Subgroups — C₆₇×#22458 #30701709

Order 17152.

1338 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×#2957	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2888	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1978	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1903	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1813	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	#568×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
20	#444×C₆₇	—	2	4	no	—
21	#443×C₆₇	—	2	4	no	—
22	C₂×#108×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	—	2	4	no	—
26	#413×C₆₇	C₂²	1	2	yes	—
27	#413×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#100×C₆₇	—	2	4	no	—
29	C₂×#100×C₆₇	C₂²	1	4	yes	complemented
30	#410×C₆₇	C₂²	1	1	yes	characteristic
31	#405×C₆₇	C₂²	1	2	yes	complemented
32	#344×C₆₇	C₂²	1	2	yes	complemented
33	#338×C₆₇	C₂²	1	2	yes	complemented
34	H(5,𝔽₂)×C₆₇	—	2	8	no	—
35	H(5,𝔽₂)×C₆₇	—	4	4	no	—
36	C₂×#41×C₆₇	—	2	2	no	—
37	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×#41×C₆₇	C₂³	1	2	yes	—
39	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	—	2	4	no	—
42	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
43	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
44	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
45	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
46	#135×C₆₇	—	2	4	no	—
47	#135×C₆₇	—	2	4	no	—
48	#135×C₆₇	D₄	1	4	yes	—
49	#135×C₆₇	D₄	1	4	yes	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×D₈×C₆₇	—	2	4	no	—
54	C₂×D₈×C₆₇	—	2	4	no	—
55	C₂×D₈×C₆₇	—	2	2	no	—
56	C₂×D₈×C₆₇	—	2	2	no	—
57	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
61	#131×C₆₇	—	2	2	no	—
62	#131×C₆₇	—	2	2	no	—
63	#131×C₆₇	—	2	8	no	—
64	#131×C₆₇	—	4	4	no	—
65	C₂×M₁₆×C₆₇	—	2	8	no	—
66	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	#117×C₆₇	C₂³	1	2	yes	—
73	#108×C₆₇	—	2	2	no	—
74	#108×C₆₇	—	2	2	no	—
75	#104×C₆₇	—	2	8	no	—
76	#102×C₆₇	—	4	4	no	—
77	#101×C₆₇	—	2	2	no	—
78	#101×C₆₇	D₄	1	2	yes	—
79	#100×C₆₇	—	2	2	no	—
80	#100×C₆₇	—	2	8	no	—
81	#100×C₆₇	D₄	1	2	yes	—
82	#99×C₆₇	—	2	2	no	—
83	#97×C₆₇	—	2	2	no	—
84	C₄×C₈×C₆₇	—	2	2	no	—
85	C₂⁴×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	—	2	2	no	—
87	#41×C₆₇	—	2	2	no	—
88	#41×C₆₇	—	2	4	no	—
89	#41×C₆₇	—	2	2	no	—
90	#41×C₆₇	—	2	8	no	—
91	#41×C₆₇	—	4	4	no	—
92	#41×C₆₇	—	4	8	no	—
93	#41×C₆₇	C₂×D₄	1	8	yes	—
94	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	8	no	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	4	4	no	—
104	C₂×D₄×C₆₇	—	4	8	no	—
105	C₂×D₄×C₆₇	—	4	16	no	—
106	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×D₄×C₆₇	#41	1	2	yes	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
115	Q₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
116	SD₁₆×C₆₇	—	2	8	no	—
117	SD₁₆×C₆₇	C₂²⋊C₄	1	8	yes	—
118	D₈×C₆₇	—	2	8	no	—
119	D₈×C₆₇	—	2	2	no	—
120	D₈×C₆₇	—	2	2	no	—
121	D₈×C₆₇	—	2	4	no	—
122	D₈×C₆₇	C₂²⋊C₄	1	2	yes	—
123	D₈×C₆₇	C₂²⋊C₄	1	2	yes	—
124	M₁₆×C₆₇	—	2	2	no	—
125	M₁₆×C₆₇	—	2	2	no	—
126	M₁₆×C₆₇	—	2	4	no	—
127	M₁₆×C₆₇	—	2	4	no	—
128	M₁₆×C₆₇	—	2	4	no	—
129	M₁₆×C₆₇	—	4	8	no	—
130	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	4	no	—
135	C₂×C₈×C₆₇	—	4	8	no	—
136	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₂²⋊C₄×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	4	8	no	—
147	C₂³×C₆₇	—	4	4	no	—
148	C₂³×C₆₇	—	16	2	no	—
149	C₂³×C₆₇	—	16	4	no	—
150	C₂³×C₆₇	C₄×D₄	1	2	yes	—
151	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	2	no	—
153	Q₈×C₆₇	—	2	2	no	—
154	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
155	D₄×C₆₇	—	2	8	no	—
156	D₄×C₆₇	—	2	4	no	—
157	D₄×C₆₇	—	2	4	no	—
158	D₄×C₆₇	—	2	2	no	—
159	D₄×C₆₇	—	2	4	no	—
160	D₄×C₆₇	—	2	2	no	—
161	D₄×C₆₇	—	4	8	no	—
162	D₄×C₆₇	—	4	8	no	—
163	D₄×C₆₇	—	4	4	no	—
164	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	8	no	—
169	C₂×C₄×C₆₇	—	4	8	no	—
170	C₂×C₄×C₆₇	—	4	2	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	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
