C₂×D₄×C₆₇	—	2	12	no	—
112	C₂×D₄×C₆₇	—	2	6	no	—
113	C₂×D₄×C₆₇	—	4	96	no	—
114	C₂×D₄×C₆₇	—	4	6	no	—
115	C₂×D₄×C₆₇	—	4	6	no	—
116	C₂×D₄×C₆₇	C₂×D₄	1	12	yes	—
117	C₂×D₄×C₆₇	C₂×D₄	1	12	yes	complemented
118	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
119	C₂×D₄×C₆₇	C₂⁴	1	4	yes	complemented
120	C₂²×C₄×C₆₇	—	2	24	no	—
121	C₂²×C₄×C₆₇	—	2	8	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	6	no	—
124	C₂²×C₄×C₆₇	—	2	12	no	—
125	C₂²×C₄×C₆₇	—	2	12	no	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
130	SD₁₆×C₆₇	—	2	32	no	—
131	D₈×C₆₇	—	2	32	no	—
132	M₁₆×C₆₇	—	2	12	no	—
133	M₁₆×C₆₇	C₂⁴	1	8	yes	complemented
134	C₂×C₈×C₆₇	—	2	12	no	—
135	C₄²×C₆₇	—	2	6	no	—
136	C₄²×C₆₇	C₂⁴	1	4	yes	complemented
137	C₂³×C₆₇	—	2	4	no	—
138	C₂³×C₆₇	—	2	32	no	—
139	C₂³×C₆₇	—	2	12	no	—
140	C₂³×C₆₇	—	2	12	no	—
141	C₂³×C₆₇	—	2	12	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	4	24	no	—
144	C₂³×C₆₇	—	4	24	no	—
145	C₂³×C₆₇	—	4	8	no	—
146	C₂³×C₆₇	—	4	8	no	—
147	C₂³×C₆₇	—	4	6	no	—
148	C₂³×C₆₇	—	4	6	no	—
149	C₂³×C₆₇	—	8	12	no	—
150	C₂³×C₆₇	—	8	12	no	—
151	C₂³×C₆₇	—	8	48	no	—
152	C₂³×C₆₇	—	8	4	no	—
153	C₂³×C₆₇	#120	1	3	yes	—
154	C₂³×C₆₇	#120	1	1	yes	characteristic
155	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
156	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
157	Q₈×C₆₇	—	2	12	no	—
158	Q₈×C₆₇	C₂²×D₄	1	8	yes	complemented
159	D₄×C₆₇	—	2	64	no	—
160	D₄×C₆₇	—	2	24	no	—
161	D₄×C₆₇	—	2	8	no	—
162	D₄×C₆₇	—	2	24	no	—
163	D₄×C₆₇	—	2	12	no	—
164	D₄×C₆₇	—	2	8	no	—
165	D₄×C₆₇	—	2	8	no	—
166	D₄×C₆₇	—	4	64	no	—
167	D₄×C₆₇	—	4	12	no	—
168	D₄×C₆₇	C₂²×D₄	1	8	yes	complemented
169	C₂×C₄×C₆₇	—	2	32	no	—
170	C₂×C₄×C₆₇	—	2	24	no	—
171	C₂×C₄×C₆₇	—	2	12	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	24	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	#120	1	6	yes	—
176	C₂×C₄×C₆₇	#120	1	6	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
180	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
181	C₈×C₆₇	—	2	8	no	—
182	C₂²×C₆₇	—	2	6	no	—
183	C₂²×C₆₇	—	2	6	no	—
184	C₂²×C₆₇	—	2	8	no	—
185	C₂²×C₆₇	—	2	8	no	—
186	C₂²×C₆₇	—	4	8	no	—
187	C₂²×C₆₇	—	4	8	no	—
188	C₂²×C₆₇	—	4	24	no	—
189	C₂²×C₆₇	—	4	24	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	8	12	no	—
192	C₂²×C₆₇	—	8	12	no	—
193	C₂²×C₆₇	—	8	32	no	—
194	C₂²×C₆₇	—	8	12	no	—
195	C₂²×C₆₇	#453	1	4	yes	direct product factor
196	C₂²×C₆₇	C₂×#120	1	3	yes	—
