Subgroups — C₆₇×#22455 #30701706

Order 17152.

1378 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2948	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2883	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1976	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1904	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1812	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1795	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1757	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	—
10	#576×C₆₇	—	2	4	no	—
11	#576×C₆₇	—	2	4	no	—
12	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	4	yes	complemented
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
14	C₂×#135×C₆₇	C₄	1	2	yes	complemented
15	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×D₈×C₆₇	C₄	1	2	yes	complemented
17	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
19	#518×C₆₇	C₂²	1	4	yes	complemented
20	#442×C₆₇	—	2	4	no	—
21	C₄×D₈×C₆₇	—	2	4	no	—
22	#427×C₆₇	C₂²	1	2	yes	complemented
23	#421×C₆₇	C₂²	1	4	yes	complemented
24	#418×C₆₇	C₂²	1	4	yes	complemented
25	#413×C₆₇	C₂²	1	1	yes	characteristic
26	#410×C₆₇	C₂²	1	2	yes	—
27	#410×C₆₇	C₂²	1	1	yes	characteristic
28	#410×C₆₇	C₂²	1	4	yes	complemented
29	C₂×#99×C₆₇	—	2	8	no	—
30	#405×C₆₇	C₂²	1	2	yes	complemented
31	#344×C₆₇	C₂²	1	2	yes	complemented
32	H(3,ℤ/4)×C₆₇	C₂²	1	2	yes	complemented
33	H(5,𝔽₂)×C₆₇	—	2	8	no	—
34	H(5,𝔽₂)×C₆₇	—	4	4	no	—
35	C₂×#41×C₆₇	—	2	2	no	—
36	C₂×#41×C₆₇	C₂×C₄	1	4	yes	complemented
37	C₂×#41×C₆₇	C₂³	1	2	yes	—
38	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₆₇	—	2	4	no	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
42	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
43	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
44	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
45	#135×C₆₇	—	2	4	no	—
46	#135×C₆₇	—	2	4	no	—
47	#135×C₆₇	—	2	2	no	—
48	#135×C₆₇	—	2	2	no	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
50	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
52	C₂×D₈×C₆₇	—	2	4	no	—
53	C₂×D₈×C₆₇	—	2	4	no	—
54	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
57	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
58	C₂×D₈×C₆₇	D₄	1	4	yes	—
59	C₂×D₈×C₆₇	D₄	1	4	yes	—
60	#131×C₆₇	—	2	2	no	—
61	#131×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
63	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
64	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	complemented
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₆₇	—	2	8	no	—
68	C₄×D₄×C₆₇	—	4	4	no	—
69	#117×C₆₇	C₂³	1	4	yes	—
70	#117×C₆₇	C₂³	1	2	yes	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
72	#107×C₆₇	—	2	2	no	—
73	#106×C₆₇	—	2	2	no	—
74	#104×C₆₇	—	4	4	no	—
75	#102×C₆₇	—	2	8	no	—
76	#101×C₆₇	—	2	1	no	—
77	#100×C₆₇	—	2	1	no	—
78	#99×C₆₇	—	2	4	no	—
79	#99×C₆₇	—	2	8	no	—
80	#99×C₆₇	D₄	1	4	yes	—
81	#97×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	—	2	2	no	—
83	C₂⁴×C₆₇	—	2	2	no	—
84	C₂⁴×C₆₇	—	2	4	no	—
85	#41×C₆₇	—	2	2	no	—
86	#41×C₆₇	—	2	2	no	—
87	#41×C₆₇	—	2	8	no	—
88	#41×C₆₇	—	2	4	no	—
89	#41×C₆₇	—	2	4	no	—
90	#41×C₆₇	—	4	4	no	—
91	#41×C₆₇	—	4	8	no	—
92	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
94	C₂×D₄×C₆₇	—	2	8	no	—
95	C₂×D₄×C₆₇	—	2	4	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	4	4	no	—
101	C₂×D₄×C₆₇	—	4	8	no	—
102	C₂×D₄×C₆₇	—	4	16	no	—
103	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
105	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