176	C₈×C₆₇	—	2	2	no	—
177	C₈×C₆₇	—	2	4	no	—
178	C₈×C₆₇	—	4	4	no	—
179	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂²×C₆₇	—	2	2	no	—
183	C₂²×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	4	1	no	—
185	C₂²×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	4	4	no	—
187	C₂²×C₆₇	—	8	2	no	—
188	C₂²×C₆₇	—	8	4	no	—
189	C₂²×C₆₇	—	16	4	no	—
190	C₂²×C₆₇	—	16	8	no	—
191	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₆₇	#546	1	2	yes	—
193	C₄×C₆₇	—	2	4	no	—
194	C₄×C₆₇	—	2	2	no	—
195	C₄×C₆₇	—	8	2	no	—
196	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
197	C₄×C₆₇	#518	1	1	yes	characteristic
198	C₄×C₆₇	D₄²	1	2	yes	—
199	#22458	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	4	no	—
204	C₂×C₆₇	—	8	4	no	—
205	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×#576	C₂×C₆₇	1	1	yes	characteristic
207	#2957	C₂×C₆₇	1	2	yes	complemented
208	#2888	C₂×C₆₇	1	2	yes	complemented
209	#1978	C₂×C₆₇	1	2	yes	complemented
210	#1903	C₂×C₆₇	1	4	yes	complemented
211	#1813	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	#22458	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	#568	C₂²×C₆₇	1	4	yes	complemented
225	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
226	#444	—	2	4	no	—
227	#443	—	2	4	no	—
228	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
229	#421	C₂²×C₆₇	1	4	yes	complemented
230	#418	C₂²×C₆₇	1	4	yes	complemented
231	#413	—	2	4	no	—
232	#413	C₂²×C₆₇	1	2	yes	—
233	#413	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#100	—	2	4	no	—
235	C₂×#100	C₂²×C₆₇	1	4	yes	complemented
236	#410	C₂²×C₆₇	1	1	yes	characteristic
237	#405	C₂²×C₆₇	1	2	yes	complemented
238	#344	C₂²×C₆₇	1	2	yes	complemented
239	#338	C₂²×C₆₇	1	2	yes	complemented
240	H(5,𝔽₂)	—	2	8	no	—
241	H(5,𝔽₂)	—	4	4	no	—
242	C₂×#41	—	2	2	no	—
243	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
244	C₂×#41	C₂³×C₆₇	1	2	yes	—
245	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	—	2	4	no	—
248	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
249	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
250	AGL(1,ℤ/8)	—	2	8	no	—
251	AGL(1,ℤ/8)	—	2	16	no	—
252	#135	—	2	4	no	—
253	#135	—	2	4	no	—
254	#135	D₄×C₆₇	1	4	yes	—
255	#135	D₄×C₆₇	1	4	yes	—
256	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
257	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
258	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
259	C₂×D₈	—	2	4	no	—
260	C₂×D₈	—	2	4	no	—
261	C₂×D₈	—	2	2	no	—
262	C₂×D₈	—	2	2	no	—
263	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
264	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
265	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
266	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
267	#131	—	2	2	no	—
268	#131	—	2	2	no	—
269	#131	—	2	8	no	—
270	#131	—	4	4	no	—
271	C₂×M₁₆	—	2	8	no	—
272	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
274	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
275	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
276	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
277	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
278	#117	C₂³×C₆₇	1	2	yes	—
279	#108	—	2	2	no	—
280	#108	—	2	2	no	—
281	#104	—	2	8	no	—
282	#102	—	4	4	no	—
283	#101	—	2	2	no	—
284	#101	D₄×C₆₇	1	2	yes	—
285	#100	—	2	2	no	—
286	#100	—	2	8	no	—
287	#100	D₄×C₆₇	1	2	yes	—
288	#99	—	2	2	no	—
289	#97	—	2	2	no	—
290	C₄×C₈	—	2	2	no	—
291	C₂⁴	—	2	4	no	—
292	C₂⁴	—	2	2	no	—
293	#41	—	2	2	no	—
294	#41	—	2	4	no	—
295	#41	—	2	2	no	—
296	#41	—	2	8	no	—
297	#41	—	4	4	no	—
298	#41	—	4	8	no	—
299	#41	C₂×D₄×C₆₇	1	8	yes	—
300	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
301	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	8	no	—
304	C₂×D₄	—	2	4	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	—	2	4	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	4	no	—
309	C₂×D₄	—	4	4	no	—
310	C₂×D₄	—	4	8	no	—
311	C₂×D₄	—	4	16	no	—