197	C₂²×C₆₇	C₂×#120	1	3	yes	—
198	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
199	C₄×C₆₇	—	2	8	no	—
200	C₄×C₆₇	—	2	8	no	—
201	C₄×C₆₇	C₂×#120	1	6	yes	—
202	C₄×C₆₇	C₂³×D₄	1	2	yes	—
203	C₂²×#453	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	3	no	—
206	C₂×C₆₇	—	4	8	no	—
207	C₂×C₆₇	—	4	8	no	—
208	C₂×C₆₇	—	8	4	no	—
209	C₂×C₆₇	C₂×#453	1	6	yes	direct product factor
210	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
211	C₂²×H(5,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic, complemented
212	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	2	yes	complemented
213	C₂²×#127	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₂×#453	C₂×C₆₇	1	24	yes	direct product factor
215	C₂²×#104	C₂×C₆₇	1	2	yes	complemented
216	C₂²×#100	C₂×C₆₇	1	1	yes	characteristic, complemented
217	C₆₇	C₂²×#453	1	1	yes	characteristic, direct product factor, Sylow
218	C₂×H(5,𝔽₂)	—	2	12	no	—
219	C₂×H(5,𝔽₂)	—	2	6	no	—
220	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	12	yes	complemented
221	C₂²×#41	—	2	2	no	—
222	C₂²×#41	C₂²×C₆₇	1	2	yes	—
223	C₂³×D₄	—	2	2	no	—
224	C₂³×D₄	—	2	2	no	—
225	C₂³×D₄	—	2	2	no	—
226	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
227	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
228	C₂×AGL(1,ℤ/8)	—	2	12	no	—
229	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	24	yes	complemented
230	C₂²×SD₁₆	—	2	2	no	—
231	C₂²×D₈	—	2	2	no	—
232	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
233	C₂×#127	—	2	6	no	—
234	C₂×#127	C₂²×C₆₇	1	12	yes	complemented
235	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic, complemented
236	#453	C₂²×C₆₇	1	64	yes	direct product factor
237	C₂×#104	C₂²×C₆₇	1	24	yes	complemented
238	C₂×#100	C₂²×C₆₇	1	12	yes	complemented
239	C₂⁵	—	2	2	no	—
240	C₂⁵	—	2	1	no	—
241	C₂⁵	D₄×C₆₇	1	2	yes	—
242	H(5,𝔽₂)	—	2	48	no	—
243	H(5,𝔽₂)	—	2	48	no	—
244	H(5,𝔽₂)	—	2	24	no	—
245	H(5,𝔽₂)	C₂³×C₆₇	1	16	yes	complemented
246	C₂×#41	—	2	24	no	—
247	C₂×#41	—	2	24	no	—
248	C₂×#41	—	2	6	no	—
249	C₂×#41	—	2	6	no	—
250	C₂×#41	D₄×C₆₇	1	12	yes	complemented
251	C₂×#41	C₂³×C₆₇	1	12	yes	—
252	C₂²×Q₈	D₄×C₆₇	1	2	yes	complemented
253	C₂²×D₄	—	2	4	no	—
254	C₂²×D₄	—	2	24	no	—
255	C₂²×D₄	—	2	24	no	—
256	C₂²×D₄	—	2	2	no	—
257	C₂²×D₄	—	2	24	no	—
258	C₂²×D₄	—	2	24	no	—
259	C₂²×D₄	—	2	24	no	—
260	C₂²×D₄	—	2	2	no	—
261	C₂²×D₄	—	2	6	no	—
262	C₂²×D₄	—	2	2	no	—