107	C₂×D₄×C₆₇	#41	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	4	8	no	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
113	Q₁₆×C₆₇	—	2	1	no	—
114	Q₁₆×C₆₇	—	2	1	no	—
115	SD₁₆×C₆₇	—	2	8	no	—
116	SD₁₆×C₆₇	—	2	4	no	—
117	D₈×C₆₇	—	2	8	no	—
118	D₈×C₆₇	—	2	1	no	—
119	D₈×C₆₇	—	2	1	no	—
120	D₈×C₆₇	C₂²⋊C₄	1	4	yes	—
121	D₈×C₆₇	C₂²⋊C₄	1	4	yes	—
122	D₈×C₆₇	C₂²⋊C₄	1	8	yes	—
123	M₁₆×C₆₇	—	2	2	no	—
124	M₁₆×C₆₇	—	2	4	no	—
125	M₁₆×C₆₇	—	2	2	no	—
126	M₁₆×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	—	4	8	no	—
139	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	4	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	2	8	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	4	8	no	—
146	C₂³×C₆₇	—	16	2	no	—
147	C₂³×C₆₇	—	16	4	no	—
148	C₂³×C₆₇	C₄×D₄	1	2	yes	—
149	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₆₇	—	2	2	no	—
151	Q₈×C₆₇	—	2	2	no	—
152	Q₈×C₆₇	—	2	2	no	—
153	D₄×C₆₇	—	2	4	no	—
154	D₄×C₆₇	—	2	2	no	—
155	D₄×C₆₇	—	2	4	no	—
156	D₄×C₆₇	—	2	2	no	—
157	D₄×C₆₇	—	2	8	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	—	4	8	no	—
160	D₄×C₆₇	—	4	8	no	—
161	D₄×C₆₇	—	4	4	no	—
162	D₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	4	8	no	—
168	C₂×C₄×C₆₇	—	4	4	no	—
169	C₂×C₄×C₆₇	—	4	4	no	—
170	C₂×C₄×C₆₇	—	16	8	no	—
171	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
176	C₈×C₆₇	—	2	2	no	—
177	C₈×C₆₇	—	4	2	no	—
178	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
181	C₂²×C₆₇	—	2	2	no	—
182	C₂²×C₆₇	—	2	8	no	—
183	C₂²×C₆₇	—	4	1	no	—
184	C₂²×C₆₇	—	4	2	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	8	8	no	—
187	C₂²×C₆₇	—	8	4	no	—
188	C₂²×C₆₇	—	8	2	no	—
189	C₂²×C₆₇	—	16	8	no	—
190	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
191	C₂²×C₆₇	D₄²	1	2	yes	—
192	C₄×C₆₇	—	2	2	no	—
193	C₄×C₆₇	—	4	4	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	8	4	no	—
196	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
197	C₄×C₆₇	#518	1	1	yes	characteristic
198	C₄×C₆₇	#546	1	2	yes	—
199	#22455	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₆₇	—	2	2	no	—
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	—	4	2	no	—
203	C₂×C₆₇	—	4	8	no	—
204	C₂×C₆₇	—	8	2	no	—
205	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×#576	C₂×C₆₇	1	1	yes	characteristic
207	#2948	C₂×C₆₇	1	2	yes	complemented
208	#2883	C₂×C₆₇	1	2	yes	complemented
209	#1976	C₂×C₆₇	1	2	yes	complemented
210	#1904	C₂×C₆₇	1	4	yes	complemented
211	#1812	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#1795	C₂×C₆₇	1	2	yes	complemented
213	#1757	C₂×C₆₇	1	1	yes	characteristic, complemented
214	C₆₇	#22455	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	—
216	#576	—	2	4	no	—
217	#576	—	2	4	no	—
218	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	4	yes	complemented
219	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
220	C₂×#135	C₄×C₆₇	1	2	yes	complemented
221	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
222	C₂²×D₈	C₄×C₆₇	1	2	yes	complemented
223	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
225	#518	C₂²×C₆₇	1	4	yes	complemented
226	#442	—	2	4	no	—
227	C₄×D₈	—	2	4	no	—
228	#427	C₂²×C₆₇	1	2	yes	complemented
229	#421	C₂²×C₆₇	1	4	yes	complemented
230	#418	C₂²×C₆₇	1	4	yes	complemented
231	#413	C₂²×C₆₇	1	1	yes	characteristic
232	#410	C₂²×C₆₇	1	2	yes	—
233	#410	C₂²×C₆₇	1	1	yes	characteristic
234	#410	C₂²×C₆₇	1	4	yes	complemented
235	C₂×#99	—	2	8	no	—
236	#405	C₂²×C₆₇	1	2	yes	complemented
237	#344	C₂²×C₆₇	1	2	yes	complemented
