Subgroups — C₂×C₆₇×#1833 #30700007

Order 17152.

786 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1833	C₂	1	8	yes	maximal, direct product factor
5	C₂×#344×C₆₇	C₂	1	2	yes	maximal
6	C₂×H(3,ℤ/4)×C₆₇	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₂²	1	2	yes	—
11	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#108×C₆₇	—	2	1	no	—
13	C₂×#108×C₆₇	—	2	1	no	—
14	C₂×#108×C₆₇	—	2	2	no	—
15	C₂×#108×C₆₇	C₂²	1	4	yes	—
16	#427×C₆₇	—	2	2	no	—
17	#427×C₆₇	C₂²	1	4	yes	—
18	C₂×#107×C₆₇	—	2	1	no	—
19	C₂×#106×C₆₇	—	2	1	no	—
20	#405×C₆₇	C₄	1	4	yes	complemented
21	#405×C₆₇	C₂²	1	4	yes	—
22	C₂×#97×C₆₇	C₄	1	2	yes	complemented
23	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	complemented
24	#344×C₆₇	C₂²	1	8	yes	—
25	H(3,ℤ/4)×C₆₇	C₂²	1	8	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₆₇	—	2	1	no	—
28	C₂×M₁₆×C₆₇	—	2	1	no	—
29	C₂×M₁₆×C₆₇	—	2	1	no	—
30	C₂×M₁₆×C₆₇	—	2	1	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	—	2	2	no	—
33	C₂×M₁₆×C₆₇	—	2	2	no	—
34	C₂×M₁₆×C₆₇	—	2	2	no	—
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
38	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	#117×C₆₇	—	2	1	no	—
52	#117×C₆₇	—	2	1	no	—
53	#117×C₆₇	—	2	2	no	—
54	#117×C₆₇	—	2	2	no	—
55	#117×C₆₇	—	2	2	no	—
56	#117×C₆₇	C₂×C₄	1	2	yes	—
57	#117×C₆₇	C₂³	1	4	yes	—
58	#117×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
65	#108×C₆₇	—	2	4	no	—
66	#108×C₆₇	—	2	4	no	—
67	#107×C₆₇	—	2	4	no	—
68	#106×C₆₇	—	2	4	no	—
69	#97×C₆₇	C₂×C₄	1	8	yes	complemented
70	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
71	C₂⁴×C₆₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
88	M₁₆×C₆₇	—	2	2	no	—
89	M₁₆×C₆₇	—	2	2	no	—
90	M₁₆×C₆₇	—	2	2	no	—
91	M₁₆×C₆₇	—	2	2	no	—
92	M₁₆×C₆₇	—	2	4	no	—
93	M₁₆×C₆₇	—	2	4	no	—
94	M₁₆×C₆₇	—	4	2	no	—
95	M₁₆×C₆₇	—	4	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	4	no	—
99	C₂×C₈×C₆₇	—	4	2	no	—
100	C₂×C₈×C₆₇	—	4	2	no	—
101	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
107	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
108	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
110	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
111	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₂²⋊C₄×C₆₇	—	2	2	no	—
124	C₂²⋊C₄×C₆₇	—	2	4	no	—
125	C₂²⋊C₄×C₆₇	—	4	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	4	no	—
128	C₄²×C₆₇	—	4	2	no	—
129	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	#117	1	1	yes	characteristic
132	C₂³×C₆₇	#117	1	1	yes	characteristic
133	C₂³×C₆₇	#124	1	1	yes	characteristic
134	C₂³×C₆₇	#124	1	1	yes	characteristic
135	C₂³×C₆₇	#125	1	1	yes	characteristic
136	C₂³×C₆₇	#125	1	1	yes	characteristic
137	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	4	4	no	—
140	C₂×C₄×C₆₇	—	4	4	no	—
141	C₂×C₄×C₆₇	—	8	4	no	—
142	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
151	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
152	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
153	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
154	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
155	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
156	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
157	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
158	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
159	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
160	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
161	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
162	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
163	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
170	C₈×C₆₇	—	2	4	no	—
171	C₈×C₆₇	—	4	4	no	—
172	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
173	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
174	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
175	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
176	C₂²×C₆₇	—	2	2	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	4	1	no	—
180	C₂²×C₆₇	—	4	1	no	—
181	C₂²×C₆₇	—	4	1	no	—
182	C₂²×C₆₇	—	4	1	no	—
183	C₂²×C₆₇	#382	1	1	yes	characteristic
184	C₂²×C₆₇	#382	1	1	yes	characteristic
185	C₂²×C₆₇	#382	1	1	yes	characteristic
186	C₂²×C₆₇	#382	1	1	yes	characteristic
187	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
188	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