263	C₂²×D₄	—	2	6	no	—
264	C₂²×D₄	—	2	3	no	—
265	C₂²×D₄	—	4	16	no	—
266	C₂²×D₄	D₄×C₆₇	1	6	yes	—
267	C₂²×D₄	D₄×C₆₇	1	2	yes	complemented
268	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
269	C₂²×D₄	C₂³×C₆₇	1	6	yes	complemented
270	C₂³×C₄	—	2	2	no	—
271	C₂³×C₄	—	2	1	no	—
272	C₂³×C₄	—	2	2	no	—
273	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
274	AGL(1,ℤ/8)	—	2	48	no	—
275	AGL(1,ℤ/8)	C₂³×C₆₇	1	32	yes	complemented
276	C₂×SD₁₆	—	2	24	no	—
277	C₂×D₈	—	2	24	no	—
278	C₂×M₁₆	—	2	6	no	—
279	C₂×M₁₆	C₂³×C₆₇	1	12	yes	complemented
280	C₂²×C₈	—	2	2	no	—
281	#127	—	2	24	no	—
282	#127	C₂³×C₆₇	1	16	yes	complemented
283	C₂×C₄²	—	2	3	no	—
284	C₂×C₄²	C₂³×C₆₇	1	6	yes	complemented
285	#104	C₂³×C₆₇	1	32	yes	complemented
286	#100	C₂³×C₆₇	1	16	yes	complemented
287	C₂⁴	—	2	24	no	—
288	C₂⁴	—	2	2	no	—
289	C₂⁴	—	2	2	no	—
290	C₂⁴	—	2	6	no	—
291	C₂⁴	—	2	6	no	—
292	C₂⁴	—	4	8	no	—
293	C₂⁴	—	4	8	no	—
294	C₂⁴	—	4	3	no	—
295	C₂⁴	—	4	1	no	—
296	C₂⁴	—	8	8	no	—
297	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
298	C₂⁴	C₂×D₄×C₆₇	1	12	yes	—
299	#41	—	2	96	no	—
300	#41	—	2	32	no	—
301	#41	—	2	24	no	—
302	#41	—	2	12	no	—
303	#41	—	2	12	no	—
304	#41	C₂×D₄×C₆₇	1	24	yes	complemented
305	#41	C₂⁴×C₆₇	1	8	yes	—
306	C₂×Q₈	—	2	6	no	—
307	C₂×Q₈	C₂×D₄×C₆₇	1	12	yes	complemented
308	C₂×D₄	—	2	48	no	—
309	C₂×D₄	—	2	96	no	—
310	C₂×D₄	—	2	96	no	—
311	C₂×D₄	—	2	12	no	—
312	C₂×D₄	—	2	12	no	—
313	C₂×D₄	—	2	32	no	—
314	C₂×D₄	—	2	32	no	—
315	C₂×D₄	—	2	32	no	—
316	C₂×D₄	—	2	12	no	—
317	C₂×D₄	—	2	6	no	—
318	C₂×D₄	—	2	12	no	—
319	C₂×D₄	—	2	12	no	—
320	C₂×D₄	—	2	12	no	—
321	C₂×D₄	—	2	12	no	—
322	C₂×D₄	—	2	12	no	—
323	C₂×D₄	—	2	6	no	—
324	C₂×D₄	—	4	96	no	—
325	C₂×D₄	—	4	6	no	—
326	C₂×D₄	—	4	6	no	—
327	C₂×D₄	C₂×D₄×C₆₇	1	12	yes	—
328	C₂×D₄	C₂×D₄×C₆₇	1	12	yes	complemented
329	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
330	C₂×D₄	C₂⁴×C₆₇	1	4	yes	complemented
331	C₂²×C₄	—	2	24	no	—
332	C₂²×C₄	—	2	8	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	6	no	—
335	C₂²×C₄	—	2	12	no	—
336	C₂²×C₄	—	2	12	no	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
338	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
339	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
341	SD₁₆	—	2	32	no	—
342	D₈	—	2	32	no	—
343	M₁₆	—	2	12	no	—
344	M₁₆	C₂⁴×C₆₇	1	8	yes	complemented
345	C₂×C₈	—	2	12	no	—
346	C₄²	—	2	6	no	—
347	C₄²	C₂⁴×C₆₇	1	4	yes	complemented