312	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
313	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
314	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
315	C₂×D₄	#41×C₆₇	1	2	yes	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
319	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
320	Q₁₆	C₂²⋊C₄×C₆₇	1	2	yes	—
321	Q₁₆	C₂²⋊C₄×C₆₇	1	2	yes	—
322	SD₁₆	—	2	8	no	—
323	SD₁₆	C₂²⋊C₄×C₆₇	1	8	yes	—
324	D₈	—	2	8	no	—
325	D₈	—	2	2	no	—
326	D₈	—	2	2	no	—
327	D₈	—	2	4	no	—
328	D₈	C₂²⋊C₄×C₆₇	1	2	yes	—
329	D₈	C₂²⋊C₄×C₆₇	1	2	yes	—
330	M₁₆	—	2	2	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	4	no	—
333	M₁₆	—	2	4	no	—
334	M₁₆	—	2	4	no	—
335	M₁₆	—	4	8	no	—
336	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	4	no	—
341	C₂×C₈	—	4	8	no	—
342	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
343	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
344	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
345	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
346	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
347	C₄⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₄²	—	2	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	2	4	no	—
352	C₂³	—	4	8	no	—
353	C₂³	—	4	4	no	—
354	C₂³	—	16	2	no	—
355	C₂³	—	16	4	no	—
356	C₂³	C₄×D₄×C₆₇	1	2	yes	—
357	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
358	Q₈	—	2	2	no	—
359	Q₈	—	2	2	no	—
360	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
361	D₄	—	2	8	no	—
362	D₄	—	2	4	no	—
363	D₄	—	2	4	no	—
364	D₄	—	2	2	no	—
365	D₄	—	2	4	no	—
366	D₄	—	2	2	no	—
367	D₄	—	4	8	no	—
368	D₄	—	4	8	no	—
369	D₄	—	4	4	no	—
370	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	4	8	no	—
376	C₂×C₄	—	4	2	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	C₂²×D₄×C₆₇	1	1	yes	characteristic
381	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
382	C₈	—	2	2	no	—
383	C₈	—	2	4	no	—
384	C₈	—	4	4	no	—
385	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
386	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
387	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	4	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	4	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	4	no	—
395	C₂²	—	16	4	no	—
396	C₂²	—	16	8	no	—
397	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
398	C₂²	#546×C₆₇	1	2	yes	—
399	C₄	—	2	4	no	—
400	C₄	—	2	2	no	—
401	C₄	—	8	2	no	—
402	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
403	C₄	#518×C₆₇	1	1	yes	characteristic
404	C₄	D₄²×C₆₇	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	2	1	no	—
407	C₂	—	4	2	no	—
408	C₂	—	4	4	no	—
409	C₂	—	8	4	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×#2957	C₂	semidirect
3. C₆₇×#2888	C₂	semidirect
4. C₆₇×#1978	C₂	semidirect
5. C₆₇×#1903	C₂	semidirect
6. C₆₇×#1813	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
18. #568×C₆₇	C₂²	semidirect
22. C₂×#108×C₆₇	C₂²	semidirect
23. #421×C₆₇	C₂²	semidirect
24. #418×C₆₇	C₂²	semidirect
29. C₂×#100×C₆₇	C₂²	semidirect
31. #405×C₆₇	C₂²	semidirect
32. #344×C₆₇	C₂²	semidirect
33. #338×C₆₇	C₂²	semidirect
199. #22458	C₆₇	direct (N × H)
207. #2957	C₂×C₆₇	semidirect
208. #2888	C₂×C₆₇	semidirect
209. #1978	C₂×C₆₇	semidirect
210. #1903	C₂×C₆₇	semidirect
211. #1813	C₂×C₆₇	semidirect
212. #1795	C₂×C₆₇	semidirect
213. #1757	C₂×C₆₇	semidirect
214. C₆₇	#22458	direct (N × H)
218. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
220. C₂×#135	C₄×C₆₇	semidirect
222. C₂²×D₈	C₄×C₆₇	semidirect
224. #568	C₂²×C₆₇	semidirect
228. C₂×#108	C₂²×C₆₇	semidirect
229. #421	C₂²×C₆₇	semidirect
230. #418	C₂²×C₆₇	semidirect
235. C₂×#100	C₂²×C₆₇	semidirect
237. #405	C₂²×C₆₇	semidirect
238. #344	C₂²×C₆₇	semidirect
239. #338	C₂²×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×#2957	2
C₆₇×#2888	2
C₆₇×#1978	2
C₆₇×#1903	2
C₆₇×#1813	2
C₆₇×#1795	2
C₆₇×#1757	2
#22458	67

Subgroups — C67×#22458 #30701709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22458 #30701709