189	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
190	C₄×C₆₇	—	4	4	no	—
191	C₄×C₆₇	—	8	4	no	—
192	C₄×C₆₇	#382	1	1	yes	characteristic
193	C₄×C₆₇	#382	1	1	yes	characteristic
194	C₄×C₆₇	#382	1	1	yes	characteristic
195	C₄×C₆₇	#382	1	1	yes	characteristic
196	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
197	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
198	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
199	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
200	C₂×#1833	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₆₇	—	2	1	no	—
202	C₂×C₆₇	—	2	1	no	—
203	C₂×C₆₇	—	2	1	no	—
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	1	no	—
206	C₂×C₆₇	—	2	1	no	—
207	C₂×C₆₇	#1833	1	2	yes	direct product factor
208	C₂×C₆₇	C₂×#382	1	1	yes	characteristic
209	C₂²×#108	C₂×C₆₇	1	1	yes	characteristic
210	C₂×#427	C₂×C₆₇	1	1	yes	characteristic
211	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
212	#1833	C₂×C₆₇	1	8	yes	direct product factor
213	C₂×#344	C₂×C₆₇	1	2	yes	—
214	C₂×H(3,ℤ/4)	C₂×C₆₇	1	2	yes	—
215	C₆₇	C₂×#1833	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
217	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
218	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
219	C₂×#117	C₂²×C₆₇	1	2	yes	—
220	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×#108	—	2	1	no	—
222	C₂×#108	—	2	1	no	—
223	C₂×#108	—	2	2	no	—
224	C₂×#108	C₂²×C₆₇	1	4	yes	—
225	#427	—	2	2	no	—
226	#427	C₂²×C₆₇	1	4	yes	—
227	C₂×#107	—	2	1	no	—
228	C₂×#106	—	2	1	no	—
229	#405	C₄×C₆₇	1	4	yes	complemented
230	#405	C₂²×C₆₇	1	4	yes	—
231	C₂×#97	C₄×C₆₇	1	2	yes	complemented
232	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	complemented
233	#344	C₂²×C₆₇	1	8	yes	—
234	H(3,ℤ/4)	C₂²×C₆₇	1	8	yes	—
235	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	2	1	no	—
239	C₂×M₁₆	—	2	1	no	—
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	2	2	no	—
242	C₂×M₁₆	—	2	2	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
246	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
247	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
250	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
251	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
252	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
253	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
254	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
255	C₂²×C₈	D₄×C₆₇	1	2	yes	—
256	C₂²×C₈	Q₈×C₆₇	1	1	yes	characteristic
257	C₂²×C₈	Q₈×C₆₇	1	1	yes	characteristic
258	C₂²×C₈	Q₈×C₆₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
260	#117	—	2	1	no	—
261	#117	—	2	1	no	—
262	#117	—	2	2	no	—
263	#117	—	2	2	no	—
264	#117	—	2	2	no	—
265	#117	C₂×C₄×C₆₇	1	2	yes	—
266	#117	C₂³×C₆₇	1	4	yes	—
267	#117	C₂³×C₆₇	1	2	yes	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₂²⋊C₄	—	2	1	no	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
274	#108	—	2	4	no	—
275	#108	—	2	4	no	—
276	#107	—	2	4	no	—
277	#106	—	2	4	no	—
278	#97	C₂×C₄×C₆₇	1	8	yes	complemented
279	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
280	C₂⁴	#41×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
288	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
289	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
291	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
293	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
297	M₁₆	—	2	2	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	2	2	no	—
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	4	no	—
302	M₁₆	—	2	4	no	—
303	M₁₆	—	4	2	no	—
304	M₁₆	—	4	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	4	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	4	2	no	—
310	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
311	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
313	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
314	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
315	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
316	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
317	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
318	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
319	C₂×C₈	C₄⋊C₄×C₆₇	1	2	yes	—
320	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