238	H(3,ℤ/4)	C₂²×C₆₇	1	2	yes	complemented
239	H(5,𝔽₂)	—	2	8	no	—
240	H(5,𝔽₂)	—	4	4	no	—
241	C₂×#41	—	2	2	no	—
242	C₂×#41	C₂×C₄×C₆₇	1	4	yes	complemented
243	C₂×#41	C₂³×C₆₇	1	2	yes	—
244	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
245	C₂²×D₄	—	2	4	no	—
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
248	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
249	AGL(1,ℤ/8)	—	2	8	no	—
250	AGL(1,ℤ/8)	—	2	16	no	—
251	#135	—	2	4	no	—
252	#135	—	2	4	no	—
253	#135	—	2	2	no	—
254	#135	—	2	2	no	—
255	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
256	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
257	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	complemented
258	C₂×D₈	—	2	4	no	—
259	C₂×D₈	—	2	4	no	—
260	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
261	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
262	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
263	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
264	C₂×D₈	D₄×C₆₇	1	4	yes	—
265	C₂×D₈	D₄×C₆₇	1	4	yes	—
266	#131	—	2	2	no	—
267	#131	—	2	2	no	—
268	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	complemented
269	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
270	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	complemented
272	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
273	C₄×D₄	—	2	8	no	—
274	C₄×D₄	—	4	4	no	—
275	#117	C₂³×C₆₇	1	4	yes	—
276	#117	C₂³×C₆₇	1	2	yes	—
277	C₂×C₂²⋊C₄	—	2	8	no	—
278	#107	—	2	2	no	—
279	#106	—	2	2	no	—
280	#104	—	4	4	no	—
281	#102	—	2	8	no	—
282	#101	—	2	1	no	—
283	#100	—	2	1	no	—
284	#99	—	2	4	no	—
285	#99	—	2	8	no	—
286	#99	D₄×C₆₇	1	4	yes	—
287	#97	—	2	2	no	—
288	C₄×C₈	—	2	2	no	—
289	C₂⁴	—	2	2	no	—
290	C₂⁴	—	2	4	no	—
291	#41	—	2	2	no	—
292	#41	—	2	2	no	—
293	#41	—	2	8	no	—
294	#41	—	2	4	no	—
295	#41	—	2	4	no	—
296	#41	—	4	4	no	—
297	#41	—	4	8	no	—
298	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
299	C₂×Q₈	#41×C₆₇	1	1	yes	characteristic
300	C₂×D₄	—	2	8	no	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	—	4	4	no	—
307	C₂×D₄	—	4	8	no	—
308	C₂×D₄	—	4	16	no	—
309	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
310	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
311	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
312	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
313	C₂×D₄	#41×C₆₇	1	1	yes	characteristic
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	4	8	no	—
317	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
318	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
319	Q₁₆	—	2	1	no	—
320	Q₁₆	—	2	1	no	—
321	SD₁₆	—	2	8	no	—
322	SD₁₆	—	2	4	no	—
323	D₈	—	2	8	no	—
324	D₈	—	2	1	no	—
325	D₈	—	2	1	no	—
326	D₈	C₂²⋊C₄×C₆₇	1	4	yes	—
327	D₈	C₂²⋊C₄×C₆₇	1	4	yes	—
328	D₈	C₂²⋊C₄×C₆₇	1	8	yes	—
329	M₁₆	—	2	2	no	—
330	M₁₆	—	2	4	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
338	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
340	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	4	8	no	—
345	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	4	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	2	8	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	4	8	no	—
352	C₂³	—	16	2	no	—
353	C₂³	—	16	4	no	—
354	C₂³	C₄×D₄×C₆₇	1	2	yes	—
355	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
356	Q₈	—	2	2	no	—
357	Q₈	—	2	2	no	—
358	Q₈	—	2	2	no	—
359	D₄	—	2	4	no	—
360	D₄	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	2	no	—