348	C₂³	—	2	4	no	—
349	C₂³	—	2	32	no	—
350	C₂³	—	2	12	no	—
351	C₂³	—	2	12	no	—
352	C₂³	—	2	12	no	—
353	C₂³	—	2	4	no	—
354	C₂³	—	4	24	no	—
355	C₂³	—	4	24	no	—
356	C₂³	—	4	8	no	—
357	C₂³	—	4	8	no	—
358	C₂³	—	4	6	no	—
359	C₂³	—	4	6	no	—
360	C₂³	—	8	12	no	—
361	C₂³	—	8	12	no	—
362	C₂³	—	8	48	no	—
363	C₂³	—	8	4	no	—
364	C₂³	#120×C₆₇	1	3	yes	—
365	C₂³	#120×C₆₇	1	1	yes	characteristic
366	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
367	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
368	Q₈	—	2	12	no	—
369	Q₈	C₂²×D₄×C₆₇	1	8	yes	complemented
370	D₄	—	2	64	no	—
371	D₄	—	2	24	no	—
372	D₄	—	2	8	no	—
373	D₄	—	2	24	no	—
374	D₄	—	2	12	no	—
375	D₄	—	2	8	no	—
376	D₄	—	2	8	no	—
377	D₄	—	4	64	no	—
378	D₄	—	4	12	no	—
379	D₄	C₂²×D₄×C₆₇	1	8	yes	complemented
380	C₂×C₄	—	2	32	no	—
381	C₂×C₄	—	2	24	no	—
382	C₂×C₄	—	2	12	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	24	no	—
385	C₂×C₄	—	2	8	no	—
386	C₂×C₄	#120×C₆₇	1	6	yes	—
387	C₂×C₄	#120×C₆₇	1	6	yes	—
388	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
389	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
390	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
391	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
392	C₈	—	2	8	no	—
393	C₂²	—	2	6	no	—
394	C₂²	—	2	6	no	—
395	C₂²	—	2	8	no	—
396	C₂²	—	2	8	no	—
397	C₂²	—	4	8	no	—
398	C₂²	—	4	8	no	—
399	C₂²	—	4	24	no	—
400	C₂²	—	4	24	no	—
401	C₂²	—	4	4	no	—
402	C₂²	—	8	12	no	—
403	C₂²	—	8	12	no	—
404	C₂²	—	8	32	no	—
405	C₂²	—	8	12	no	—
406	C₂²	#453×C₆₇	1	4	yes	direct product factor
407	C₂²	C₂×#120×C₆₇	1	3	yes	—
408	C₂²	C₂×#120×C₆₇	1	3	yes	—
409	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
410	C₄	—	2	8	no	—
411	C₄	—	2	8	no	—
412	C₄	C₂×#120×C₆₇	1	6	yes	—
413	C₄	C₂³×D₄×C₆₇	1	2	yes	—
414	C₂	—	2	1	no	—
415	C₂	—	2	3	no	—
416	C₂	—	4	8	no	—
417	C₂	—	4	8	no	—
418	C₂	—	8	4	no	—
419	C₂	C₂×#453×C₆₇	1	6	yes	direct product factor
420	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×H(5,𝔽₂)×C₆₇	C₂	semidirect
2. C₂²×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
3. C₂²×#127×C₆₇	C₂	semidirect
4. C₂×#453×C₆₇	C₂	direct (N × H)
5. C₂²×#104×C₆₇	C₂	semidirect
6. C₂²×#100×C₆₇	C₂	semidirect
9. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
16. C₂³×D₄×C₆₇	C₂²	semidirect
18. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
21. C₂²×M₁₆×C₆₇	C₂²	semidirect
23. C₂×#127×C₆₇	C₂²	semidirect