321	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
322	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
323	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
324	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
325	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
326	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	4	2	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₂²⋊C₄	—	2	2	no	—
333	C₂²⋊C₄	—	2	4	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	4	no	—
337	C₄²	—	4	2	no	—
338	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
339	C₂³	—	4	2	no	—
340	C₂³	#117×C₆₇	1	1	yes	characteristic
341	C₂³	#117×C₆₇	1	1	yes	characteristic
342	C₂³	#124×C₆₇	1	1	yes	characteristic
343	C₂³	#124×C₆₇	1	1	yes	characteristic
344	C₂³	#125×C₆₇	1	1	yes	characteristic
345	C₂³	#125×C₆₇	1	1	yes	characteristic
346	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	4	4	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	8	4	no	—
351	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
355	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
356	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
357	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
358	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
359	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
360	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
361	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
362	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
363	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
364	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
365	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
366	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
367	C₂×C₄	#125×C₆₇	1	1	yes	characteristic
368	C₂×C₄	#125×C₆₇	1	1	yes	characteristic
369	C₂×C₄	#125×C₆₇	1	1	yes	characteristic
370	C₂×C₄	#125×C₆₇	1	1	yes	characteristic
371	C₂×C₄	#125×C₆₇	1	1	yes	characteristic
372	C₂×C₄	#125×C₆₇	1	1	yes	characteristic
373	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
376	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
377	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
378	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
379	C₈	—	2	4	no	—
380	C₈	—	4	4	no	—
381	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
382	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
383	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
384	C₈	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	#382×C₆₇	1	1	yes	characteristic
393	C₂²	#382×C₆₇	1	1	yes	characteristic
394	C₂²	#382×C₆₇	1	1	yes	characteristic
395	C₂²	#382×C₆₇	1	1	yes	characteristic
396	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
397	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
398	C₂²	C₂×#125×C₆₇	1	1	yes	characteristic
399	C₄	—	4	4	no	—
400	C₄	—	8	4	no	—
401	C₄	#382×C₆₇	1	1	yes	characteristic
402	C₄	#382×C₆₇	1	1	yes	characteristic
403	C₄	#382×C₆₇	1	1	yes	characteristic
404	C₄	#382×C₆₇	1	1	yes	characteristic
405	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
406	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
407	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
408	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	C₆₇×#1833	1	2	yes	direct product factor
416	C₂	C₂×#382×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#117
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	#382
prime radical	C₂⁴×C₆₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1833	C₂	direct (N × H)
20. #405×C₆₇	C₄	semidirect
22. C₂×#97×C₆₇	C₄	semidirect
23. C₂×C₄×C₈×C₆₇	C₄	semidirect
69. #97×C₆₇	C₂×C₄	semidirect
70. C₄×C₈×C₆₇	C₂×C₄	semidirect
200. C₂×#1833	C₆₇	direct (N × H)
207. C₂×C₆₇	#1833	direct (N × H)
212. #1833	C₂×C₆₇	direct (N × H)
215. C₆₇	C₂×#1833	direct (N × H)
229. #405	C₄×C₆₇	semidirect
231. C₂×#97	C₄×C₆₇	semidirect
232. C₂×C₄×C₈	C₄×C₆₇	semidirect
278. #97	C₂×C₄×C₆₇	semidirect
279. C₄×C₈	C₂×C₄×C₆₇	semidirect
415. C₂	C₆₇×#1833	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×#108×C₆₇	2
C₂×#427×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#1833	2
C₂×#344×C₆₇	2
C₂×H(3,ℤ/4)×C₆₇	2
C₂×#1833	67

Subgroups — C2×C67×#1833 #30700007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1833 #30700007