363	D₄	—	2	8	no	—
364	D₄	—	2	2	no	—
365	D₄	—	4	8	no	—
366	D₄	—	4	8	no	—
367	D₄	—	4	4	no	—
368	D₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	4	8	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	4	4	no	—
376	C₂×C₄	—	16	8	no	—
377	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
378	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
380	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
381	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
385	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
386	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	8	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	8	8	no	—
393	C₂²	—	8	4	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	16	8	no	—
396	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
397	C₂²	D₄²×C₆₇	1	2	yes	—
398	C₄	—	2	2	no	—
399	C₄	—	4	4	no	—
400	C₄	—	4	2	no	—
401	C₄	—	8	4	no	—
402	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
403	C₄	#518×C₆₇	1	1	yes	characteristic
404	C₄	#546×C₆₇	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	2	1	no	—
407	C₂	—	4	2	no	—
408	C₂	—	4	8	no	—
409	C₂	—	8	2	no	—
410	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#576×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2948	C₂	semidirect
3. C₆₇×#2883	C₂	semidirect
4. C₆₇×#1976	C₂	semidirect
5. C₆₇×#1904	C₂	semidirect
6. C₆₇×#1812	C₂	semidirect
7. C₆₇×#1795	C₂	semidirect
8. C₆₇×#1757	C₂	semidirect
12. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
14. C₂×#135×C₆₇	C₄	semidirect
16. C₂²×D₈×C₆₇	C₄	semidirect
19. #518×C₆₇	C₂²	semidirect
22. #427×C₆₇	C₂²	semidirect
23. #421×C₆₇	C₂²	semidirect
24. #418×C₆₇	C₂²	semidirect
28. #410×C₆₇	C₂²	semidirect
30. #405×C₆₇	C₂²	semidirect
31. #344×C₆₇	C₂²	semidirect
32. H(3,ℤ/4)×C₆₇	C₂²	semidirect
36. C₂×#41×C₆₇	C₂×C₄	semidirect
41. C₂²×D₄×C₆₇	C₂×C₄	semidirect
49. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
50. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
51. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
56. C₂×D₈×C₆₇	C₂×C₄	semidirect
57. C₂×D₈×C₆₇	C₂×C₄	semidirect
62. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
65. C₂²×C₈×C₆₇	C₂×C₄	semidirect
199. #22455	C₆₇	direct (N × H)
207. #2948	C₂×C₆₇	semidirect
208. #2883	C₂×C₆₇	semidirect
209. #1976	C₂×C₆₇	semidirect
210. #1904	C₂×C₆₇	semidirect
211. #1812	C₂×C₆₇	semidirect
212. #1795	C₂×C₆₇	semidirect
213. #1757	C₂×C₆₇	semidirect
214. C₆₇	#22455	direct (N × H)
218. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
220. C₂×#135	C₄×C₆₇	semidirect
222. C₂²×D₈	C₄×C₆₇	semidirect
225. #518	C₂²×C₆₇	semidirect
228. #427	C₂²×C₆₇	semidirect
229. #421	C₂²×C₆₇	semidirect
230. #418	C₂²×C₆₇	semidirect
234. #410	C₂²×C₆₇	semidirect
236. #405	C₂²×C₆₇	semidirect
237. #344	C₂²×C₆₇	semidirect
238. H(3,ℤ/4)	C₂²×C₆₇	semidirect
242. C₂×#41	C₂×C₄×C₆₇	semidirect
247. C₂²×D₄	C₂×C₄×C₆₇	semidirect
255. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
256. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
257. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
262. C₂×D₈	C₂×C₄×C₆₇	semidirect
263. C₂×D₈	C₂×C₄×C₆₇	semidirect
268. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
271. C₂²×C₈	C₂×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×#576×C₆₇	2
C₆₇×#2948	2
C₆₇×#2883	2
C₆₇×#1976	2
C₆₇×#1904	2
C₆₇×#1812	2
C₆₇×#1795	2
C₆₇×#1757	2
#22455	67

Subgroups — C67×#22455 #30701706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22455 #30701706