24. C₂²×C₄²×C₆₇	C₂²	semidirect
25. #453×C₆₇	C₂²	direct (N × H)
26. C₂×#104×C₆₇	C₂²	semidirect
27. C₂×#100×C₆₇	C₂²	semidirect
34. H(5,𝔽₂)×C₆₇	C₂³	semidirect
39. C₂×#41×C₆₇	D₄	semidirect
41. C₂²×Q₈×C₆₇	D₄	semidirect
56. C₂²×D₄×C₆₇	D₄	semidirect
58. C₂²×D₄×C₆₇	C₂³	semidirect
64. AGL(1,ℤ/8)×C₆₇	C₂³	semidirect
68. C₂×M₁₆×C₆₇	C₂³	semidirect
71. #127×C₆₇	C₂³	semidirect
73. C₂×C₄²×C₆₇	C₂³	semidirect
74. #104×C₆₇	C₂³	semidirect
75. #100×C₆₇	C₂³	semidirect
93. #41×C₆₇	C₂×D₄	semidirect
96. C₂×Q₈×C₆₇	C₂×D₄	semidirect
117. C₂×D₄×C₆₇	C₂×D₄	semidirect
119. C₂×D₄×C₆₇	C₂⁴	semidirect
133. M₁₆×C₆₇	C₂⁴	semidirect
136. C₄²×C₆₇	C₂⁴	semidirect
158. Q₈×C₆₇	C₂²×D₄	semidirect
168. D₄×C₆₇	C₂²×D₄	semidirect
195. C₂²×C₆₇	#453	direct (N × H)
203. C₂²×#453	C₆₇	direct (N × H)
209. C₂×C₆₇	C₂×#453	direct (N × H)
211. C₂²×H(5,𝔽₂)	C₂×C₆₇	semidirect
212. C₂²×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
213. C₂²×#127	C₂×C₆₇	semidirect
214. C₂×#453	C₂×C₆₇	direct (N × H)
215. C₂²×#104	C₂×C₆₇	semidirect
216. C₂²×#100	C₂×C₆₇	semidirect
217. C₆₇	C₂²×#453	direct (N × H)
220. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
227. C₂³×D₄	C₂²×C₆₇	semidirect
229. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
232. C₂²×M₁₆	C₂²×C₆₇	semidirect
234. C₂×#127	C₂²×C₆₇	semidirect
235. C₂²×C₄²	C₂²×C₆₇	semidirect
236. #453	C₂²×C₆₇	direct (N × H)
237. C₂×#104	C₂²×C₆₇	semidirect
238. C₂×#100	C₂²×C₆₇	semidirect
245. H(5,𝔽₂)	C₂³×C₆₇	semidirect
250. C₂×#41	D₄×C₆₇	semidirect
252. C₂²×Q₈	D₄×C₆₇	semidirect
267. C₂²×D₄	D₄×C₆₇	semidirect
269. C₂²×D₄	C₂³×C₆₇	semidirect
275. AGL(1,ℤ/8)	C₂³×C₆₇	semidirect
279. C₂×M₁₆	C₂³×C₆₇	semidirect
282. #127	C₂³×C₆₇	semidirect
284. C₂×C₄²	C₂³×C₆₇	semidirect
285. #104	C₂³×C₆₇	semidirect
286. #100	C₂³×C₆₇	semidirect
304. #41	C₂×D₄×C₆₇	semidirect
307. C₂×Q₈	C₂×D₄×C₆₇	semidirect
328. C₂×D₄	C₂×D₄×C₆₇	semidirect
330. C₂×D₄	C₂⁴×C₆₇	semidirect
344. M₁₆	C₂⁴×C₆₇	semidirect
347. C₄²	C₂⁴×C₆₇	semidirect
369. Q₈	C₂²×D₄×C₆₇	semidirect
379. D₄	C₂²×D₄×C₆₇	semidirect
406. C₂²	#453×C₆₇	direct (N × H)
419. C₂	C₂×#453×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×H(5,𝔽₂)×C₆₇	2
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂²×#127×C₆₇	2
C₂×#453×C₆₇	2
C₂²×#104×C₆₇	2
C₂²×#100×C₆₇	2
C₂²×#453	67

Subgroups — C22×#453×C67 #30739629

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#453×C₆₇ #30739629